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Multi-objective optimization

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Multi-objective optimization or Pareto optimization (also known as multi-objective programming, vector optimization, multicriteria optimization, or multiattribute optimization) is an area of <a href="MCDM" class="mw-redirect" title="MCDM">multiple-criteria decision making</a> that is concerned with <a href="Mathematical_optimization" title="Mathematical optimization">mathematical optimization problems</a> involving more than one <a href="Loss_function" title="Loss function">objective function</a> to be optimized simultaneously. Multi-objective is a type of <a href="Vector_optimization" title="Vector optimization">vector optimization</a> that has been applied in many fields of science, including engineering, economics and logistics where optimal decisions need to be taken in the presence of <a href="Trade-off" title="Trade-off">trade-offs</a> between two or more conflicting objectives. Minimizing cost while maximizing comfort while buying a car, and maximizing performance whilst minimizing fuel consumption and emission of pollutants of a vehicle are examples of multi-objective optimization problems involving two and three objectives, respectively. In practical problems, there can be more than three objectives.

For a multi-objective optimization problem, it is not guaranteed that a single solution simultaneously optimizes each objective. The objective functions are said to be conflicting. A solution is called <a href="Maxima_of_a_point_set" title="Maxima of a point set">nondominated</a>, Pareto optimal, <a href="Pareto_efficient" class="mw-redirect" title="Pareto efficient">Pareto efficient</a> or noninferior, if none of the objective functions can be improved in value without degrading some of the other objective values. Without additional <a href="Subjectivity" class="mw-redirect" title="Subjectivity">subjective</a> preference information, there may exist a (possibly infinite) number of Pareto optimal solutions, all of which are considered equally good. Researchers study multi-objective optimization problems from different viewpoints and, thus, there exist different solution philosophies and goals when setting and solving them. The goal may be to find a representative set of Pareto optimal solutions, and/or quantify the trade-offs in satisfying the different objectives, and/or finding a single solution that satisfies the subjective preferences of a human decision maker (DM).

Bicriteria optimization denotes the special case in which there are two objective functions.

There is a direct relationship between <a href="Multitask_optimization" class="mw-redirect" title="Multitask optimization">multitask optimization</a> and multi-objective optimization.<a href="#cite_note-1">[1]</a>

<div class="mw-heading mw-heading2"><h2 id="Introduction">Introduction</h2></div>

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</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="Pareto_order" class="mw-redirect" title="Pareto order">Pareto order</a></div>

A multi-objective optimization problem is an <a href="Optimization_problem" title="Optimization problem">optimization problem</a> that involves multiple objective functions.<a href="#cite_note-Miettinen1999-2">[2]</a><a href="#cite_note-HwangMasud1979-3">[3]</a><a href="#cite_note-hassanzadeh-4">[4]</a> In mathematical terms, a multi-objective optimization problem can be formulated as

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<annotation encoding="application/x-tex">{\displaystyle \min _{x\in X}(f_{1}(x),f_{2}(x),\ldots ,f_{k}(x))}</annotation>

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</math><img src="./9958be1c4fe3f3640b6112b15c18e6f86640a41b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:27.929ex; height:4.009ex;" alt="{\displaystyle \min _{x\in X}(f_{1}(x),f_{2}(x),\ldots ,f_{k}(x))}" loading="lazy"></dd></dl>

where the integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\geq 2}">

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<annotation encoding="application/x-tex">{\displaystyle k\geq 2}</annotation>

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</math><img src="./c797a67c0a51167d373c013a9a020f4568a11754.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.472ex; height:2.343ex;" alt="{\displaystyle k\geq 2}" loading="lazy"> is the number of objectives and the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is the <a href="Feasible_set" class="mw-redirect" title="Feasible set">feasible set</a> of decision vectors, which is typically <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\subseteq \mathbb {R} ^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle X\subseteq \mathbb {R} ^{n}}</annotation>

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</math><img src="./4538114b7e232be1bd1d0d774e97f5c43236518a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.975ex; height:2.509ex;" alt="{\displaystyle X\subseteq \mathbb {R} ^{n}}" loading="lazy"> but it depends on the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-dimensional application domain. The feasible set is typically defined by some constraint functions. In addition, the vector-valued objective function is often defined as

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}f:X&\to \mathbb {R} ^{k}\\x&\mapsto {\begin{pmatrix}f_{1}(x)\\\vdots \\f_{k}(x)\end{pmatrix}}\end{aligned}}}</annotation>

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</math><img src="./39ac342b7e707434438be9d5f9cb1faa30058075.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:19.747ex; height:14.009ex;" alt="{\displaystyle {\begin{aligned}f:X&\to \mathbb {R} ^{k}\\x&\mapsto {\begin{pmatrix}f_{1}(x)\\\vdots \\f_{k}(x)\end{pmatrix}}\end{aligned}}}" loading="lazy"></dd></dl>

If some objective function is to be maximized, it is equivalent to minimize its negative or its inverse. We denote <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y\subseteq \mathbb {R} ^{k}}">

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</math><img src="./d01e825f39834f7aaf9c0533f0e2fd68f636cd14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.639ex; height:2.843ex;" alt="{\displaystyle Y\subseteq \mathbb {R} ^{k}}" loading="lazy"> the image of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">; <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{*}\in X}">

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<annotation encoding="application/x-tex">{\displaystyle x^{*}\in X}</annotation>

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</math><img src="./2a7c7bc14dac4bd00697dfef4741db3f8e57f2d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.205ex; height:2.343ex;" alt="{\displaystyle x^{*}\in X}" loading="lazy"> a <a href="Feasible_solution" class="mw-redirect" title="Feasible solution">feasible solution</a> or feasible decision; and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z^{*}=f(x^{*})\in \mathbb {R} ^{k}}">

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<annotation encoding="application/x-tex">{\displaystyle z^{*}=f(x^{*})\in \mathbb {R} ^{k}}</annotation>

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</math><img src="./f85e43bb8658e21ee6f2c65042e58e2a5e81a499.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.322ex; height:3.176ex;" alt="{\displaystyle z^{*}=f(x^{*})\in \mathbb {R} ^{k}}" loading="lazy">an objective vector or an outcome.

In multi-objective optimization, there does not typically exist a feasible solution that minimizes all objective functions simultaneously. Therefore, attention is paid to <a href="Pareto_optimality" class="mw-redirect" title="Pareto optimality">Pareto optimal</a> solutions; that is, solutions that cannot be improved in any of the objectives without degrading at least one of the other objectives. In mathematical terms, a feasible solution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}\in X}">

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</math><img src="./f67557f522e10f6156f21eb38aaba83a4df0738e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.205ex; height:2.509ex;" alt="{\displaystyle x_{2}\in X}" loading="lazy">, if

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<annotation encoding="application/x-tex">{\displaystyle \forall i\in \{1,\dots ,k\},f_{i}(x_{1})\leq f_{i}(x_{2})}</annotation>

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</math><img src="./57f07a9164ea701fe3de90892021b16315b1367a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.209ex; height:2.843ex;" alt="{\displaystyle \forall i\in \{1,\dots ,k\},f_{i}(x_{1})\leq f_{i}(x_{2})}" loading="lazy">, and</li>

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<annotation encoding="application/x-tex">{\displaystyle \exists i\in \{1,\dots ,k\},f_{i}(x_{1})<f_{i}(x_{2})}</annotation>

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A solution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{*}\in X}">

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<annotation encoding="application/x-tex">{\displaystyle x^{*}\in X}</annotation>

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</math><img src="./2a7c7bc14dac4bd00697dfef4741db3f8e57f2d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.205ex; height:2.343ex;" alt="{\displaystyle x^{*}\in X}" loading="lazy"> (and the corresponding outcome <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x^{*})}">

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<annotation encoding="application/x-tex">{\displaystyle f(x^{*})}</annotation>

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</math><img src="./2c95e0b2fb6edd57d74bf0590657f2d9bdf4a2b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.472ex; height:2.843ex;" alt="{\displaystyle f(x^{*})}" loading="lazy">) is called Pareto optimal if there does not exist another solution that dominates it. The set of Pareto optimal outcomes, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{*}}">

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<annotation encoding="application/x-tex">{\displaystyle X^{*}}</annotation>

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</math><img src="./01924e6e5570e2631081fea6c6981b4872d3e04b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.051ex; height:2.343ex;" alt="{\displaystyle X^{*}}" loading="lazy">, is often called the <a href="Pareto_front" title="Pareto front">Pareto front</a>, Pareto frontier, or Pareto boundary.

The Pareto front of a multi-objective optimization problem is bounded by a so-called <a href="Nadir" title="Nadir">nadir</a> objective vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z^{nadir}}">

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<annotation encoding="application/x-tex">{\displaystyle z^{nadir}}</annotation>

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</math><img src="./144d08c85a5ca6924b9651893eb7ec27af7663a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.347ex; height:2.676ex;" alt="{\displaystyle z^{nadir}}" loading="lazy">and an ideal objective vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z^{ideal}}">

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<annotation encoding="application/x-tex">{\displaystyle z^{ideal}}</annotation>

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</math><img src="./1cd1966b8f0ccb50f4ff183dda66c1f39f74dd1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.876ex; height:2.676ex;" alt="{\displaystyle z^{ideal}}" loading="lazy">, if these are finite. The nadir objective vector is defined as

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<annotation encoding="application/x-tex">{\displaystyle z^{nadir}={\begin{pmatrix}\sup _{x^{*}\in X^{*}}f_{1}(x^{*})\\\vdots \\\sup _{x^{*}\in X^{*}}f_{k}(x^{*})\end{pmatrix}}}</annotation>

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</math><img src="./9161daccecceaa3349ef0f05f04ea3382027997a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.838ex; width:28.917ex; height:10.843ex;" alt="{\displaystyle z^{nadir}={\begin{pmatrix}\sup _{x^{*}\in X^{*}}f_{1}(x^{*})\\\vdots \\\sup _{x^{*}\in X^{*}}f_{k}(x^{*})\end{pmatrix}}}" loading="lazy"></dd></dl>

and the ideal objective vector as

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<annotation encoding="application/x-tex">{\displaystyle z^{ideal}={\begin{pmatrix}\inf _{x^{*}\in X^{*}}f_{1}(x^{*})\\\vdots \\\inf _{x^{*}\in X^{*}}f_{k}(x^{*})\end{pmatrix}}}</annotation>

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</math><img src="./60a3dd9f9ef99617644794d5fd831e2c35538351.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.838ex; width:27.596ex; height:10.843ex;" alt="{\displaystyle z^{ideal}={\begin{pmatrix}\inf _{x^{*}\in X^{*}}f_{1}(x^{*})\\\vdots \\\inf _{x^{*}\in X^{*}}f_{k}(x^{*})\end{pmatrix}}}" loading="lazy"></dd></dl>

In other words, the components of the nadir and ideal objective vectors define the upper and lower bounds of the objective function of Pareto optimal solutions. In practice, the nadir objective vector can only be approximated as, typically, the whole Pareto optimal set is unknown. In addition, a utopian objective vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z^{utop}}">

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<annotation encoding="application/x-tex">{\displaystyle z^{utop}}</annotation>

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</math><img src="./36bb60704fac35caff3db937e915a69b1d5106fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.481ex; height:2.509ex;" alt="{\displaystyle z^{utop}}" loading="lazy">, such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{i}^{utop}=z_{i}^{ideal}-\epsilon ,\forall i\in \{1,\dots ,k\}}">

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<annotation encoding="application/x-tex">{\displaystyle z_{i}^{utop}=z_{i}^{ideal}-\epsilon ,\forall i\in \{1,\dots ,k\}}</annotation>

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</math><img src="./04970348bc519824cc013db36b22be1b793d2890.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:32.086ex; height:3.343ex;" alt="{\displaystyle z_{i}^{utop}=z_{i}^{ideal}-\epsilon ,\forall i\in \{1,\dots ,k\}}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon >0}">

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</math><img src="./568095ad3924314374a5ab68fae17343661f2a71.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.205ex; height:2.176ex;" alt="{\displaystyle \epsilon >0}" loading="lazy"> is a small constant, is often defined because of numerical reasons.

<div class="mw-heading mw-heading2"><h2 id="Examples_of_applications">Examples of applications</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Economics">Economics</h3></div>

In <a href="Economics" title="Economics">economics</a>, many problems involve multiple objectives along with constraints on what combinations of those objectives are attainable. For example, consumer's <a href="Demand" title="Demand">demand</a> for various goods is determined by the process of maximization of the <a href="Utility" title="Utility">utilities</a> derived from those goods, subject to a constraint based on how much income is available to spend on those goods and on the prices of those goods. This constraint allows more of one good to be purchased only at the sacrifice of consuming less of another good; therefore, the various objectives (more consumption of each good is preferred) are in conflict with each other. A common method for analyzing such a problem is to use a graph of <a href="Indifference_curve" title="Indifference curve">indifference curves</a>, representing preferences, and a budget constraint, representing the trade-offs that the consumer is faced with.

Another example involves the <a href="Production_possibilities_frontier" class="mw-redirect" title="Production possibilities frontier">production possibilities frontier</a>, which specifies what combinations of various types of goods can be produced by a society with certain amounts of various resources. The frontier specifies the trade-offs that the society is faced with — if the society is fully utilizing its resources, more of one good can be produced only at the expense of producing less of another good. A society must then use some process to choose among the possibilities on the frontier.

<a href="Macroeconomics#Macroeconomic_policy" title="Macroeconomics">Macroeconomic policy</a>-making is a context requiring multi-objective optimization. Typically a <a href="Central_bank" title="Central bank">central bank</a> must choose a stance for <a href="Monetary_policy" title="Monetary policy">monetary policy</a> that balances competing objectives — low <a href="Inflation" title="Inflation">inflation</a>, low <a href="Unemployment" title="Unemployment">unemployment</a>, low <a href="Balance_of_trade" title="Balance of trade">balance of trade</a> deficit, etc. To do this, the central bank uses a <a href="Economic_model" title="Economic model">model of the economy</a> that quantitatively describes the various causal linkages in the economy; it <a href="Simulation" title="Simulation">simulates</a> the model repeatedly under various possible stances of monetary policy, in order to obtain a menu of possible predicted outcomes for the various variables of interest. Then in principle it can use an aggregate objective function to rate the alternative sets of predicted outcomes, although in practice central banks use a non-quantitative, judgement-based, process for ranking the alternatives and making the policy choice.

<div class="mw-heading mw-heading3"><h3 id="Finance">Finance</h3></div>

In <a href="Finance" title="Finance">finance</a>, a common problem is to choose a portfolio when there are two conflicting objectives — the desire to have the <a href="Expected_value" title="Expected value">expected value</a> of portfolio returns be as high as possible, and the desire to have <a href="Financial_risk" title="Financial risk">risk</a>, often measured by the <a href="Standard_deviation" title="Standard deviation">standard deviation</a> of portfolio returns, be as low as possible. This problem is often represented by a graph in which the <a href="Efficient_frontier" title="Efficient frontier">efficient frontier</a> shows the best combinations of risk and expected return that are available, and in which indifference curves show the investor's preferences for various risk-expected return combinations. The problem of optimizing a function of the expected value (first <a href="Moment_(mathematics)" title="Moment (mathematics)">moment</a>) and the standard deviation (square root of the second central moment) of portfolio return is called a <a href="Two-moment_decision_model" title="Two-moment decision model">two-moment decision model</a>.

<div class="mw-heading mw-heading3"><h3 id="Optimal_control">Optimal control</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="Optimal_control" title="Optimal control">Optimal control</a>, <a href="Dynamic_programming" title="Dynamic programming">Dynamic programming</a>, and <a href="Linear-quadratic_regulator" class="mw-redirect" title="Linear-quadratic regulator">Linear-quadratic regulator</a></div>

In <a href="Engineering" title="Engineering">engineering</a> and <a href="Economics" title="Economics">economics</a>, many problems involve multiple objectives which are not describable as the-more-the-better or the-less-the-better; instead, there is an ideal target value for each objective, and the desire is to get as close as possible to the desired value of each objective. For example, energy systems typically have a trade-off between performance and cost<a href="#cite_note-5">[5]</a><a href="#cite_note-6">[6]</a> or one might want to adjust a rocket's fuel usage and orientation so that it arrives both at a specified place and at a specified time; or one might want to conduct <a href="Open_market_operations" class="mw-redirect" title="Open market operations">open market operations</a> so that both the <a href="Inflation_rate" class="mw-redirect" title="Inflation rate">inflation rate</a> and the <a href="Unemployment_rate" class="mw-redirect" title="Unemployment rate">unemployment rate</a> are as close as possible to their desired values.

Often such problems are subject to linear equality constraints that prevent all objectives from being simultaneously perfectly met, especially when the number of controllable variables is less than the number of objectives and when the presence of random shocks generates uncertainty. Commonly a multi-objective <a href="Quadratic_function#Bivariate_(two_variable)_quadratic_function" title="Quadratic function">quadratic objective function</a> is used, with the cost associated with an objective rising quadratically with the distance of the objective from its ideal value. Since these problems typically involve adjusting the controlled variables at various points in time and/or evaluating the objectives at various points in time, <a href="Intertemporal_optimization" class="mw-redirect" title="Intertemporal optimization">intertemporal optimization</a> techniques are employed.<a href="#cite_note-7">[7]</a>

<div class="mw-heading mw-heading3"><h3 id="Optimal_design">Optimal design</h3></div>

Product and process design can be largely improved using modern modeling, simulation, and optimization techniques. The key question in optimal design is measuring what is good or desirable about a design. Before looking for optimal designs, it is important to identify characteristics that contribute the most to the overall value of the design. A good design typically involves multiple criteria/objectives such as capital cost/investment, operating cost, profit, quality and/or product recovery, efficiency, process safety, operation time, etc. Therefore, in practical applications, the performance of process and product design is often measured with respect to multiple objectives. These objectives are typically conflicting, i.e., achieving the optimal value for one objective requires some compromise on one or more objectives.

For example, when designing a paper mill, one can seek to decrease the amount of capital invested in a paper mill and enhance the quality of paper simultaneously. If the design of a paper mill is defined by large storage volumes and paper quality is defined by quality parameters, then the problem of optimal design of a paper mill can include objectives such as i) minimization of expected variation of those quality parameters from their nominal values, ii) minimization of the expected time of breaks and iii) minimization of the investment cost of storage volumes. Here, the maximum volume of towers is a design variable. This example of optimal design of a paper mill is a simplification of the model used in.<a href="#cite_note-RoRiPi11-8">[8]</a> Multi-objective design optimization has also been implemented in engineering systems in the circumstances such as control cabinet layout optimization,<a href="#cite_note-9">[9]</a> airfoil shape optimization using scientific workflows,<a href="#cite_note-10">[10]</a> design of nano-<a href="CMOS" title="CMOS">CMOS</a>,<a href="#cite_note-11">[11]</a> <a href="System_on_a_chip" title="System on a chip">system on chip</a> design, design of solar-powered irrigation systems,<a href="#cite_note-12">[12]</a> optimization of sand mould systems,<a href="#cite_note-13">[13]</a><a href="#cite_note-14">[14]</a> engine design,<a href="#cite_note-15">[15]</a><a href="#cite_note-16">[16]</a> optimal sensor deployment<a href="#cite_note-17">[17]</a> and optimal controller design.<a href="#cite_note-18">[18]</a><a href="#cite_note-19">[19]</a>

<div class="mw-heading mw-heading3"><h3 id="Process_optimization"> Process optimization</h3></div>

Multi-objective optimization has been increasingly employed in <a href="Chemical_engineering" title="Chemical engineering">chemical engineering</a> and <a href="Manufacturing" title="Manufacturing">manufacturing</a>. In 2009, Fiandaca and Fraga used the multi-objective genetic algorithm (MOGA) to optimize the pressure swing adsorption process (cyclic separation process). The design problem involved the dual maximization of nitrogen recovery and nitrogen purity. The results approximated the Pareto frontier well with acceptable trade-offs between the objectives.<a href="#cite_note-20">[20]</a>

In 2010, Sendín et al. solved a multi-objective problem for the thermal processing of food. They tackled two case studies (bi-objective and triple-objective problems) with nonlinear dynamic models. They used a hybrid approach consisting of the weighted Tchebycheff and the Normal Boundary Intersection approach. The novel hybrid approach was able to construct a Pareto optimal set for the thermal processing of foods.<a href="#cite_note-21">[21]</a>

In 2013, Ganesan et al. carried out the multi-objective optimization of the combined carbon dioxide reforming and partial oxidation of methane. The objective functions were methane conversion, carbon monoxide selectivity, and hydrogen to carbon monoxide ratio. Ganesan used the Normal Boundary Intersection (NBI) method in conjunction with two swarm-based techniques (Gravitational Search Algorithm (GSA) and Particle Swarm Optimization (PSO)) to tackle the problem.<a href="#cite_note-22">[22]</a> Applications involving chemical extraction<a href="#cite_note-23">[23]</a> and bioethanol production processes<a href="#cite_note-24">[24]</a> have posed similar multi-objective problems.

In 2013, Abakarov et al. proposed an alternative technique to solve multi-objective optimization problems arising in food engineering.<a href="#cite_note-25">[25]</a> The Aggregating Functions Approach, the Adaptive Random Search Algorithm, and the Penalty Functions Approach were used to compute the initial set of the non-dominated or Pareto-optimal solutions. The <a href="Analytic_Hierarchy_Process" class="mw-redirect" title="Analytic Hierarchy Process">Analytic Hierarchy Process</a> and Tabular Method were used simultaneously for choosing the best alternative among the computed subset of non-dominated solutions for osmotic dehydration processes.<a href="#cite_note-26">[26]</a>

In 2018, Pearce et al. formulated task allocation to human and robotic workers as a multi-objective optimization problem, considering production time and the ergonomic impact on the human worker as the two objectives considered in the formulation. Their approach used a <a href="Linear_programming" title="Linear programming">Mixed-Integer Linear Program</a> to solve the optimization problem for a weighted sum of the two objectives to calculate a set of <a href="Pareto_efficiency" title="Pareto efficiency">Pareto optimal</a> solutions. Applying the approach to several manufacturing tasks showed improvements in at least one objective in most tasks and in both objectives in some of the processes.<a href="#cite_note-27">[27]</a>

<div class="mw-heading mw-heading3"><h3 id="Radio_resource_management">Radio resource management</h3></div>

The purpose of <a href="Radio_resource_management" title="Radio resource management">radio resource management</a> is to satisfy the data rates that are requested by the users of a cellular network.<a href="#cite_note-fnt2013-28">[28]</a> The main resources are time intervals, frequency blocks, and transmit powers. Each user has its own objective function that, for example, can represent some combination of the data rate, latency, and energy efficiency. These objectives are conflicting since the frequency resources are very scarce, thus there is a need for tight spatial <a href="Frequency_reuse" class="mw-redirect" title="Frequency reuse">frequency reuse</a> which causes immense inter-user interference if not properly controlled. <a href="Multi-user_MIMO" title="Multi-user MIMO">Multi-user MIMO</a> techniques are nowadays used to reduce the interference by adaptive <a href="Precoding" title="Precoding">precoding</a>. The network operator would like to both bring great coverage and high data rates, thus the operator would like to find a Pareto optimal solution that balance the total network data throughput and the user fairness in an appropriate subjective manner.

Radio resource management is often solved by scalarization; that is, selection of a network utility function that tries to balance throughput and user fairness. The choice of utility function has a large impact on the computational complexity of the resulting single-objective optimization problem.<a href="#cite_note-fnt2013-28">[28]</a> For example, the common utility of weighted sum rate gives an <a href="NP-hard" class="mw-redirect" title="NP-hard">NP-hard</a> problem with a complexity that scales exponentially with the number of users, while the weighted max-min fairness utility results in a quasi-convex optimization problem with only a polynomial scaling with the number of users.<a href="#cite_note-luo2008-29">[29]</a>

<div class="mw-heading mw-heading3"><h3 id="Electric_power_systems">Electric power systems</h3></div>

Reconfiguration, by exchanging the functional links between the elements of the system, represents one of the most important measures which can improve the operational performance of a distribution system. The problem of optimization through the reconfiguration of a power distribution system, in terms of its definition, is a historical single objective problem with constraints. Since 1975, when Merlin and Back <a href="#cite_note-30">[30]</a> introduced the idea of distribution system reconfiguration for active power loss reduction, until nowadays, a lot of researchers have proposed diverse methods and algorithms to solve the reconfiguration problem as a single objective problem. Some authors have proposed Pareto optimality based approaches (including active power losses and reliability indices as objectives). For this purpose, different artificial intelligence based methods have been used: microgenetic,<a href="#cite_note-31">[31]</a> branch exchange,<a href="#cite_note-32">[32]</a> particle swarm optimization <a href="#cite_note-33">[33]</a> and non-dominated sorting genetic algorithm.<a href="#cite_note-34">[34]</a>

<div class="mw-heading mw-heading3"><h3 id="Inspection_of_infrastructure">Inspection of infrastructure</h3></div>

Autonomous inspection of infrastructure has the potential to reduce costs, risks and environmental impacts, as well as ensuring better periodic maintenance of inspected assets. Typically, planning such missions has been viewed as a single-objective optimization problem, where one aims to minimize the energy or time spent in inspecting an entire target structure.<a href="#cite_note-GalceranCarreras2013-35">[35]</a> For complex, real-world structures, however, covering 100% of an inspection target is not feasible, and generating an inspection plan may be better viewed as a multiobjective optimization problem, where one aims to both maximize inspection coverage and minimize time and costs. A recent study has indicated that multiobjective inspection planning indeed has the potential to outperform traditional methods on complex structures<a href="#cite_note-EllefsenLepikson2017-36">[36]</a>

<div class="mw-heading mw-heading2"><h2 id="Solution">Solution</h2></div>

As multiple <a href="Pareto_optimality" class="mw-redirect" title="Pareto optimality">Pareto optimal</a> solutions for multi-objective optimization problems usually exist, what it means to solve such a problem is not as straightforward as it is for a conventional single-objective optimization problem. Therefore, different researchers have defined the term "solving a multi-objective optimization problem" in various ways. This section summarizes some of them and the contexts in which they are used. Many methods convert the original problem with multiple objectives into a single-objective <a href="Optimization_problem" title="Optimization problem">optimization problem</a>. This is called a scalarized problem. If the Pareto optimality of the single-objective solutions obtained can be guaranteed, the scalarization is characterized as done neatly.

Solving a multi-objective optimization problem is sometimes understood as approximating or computing all or a representative set of Pareto optimal solutions.<a href="#cite_note-Ehrgott2005-37">[37]</a><a href="#cite_note-CoelloLamont2007-38">[38]</a>

When <a href="Multiple-criteria_decision_analysis" title="Multiple-criteria decision analysis">decision making</a> is emphasized, the objective of solving a multi-objective optimization problem is referred to as supporting a decision maker in finding the most preferred Pareto optimal solution according to their subjective preferences.<a href="#cite_note-Miettinen1999-2">[2]</a><a href="#cite_note-BrankeDeb2008-39">[39]</a> The underlying assumption is that one solution to the problem must be identified to be implemented in practice. Here, a human <a href="Decision_maker" class="mw-redirect" title="Decision maker">decision maker</a> (DM) plays an important role. The DM is expected to be an expert in the problem domain.

The most preferred results can be found using different philosophies. Multi-objective optimization methods can be divided into four classes.<a href="#cite_note-HwangMasud1979-3">[3]</a>

<ol><li>In so-called no-preference methods, no DM is expected to be available, but a neutral compromise solution is identified without preference information.<a href="#cite_note-Miettinen1999-2">[2]</a> The other classes are so-called a priori, a posteriori, and interactive methods, and they all involve preference information from the DM in different ways.</li>

<li>In a priori methods, preference information is first asked from the DM, and then a solution best satisfying these preferences is found.</li>

<li>In a posteriori methods, a representative set of Pareto optimal solutions is first found, and then the DM must choose one of them.</li>

<li>In interactive methods, the decision maker is allowed to search for the most preferred solution iteratively. In each iteration of the interactive method, the DM is shown Pareto optimal solution(s) and describes how the solution(s) could be improved. The information given by the DM is then taken into account while generating new Pareto optimal solution(s) for the DM to study in the next iteration. In this way, the DM learns about the feasibility of their wishes and can concentrate on solutions that are interesting to them. The DM may stop the search whenever they want to.</li></ol>

More information and examples of different methods in the four classes are given in the following sections.

<div class="mw-heading mw-heading2"><h2 id="No-preference_methods">No-preference methods</h2></div>

When a decision maker does not explicitly articulate any preference information, the multi-objective optimization method can be classified as a no-preference method.<a href="#cite_note-HwangMasud1979-3">[3]</a> A well-known example is the method of global criterion,<a href="#cite_note-Zeleny1973-40">[40]</a> in which a scalarized problem of the form

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is solved. In the above problem, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\cdot \|}">

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<annotation encoding="application/x-tex">{\displaystyle L_{1}}</annotation>

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</math><img src="./0e79dc1b001f8b923df475ed14de023cbc456013.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.637ex; height:2.509ex;" alt="{\displaystyle L_{1}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{2}}">

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</math><img src="./c6a952cfe42c86b7741f55a817da0e251793a358.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.637ex; height:2.509ex;" alt="{\displaystyle L_{2}}" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{\infty }}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle L_{\infty }}</annotation>

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</math><img src="./a5ecf05fae003abbd91bc8c749d5a8a807d6efd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.458ex; height:2.509ex;" alt="{\displaystyle L_{\infty }}" loading="lazy">.<a href="#cite_note-Miettinen1999-2">[2]</a> The method of global criterion is sensitive to the scaling of the objective functions. Thus, it is recommended that the objectives be normalized into a uniform, dimensionless scale.<a href="#cite_note-Miettinen1999-2">[2]</a><a href="#cite_note-BrankeDeb2008-39">[39]</a>

<div class="mw-heading mw-heading2"><h2 id="A_priori_methods">A priori methods</h2></div>

A priori methods require that sufficient preference information is expressed before the solution process.<a href="#cite_note-HwangMasud1979-3">[3]</a> Well-known examples of a priori methods include the utility function method, <a href="Lexicographical_order" class="mw-redirect" title="Lexicographical order">lexicographic</a> method, and <a href="Goal_programming" title="Goal programming">goal programming</a>.

<div class="mw-heading mw-heading3"><h3 id="Utility_function_method">Utility function method</h3></div>

The utility function method assumes the decision maker's <a href="Utility" title="Utility">utility function</a> is available. A mapping <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u\colon Y\rightarrow \mathbb {R} }">

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but in practice, it is very difficult to construct a utility function that would accurately represent the decision maker's preferences,<a href="#cite_note-Miettinen1999-2">[2]</a> particularly since the Pareto front is unknown before the optimization begins.

<div class="mw-heading mw-heading3"><h3 id="Lexicographic_method">Lexicographic method</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Lexicographic_optimization" title="Lexicographic optimization">Lexicographic optimization</a></div>

The lexicographic method assumes that the objectives can be ranked in the order of importance. We assume that the objective functions are in the order of importance so that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{1}}">

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</math><img src="./8a585492f646ca803bc408103a0c705dd67ab8b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.228ex; height:2.509ex;" alt="{\displaystyle f_{k}}" loading="lazy"> the least important to the decision maker. Subject to this assumption, various methods can be used to attain the lexicographically optimal solution. Note that a goal or target value is not specified for any objective here, which makes it different from the Lexicographic <a href="Goal_Programming" class="mw-redirect" title="Goal Programming">Goal Programming</a> method.

<div class="mw-heading mw-heading3"><h3 id="Scalarizing">Scalarizing</h3></div>

Scalarizing a multi-objective optimization problem is an a priori method, which means formulating a single-objective optimization problem such that optimal solutions to the single-objective optimization problem are Pareto optimal solutions to the multi-objective optimization problem.<a href="#cite_note-HwangMasud1979-3">[3]</a> In addition, it is often required that every Pareto optimal solution can be reached with some parameters of the scalarization.<a href="#cite_note-HwangMasud1979-3">[3]</a> With different parameters for the scalarization, different Pareto optimal solutions are produced. A general formulation for a scalarization of a multi-objective optimization problem is

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<annotation encoding="application/x-tex">{\displaystyle {\begin{array}{ll}\min &g(f_{1}(x),\ldots ,f_{k}(x),\theta )\\{\text{s.t.}}&x\in X_{\theta }\end{array}}}</annotation>

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</math><img src="./110e8303495d7b8d8073450db281cb79b5976fb6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.877ex; height:6.176ex;" alt="{\displaystyle {\begin{array}{ll}\min &g(f_{1}(x),\ldots ,f_{k}(x),\theta )\\{\text{s.t.}}&x\in X_{\theta }\end{array}}}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }">

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</math><img src="./6e5ab2664b422d53eb0c7df3b87e1360d75ad9af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" loading="lazy"> is a vector parameter, the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{\theta }\subseteq X}">

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Very well-known examples are:

<ul><li>linear scalarization</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle \min _{x\in X}\sum _{i=1}^{k}w_{i}f_{i}(x)}</annotation>

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</math><img src="./78c765cc8c5d3a5f3741b5cc515a923bdd4f2cbc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.546ex; height:7.343ex;" alt="{\displaystyle \min _{x\in X}\sum _{i=1}^{k}w_{i}f_{i}(x)}" loading="lazy"></dd></dl></dd>

<dd>where the weights of the objectives <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{i}>0}">

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</math><img src="./ae83301aafa900ab58d3d6b84589d97653824986.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.725ex; height:2.509ex;" alt="{\displaystyle w_{i}>0}" loading="lazy"> are the parameters of the scalarization.</dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \epsilon }</annotation>

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</math><img src="./c3837cad72483d97bcdde49c85d3b7b859fb3fd2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.944ex; height:1.676ex;" alt="{\displaystyle \epsilon }" loading="lazy">-constraint method (see, e.g.<a href="#cite_note-Miettinen1999-2">[2]</a>)</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{array}{ll}\min &f_{j}(x)\\{\text{s.t.}}&x\in X\\&f_{i}(x)\leq \epsilon _{i}{\text{ for }}i\in \{1,\ldots ,k\}\setminus \{j\}\end{array}}}</annotation>

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</math><img src="./4567ec515782109a16f68f8c01d16085cf94dd86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:39.814ex; height:9.676ex;" alt="{\displaystyle {\begin{array}{ll}\min &f_{j}(x)\\{\text{s.t.}}&x\in X\\&f_{i}(x)\leq \epsilon _{i}{\text{ for }}i\in \{1,\ldots ,k\}\setminus \{j\}\end{array}}}" loading="lazy"></dd></dl></dd>

<dd>where upper bounds <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon _{j}}">

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</math><img src="./e1d007888875e2294ff7ef04b13349b27ec46e5a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.854ex; height:2.343ex;" alt="{\displaystyle \epsilon _{j}}" loading="lazy"> are parameters as above and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{j}}">

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</math><img src="./acc195ab3f9d65994b47774eb013601d09217aee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.049ex; height:2.843ex;" alt="{\displaystyle f_{j}}" loading="lazy"> is the objective to be minimized.</dd></dl>

Somewhat more advanced examples are the following:

<ul><li>achievement scalarizing problems of Wierzbicki<a href="#cite_note-Wierzbicki1982-41">[41]</a></li></ul>

<dl><dd>One example of the achievement scalarizing problems can be formulated as

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\min &\max _{i=1,\ldots ,k}\left[{\frac {f_{i}(x)-{\bar {z}}_{i}}{z_{i}^{nadir}-z_{i}^{utop}}}\right]+\rho \sum _{i=1}^{k}{\frac {f_{i}(x)}{z_{i}^{nadir}-z_{i}^{utop}}}\\{\text{s.t. }}&x\in S\end{aligned}}}</annotation>

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</math><img src="./2976aa9f25ed980fbecc3dd76bd262ad9cef3646.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:49.656ex; height:10.509ex;" alt="{\displaystyle {\begin{aligned}\min &\max _{i=1,\ldots ,k}\left[{\frac {f_{i}(x)-{\bar {z}}_{i}}{z_{i}^{nadir}-z_{i}^{utop}}}\right]+\rho \sum _{i=1}^{k}{\frac {f_{i}(x)}{z_{i}^{nadir}-z_{i}^{utop}}}\\{\text{s.t. }}&x\in S\end{aligned}}}" loading="lazy"></dd></dl></dd>

<dd>where the term <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho \sum _{i=1}^{k}{\frac {f_{i}(x)}{z_{i}^{nadir}-z_{i}^{utop}}}}">

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</math><img src="./dc0edd6d48173d1a981d1303c620286cfe82c8f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:18.836ex; height:7.509ex;" alt="{\displaystyle \rho \sum _{i=1}^{k}{\frac {f_{i}(x)}{z_{i}^{nadir}-z_{i}^{utop}}}}" loading="lazy"> is called the augmentation term, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho >0}">

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</math><img src="./11bd697f113e3e1bd7c76f2f441fd102eca99cab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.463ex; height:2.676ex;" alt="{\displaystyle \rho >0}" loading="lazy"> is a small constant, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z^{nadir}}">

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</math><img src="./144d08c85a5ca6924b9651893eb7ec27af7663a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.347ex; height:2.676ex;" alt="{\displaystyle z^{nadir}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z^{utop}}">

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</math><img src="./36bb60704fac35caff3db937e915a69b1d5106fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.481ex; height:2.509ex;" alt="{\displaystyle z^{utop}}" loading="lazy"> are the nadir and utopian vectors, respectively. In the above problem, the parameter is the so-called reference point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {z}}}">

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</math><img src="./52dd0599595d539f7d757ec21da6c6e6ac3ad427.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.296ex; height:2.009ex;" alt="{\displaystyle {\bar {z}}}" loading="lazy"> representing objective function values preferred by the decision maker.</dd></dl>

<ul><li>Sen's multi-objective programming<a href="#cite_note-42">[42]</a></li></ul>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{array}{ll}\max &{\frac {\sum _{j=1}^{r}Z_{j}}{W_{j}}}-{\frac {\sum _{j=r+1}^{s}Z_{j}}{W_{r+1}}}\\{\text{s.t. }}&AX=b\\&X\geq 0\end{array}}}</annotation>

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</math><img src="./18c0d632c092cc8bd379e4034e9da8f3d849d982.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.338ex; width:26.31ex; height:11.843ex;" alt="{\displaystyle {\begin{array}{ll}\max &{\frac {\sum _{j=1}^{r}Z_{j}}{W_{j}}}-{\frac {\sum _{j=r+1}^{s}Z_{j}}{W_{r+1}}}\\{\text{s.t. }}&AX=b\\&X\geq 0\end{array}}}" loading="lazy"></dd></dl></dd>

<dd>where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{j}}">

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</math><img src="./01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy">.</dd></dl>

<ul><li>hypervolume/Chebyshev scalarization<a href="#cite_note-Golovin2021-43">[43]</a></li></ul>

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</math><img src="./2e1ebed84319ddab23aa9814e9418edf71583db5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:14.889ex; height:6.009ex;" alt="{\displaystyle \min _{x\in X}\max _{i}{\frac {f_{i}(x)}{w_{i}}}}" loading="lazy"></dd></dl></dd>

<dd>where the weights of the objectives <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{i}>0}">

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</math><img src="./ae83301aafa900ab58d3d6b84589d97653824986.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.725ex; height:2.509ex;" alt="{\displaystyle w_{i}>0}" loading="lazy"> are the parameters of the scalarization. If the parameters/weights are drawn uniformly in the positive orthant, it is shown that this scalarization provably converges to the <a href="Pareto_front" title="Pareto front">Pareto front</a>,<a href="#cite_note-Golovin2021-43">[43]</a> even when the front is non-convex.</dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Smooth_Chebyshev_(Tchebycheff)_scalarization">Smooth Chebyshev (Tchebycheff) scalarization</h3></div>

The smooth Chebyshev scalarization;<a href="#cite_note-Lin2024-44">[44]</a> also called smooth Tchebycheff scalarisation (STCH); replaces the non-differentiable max-operator of the classical Chebyshev scalarization with a smooth logarithmic soft-max, making standard gradient-based optimization applicable. Unlike typical scalarization methods, it guarantees exploration of the entire Pareto front, convex or concave.

<dl><dt>Definition</dt></dl>

For a minimization problem with objective functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{1},\dots ,f_{k}}">

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</msup>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle z^{\mathrm {ideal} }\in \mathbb {R} ^{k}}</annotation>

</semantics>

</math><img src="./78dd7a700bdb4ddb1f3924d8b9dddcdaf215bc46.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.311ex; height:2.676ex;" alt="{\displaystyle z^{\mathrm {ideal} }\in \mathbb {R} ^{k}}" loading="lazy">, the smooth Chebyshev scalarising function is

<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{u}^{\mathrm {STCH} }\!{\bigl (}x\mid {\boldsymbol {\lambda }}{\bigr )}=u\,\ln \!{\Bigl (}\sum _{i=1}^{k}\exp \!{\bigl (}{\tfrac {\lambda _{i}\,[\,f_{i}(x)-z_{i}^{\mathrm {ideal} }\,]}{u}}{\bigr )}{\Bigr )},\qquad u>0,\;{\boldsymbol {\lambda }}\in \Delta _{k-1},}">

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</munderover>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mrow>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g_{u}^{\mathrm {STCH} }\!{\bigl (}x\mid {\boldsymbol {\lambda }}{\bigr )}=u\,\ln \!{\Bigl (}\sum _{i=1}^{k}\exp \!{\bigl (}{\tfrac {\lambda _{i}\,[\,f_{i}(x)-z_{i}^{\mathrm {ideal} }\,]}{u}}{\bigr )}{\Bigr )},\qquad u>0,\;{\boldsymbol {\lambda }}\in \Delta _{k-1},}</annotation>

</semantics>

</math><img src="./66db951e5d31d1fcaa06a8b55edc819b47065f89.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:68.393ex; height:7.343ex;" alt="{\displaystyle g_{u}^{\mathrm {STCH} }\!{\bigl (}x\mid {\boldsymbol {\lambda }}{\bigr )}=u\,\ln \!{\Bigl (}\sum _{i=1}^{k}\exp \!{\bigl (}{\tfrac {\lambda _{i}\,[\,f_{i}(x)-z_{i}^{\mathrm {ideal} }\,]}{u}}{\bigr )}{\Bigr )},\qquad u>0,\;{\boldsymbol {\lambda }}\in \Delta _{k-1},}" loading="lazy">

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u}</annotation>

</semantics>

</math><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"> is the smoothing parameter and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\lambda }}=(\lambda _{1},\dots ,\lambda _{k})}">

</mrow>

<msub>

</mrow>

</msub>

<msub>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\lambda }}=(\lambda _{1},\dots ,\lambda _{k})}</annotation>

</semantics>

</math><img src="./8630a32aa104e83cd89371e8875f8d6f68f532e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.499ex; height:2.843ex;" alt="{\displaystyle {\boldsymbol {\lambda }}=(\lambda _{1},\dots ,\lambda _{k})}" loading="lazy"> is a weight vector on the probability simplex <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{k-1}}">

<msub>

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</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta _{k-1}}</annotation>

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</math><img src="./e6233cdb87784669a569cdc03bef0724b1e505f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.125ex; height:2.509ex;" alt="{\displaystyle \Delta _{k-1}}" loading="lazy">.

As <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u\to 0^{+}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u\to 0^{+}}</annotation>

</semantics>

</math><img src="./8c49bbdc3e0d47c3c2a9a587c11d927d181134a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.617ex; height:2.509ex;" alt="{\displaystyle u\to 0^{+}}" loading="lazy"> this converges to the classical (non-smooth) Chebyshev form

<msup>

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</msup>

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</munder>

<msub>

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<msub>

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</msub>

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<annotation encoding="application/x-tex">{\displaystyle g^{\mathrm {TCH} }\!{\bigl (}x\mid {\boldsymbol {\lambda }}{\bigr )}=\max _{i}\lambda _{i}\,[\,f_{i}(x)-z_{i}^{\mathrm {ideal} }\,].}</annotation>

</semantics>

</math><img src="./48dff0b3532f3a178b0958eb9738138417fedcb6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:37.215ex; height:4.343ex;" alt="{\displaystyle g^{\mathrm {TCH} }\!{\bigl (}x\mid {\boldsymbol {\lambda }}{\bigr )}=\max _{i}\lambda _{i}\,[\,f_{i}(x)-z_{i}^{\mathrm {ideal} }\,].}" loading="lazy">

The parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u}</annotation>

</semantics>

</math><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"> controls the trade-off between differentiability and approximation accuracy: smaller values yield a closer match to the classical Chebyshev scalarisation but reduce the <a href="Lipschitz_continuity#Lipschitz_constant" title="Lipschitz continuity">Lipschitz constant</a> of the gradient, while larger values give a smoother surface at the cost of looser approximation.

<dl><dt>Properties</dt></dl>

<ul><li>Smoothness and complexity — <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{u}^{\mathrm {STCH} }}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g_{u}^{\mathrm {STCH} }}</annotation>

</semantics>

</math><img src="./3e78a65ab071e532f4dceaadc202fe125ad1b23d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.87ex; height:2.843ex;" alt="{\displaystyle g_{u}^{\mathrm {STCH} }}" loading="lazy"> is continuously differentiable with an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

</semantics>

</math><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy">-Lipschitz gradient. When every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{i}}</annotation>

</semantics>

</math><img src="./65da883ca3d16b461e46c94777b0d9c4aa010e79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.939ex; height:2.509ex;" alt="{\displaystyle f_{i}}" loading="lazy"> is convex the function is convex, and an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation>

</semantics>

</math><img src="./a30c89172e5b88edbd45d3e2772c7f5e562e5173.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }" loading="lazy">-optimal point is reachable in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}(1/\varepsilon )}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}(1/\varepsilon )}</annotation>

</semantics>

</math><img src="./8506769fdc7f4fe21884f2ea8a5d0f290ff75c92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.068ex; height:2.843ex;" alt="{\displaystyle {\mathcal {O}}(1/\varepsilon )}" loading="lazy"> first-order iterations; sub-gradient descent on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g^{\mathrm {TCH} }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g^{\mathrm {TCH} }}</annotation>

</semantics>

</math><img src="./42d2d1bb484f98e1a48447b12667306be890eae2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.956ex; height:3.009ex;" alt="{\displaystyle g^{\mathrm {TCH} }}" loading="lazy"> needs <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}(1/\varepsilon ^{2})}">

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}(1/\varepsilon ^{2})}</annotation>

</semantics>

</math><img src="./31482bfd7cc3e09ae4cdfa033a9e8ff1ade784bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.122ex; height:3.176ex;" alt="{\displaystyle {\mathcal {O}}(1/\varepsilon ^{2})}" loading="lazy"> iterations.<a href="#cite_note-Lin2024-44">[44]</a></li>

<li>Pareto optimality — For any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u>0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u>0}</annotation>

</semantics>

</math><img src="./e02252ee214e4ce9f8e8add664b786cbc55dd40c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle u>0}" loading="lazy"> every minimizer of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{u}^{\mathrm {STCH} }(\cdot \mid {\boldsymbol {\lambda }})}">

</mrow>

</mrow>

</msubsup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g_{u}^{\mathrm {STCH} }(\cdot \mid {\boldsymbol {\lambda }})}</annotation>

</semantics>

</math><img src="./adbbdd6d71167b82b107c6f1fcbfbe061481ed11.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.823ex; height:3.009ex;" alt="{\displaystyle g_{u}^{\mathrm {STCH} }(\cdot \mid {\boldsymbol {\lambda }})}" loading="lazy"> is weakly Pareto-optimal; if all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{i}>0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda _{i}>0}</annotation>

</semantics>

</math><img src="./514b5117b652e1a49842c886d374760d6d08b8a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.416ex; height:2.509ex;" alt="{\displaystyle \lambda _{i}>0}" loading="lazy"> (or the minimizer is unique) it is Pareto-optimal.<a href="#cite_note-Lin2024-44">[44]</a></li>

<li>Exhaustiveness — There exists a threshold <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u^{*}>0}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u^{*}>0}</annotation>

</semantics>

</math><img src="./7becd00310c2b5b7af777ea6646e0ff26a5a7d12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.645ex; height:2.343ex;" alt="{\displaystyle u^{*}>0}" loading="lazy"> such that, for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<u<u^{*}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0<u<u^{*}}</annotation>

</semantics>

</math><img src="./e622528ff4348769e368ddd2546a8499dab67548.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.073ex; height:2.343ex;" alt="{\displaystyle 0<u<u^{*}}" loading="lazy">, every Pareto-optimal point can be obtained as a minimizer of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{u}^{\mathrm {STCH} }}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g_{u}^{\mathrm {STCH} }}</annotation>

</semantics>

</math><img src="./3e78a65ab071e532f4dceaadc202fe125ad1b23d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.87ex; height:2.843ex;" alt="{\displaystyle g_{u}^{\mathrm {STCH} }}" loading="lazy"> for some weight vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\lambda }}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\lambda }}}</annotation>

</semantics>

</math><img src="./078e03b306faa99784d1135b59cd537aacbf5e27.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.56ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {\lambda }}}" loading="lazy">; when the Pareto front is convex this holds for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u>0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u>0}</annotation>

</semantics>

</math><img src="./e02252ee214e4ce9f8e8add664b786cbc55dd40c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle u>0}" loading="lazy">.<a href="#cite_note-Lin2024-44">[44]</a></li></ul>

For example, <a href="Portfolio_optimization" title="Portfolio optimization">portfolio optimization</a> is often conducted in terms of <a href="Modern_portfolio_theory" title="Modern portfolio theory">mean-variance analysis</a>. In this context, the efficient set is a subset of the portfolios parametrized by the portfolio mean return <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{P}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu _{P}}</annotation>

</semantics>

</math><img src="./bc5d5a647544de16d201a519cb67a16c1c00a700.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.868ex; height:2.176ex;" alt="{\displaystyle \mu _{P}}" loading="lazy"> in the problem of choosing portfolio shares to minimize the portfolio's variance of return <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{P}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma _{P}}</annotation>

</semantics>

</math><img src="./8e9872eb650117545176b9fde54569d40d52946b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.794ex; height:2.009ex;" alt="{\displaystyle \sigma _{P}}" loading="lazy"> subject to a given value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{P}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu _{P}}</annotation>

</semantics>

</math><img src="./bc5d5a647544de16d201a519cb67a16c1c00a700.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.868ex; height:2.176ex;" alt="{\displaystyle \mu _{P}}" loading="lazy">; see <a href="Mutual_fund_separation_theorem#Portfolio_separation_in_mean-variance_analysis" title="Mutual fund separation theorem">Mutual fund separation theorem</a> for details. Alternatively, the efficient set can be specified by choosing the portfolio shares to maximize the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{P}-b\sigma _{P}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu _{P}-b\sigma _{P}}</annotation>

</semantics>

</math><img src="./e4b8e8a40cae02fd2a66002c61ac5ebcc3495c71.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.5ex; height:2.676ex;" alt="{\displaystyle \mu _{P}-b\sigma _{P}}" loading="lazy">; the set of efficient portfolios consists of the solutions as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b}</annotation>

</semantics>

</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"> ranges from zero to infinity.

Some of the above scalarizations involve invoking the <a href="Minimax" title="Minimax">minimax</a> principle, where always the worst of the different objectives is optimized.<a href="#cite_note-45">[45]</a>

<div class="mw-heading mw-heading2"><h2 id="A_posteriori_methods">A posteriori methods</h2></div>

A posteriori methods aim at producing all the Pareto optimal solutions or a representative subset of the Pareto optimal solutions. Most a posteriori methods fall into either one of the following three classes:

<ul><li><a href="Mathematical_programming" class="mw-redirect" title="Mathematical programming">Mathematical programming</a>-based a posteriori methods where an algorithm is run repeatedly, each run producing one Pareto optimal solution;</li>

<li><a href="Evolutionary_algorithm" title="Evolutionary algorithm">Evolutionary algorithms</a> where one run of the algorithm produces a set of Pareto optimal solutions;</li>

<li><a href="Deep_learning" title="Deep learning">Deep learning</a> methods where a model is first trained on a subset of solutions and then queried to provide other solutions on the Pareto front.</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Mathematical_programming">Mathematical programming</h3></div>

Well-known examples of mathematical programming-based a posteriori methods are the Normal Boundary Intersection (NBI),<a href="#cite_note-doi10.1137/S1052623496307510-46">[46]</a> Modified Normal Boundary Intersection (NBIm),<a href="#cite_note-S._Motta-47">[47]</a> Normal Constraint (NC),<a href="#cite_note-ReferenceA-48">[48]</a><a href="#cite_note-ReferenceB-49">[49]</a> Successive Pareto Optimization (SPO),<a href="#cite_note-ReferenceC-50">[50]</a> and Directed Search Domain (DSD)<a href="#cite_note-51">[51]</a> methods, which solve the multi-objective optimization problem by constructing several scalarizations. The solution to each scalarization yields a Pareto optimal solution, whether locally or globally. The scalarizations of the NBI, NBIm, NC, and DSD methods are constructed to obtain evenly distributed Pareto points that give a good approximation of the real set of Pareto points.

<div class="mw-heading mw-heading3"><h3 id="Evolutionary_algorithms">Evolutionary algorithms</h3></div>

<a href="Evolutionary_algorithms" class="mw-redirect" title="Evolutionary algorithms">Evolutionary algorithms</a> are popular approaches to generating Pareto optimal solutions to a multi-objective optimization problem. Most evolutionary multi-objective optimization (EMO) algorithms apply Pareto-based ranking schemes. Evolutionary algorithms such as the Non-dominated Sorting Genetic Algorithm-II (NSGA-II),<a href="#cite_note-doi10.1109/4235.996017-52">[52]</a> its extended version NSGA-III,<a href="#cite_note-53">[53]</a><a href="#cite_note-54">[54]</a> Strength Pareto Evolutionary Algorithm 2 (SPEA-2)<a href="#cite_note-55">[55]</a> and multiobjective <a href="Differential_evolution" title="Differential evolution">differential evolution</a> variants have become standard approaches, although some schemes based on <a href="Particle_swarm_optimization#Variants" title="Particle swarm optimization">particle swarm optimization</a> and <a href="Simulated_annealing" title="Simulated annealing">simulated annealing</a><a href="#cite_note-56">[56]</a> are significant. The main advantage of evolutionary algorithms, when applied to solve multi-objective optimization problems, is the fact that they typically generate sets of solutions, allowing computation of an approximation of the entire Pareto front. The main disadvantage of evolutionary algorithms is their lower speed and the Pareto optimality of the solutions cannot be guaranteed; it is only known that none of the generated solutions is dominated by another.

Another paradigm for multi-objective optimization based on novelty using evolutionary algorithms was recently improved upon.<a href="#cite_note-vargas2015-57">[57]</a> This paradigm searches for novel solutions in objective space (i.e., novelty search<a href="#cite_note-58">[58]</a> on objective space) in addition to the search for non-dominated solutions. Novelty search is like stepping stones guiding the search to previously unexplored places. It is especially useful in overcoming bias and plateaus as well as guiding the search in many-objective optimization problems.

<div class="mw-heading mw-heading3"><h3 id="Deep_learning_methods">Deep learning methods</h3></div>

<a href="Deep_learning" title="Deep learning">Deep learning</a> conditional methods are new approaches to generating several Pareto optimal solutions. The idea is to use the generalization capacity of deep neural networks to learn a model of the entire Pareto front from a limited number of example trade-offs along that front, a task called Pareto Front Learning.<a href="#cite_note-:0-59">[59]</a> Several approaches address this setup, including using hypernetworks<a href="#cite_note-:0-59">[59]</a> and using Stein variational gradient descent.<a href="#cite_note-60">[60]</a>

<div class="mw-heading mw-heading3"><h3 id="List_of_methods">List of methods</h3></div>

Commonly known a posteriori methods are listed below:

<ul><li>ε-constraint method<a href="#cite_note-Mavrotas2009-61">[61]</a><a href="#cite_note-CarvalhoRibeiro2020-62">[62]</a></li>

<li>Pareto-Hypernetworks <a href="#cite_note-:0-59">[59]</a></li>

<li>Multi-objective Branch-and-Bound<a href="#cite_note-MavrotasDiakoulaki2005-63">[63]</a><a href="#cite_note-VincentSeipp2013-64">[64]</a><a href="#cite_note-PrzybylskiGandibleux2017-65">[65]</a></li>

<li>Normal Boundary Intersection (NBI)<a href="#cite_note-doi10.1137/S1052623496307510-46">[46]</a></li>

<li>Modified Normal Boundary Intersection (NBIm)<a href="#cite_note-S._Motta-47">[47]</a></li>

<li>Normal Constraint (NC)<a href="#cite_note-ReferenceA-48">[48]</a><a href="#cite_note-ReferenceB-49">[49]</a></li>

<li>Successive Pareto Optimization (SPO)<a href="#cite_note-ReferenceC-50">[50]</a></li>

<li>Directed Search Domain (DSD)</li>

<li>PGEN (Pareto surface generation for convex multi-objective instances)<a href="#cite_note-66">[66]</a></li>

<li><a href="IOSO" title="IOSO">IOSO</a> (Indirect Optimization on the basis of Self-Organization)</li>

<li>SMS-EMOA (S-metric selection evolutionary multi-objective algorithm)<a href="#cite_note-SMS-EMOA-67">[67]</a></li>

<li>Approximation-Guided Evolution (first algorithm to directly implement and optimize the formal concept of <a href="Approximation_algorithm" title="Approximation algorithm">approximation</a> from theoretical computer science)<a href="#cite_note-AGE-68">[68]</a></li>

<li><a href="Reactive_Search_Optimization" class="mw-redirect" title="Reactive Search Optimization">Reactive Search Optimization</a> (using machine learning for adapting strategies and objectives),<a href="#cite_note-69">[69]</a><a href="#cite_note-70">[70]</a> implemented in <a href="LIONsolver" title="LIONsolver">LIONsolver</a></li>

<li><a href="Benson's_algorithm" title="Benson's algorithm">Benson's algorithm</a> for <a href="Multi-objective_linear_programming" title="Multi-objective linear programming">multi-objective linear programs</a> and for multi-objective convex programs</li>

<li><a href="Particle_swarm_optimization#Variants" title="Particle swarm optimization">Multi-objective particle swarm optimization</a></li>

<li>Subpopulation Algorithm based on Novelty<a href="#cite_note-vargas2015-57">[57]</a></li>

<li>MOEA/D (Multi-Objective Evolutionary Algorithm based on Decomposition)</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Interactive_methods">Interactive methods</h2></div>

In interactive methods of optimizing multiple objective problems, the solution process is iterative and the decision maker continuously interacts with the method when searching for the most preferred solution (see e.g., Miettinen 1999,<a href="#cite_note-Miettinen1999-2">[2]</a> Miettinen 2008<a href="#cite_note-Miettinen2008-71">[71]</a>). In other words, the decision maker is expected to express preferences at each iteration to get Pareto optimal solutions that are of interest to the decision maker and learn what kind of solutions are attainable.

The following steps are commonly present in interactive methods of optimization:<a href="#cite_note-Miettinen2008-71">[71]</a>

<ol><li>initialize (e.g., calculate ideal and approximated nadir objective vectors and show them to the decision maker)</li>

<li>generate a Pareto optimal starting point (by using e.g., some no-preference method or solution given by the decision maker)</li>

<li>ask for preference information from the decision maker (e.g., aspiration levels or number of new solutions to be generated)</li>

<li>generate new Pareto optimal solution(s) according to the preferences and show it/them and possibly some other information about the problem to the decision maker</li>

<li>if several solutions were generated, ask the decision maker to select the best solution so far</li>

<li>stop (if the decision maker wants to; otherwise, go to step 3).</li></ol>

The above aspiration levels refer to desirable objective function values forming a reference point. Instead of mathematical convergence, often used as a stopping criterion in <a href="Mathematical_optimization" title="Mathematical optimization">mathematical optimization</a> methods, psychological convergence is often emphasized in interactive methods. Generally speaking, a method is terminated when the decision maker is confident that he/she has found the most preferred solution available.

<div class="mw-heading mw-heading3"><h3 id="Types_of_preference_information">Types of preference information</h3></div>

There are different interactive methods involving different types of preference information. Three types can be identified based on

<ol><li>trade-off information,</li>

<li>reference points, and</li>

<li>classification of objective functions.<a href="#cite_note-Miettinen2008-71">[71]</a></li></ol>

On the other hand, a fourth type of generating a small sample of solutions is included in:<a href="#cite_note-Luque2011-72">[72]</a><a href="#cite_note-Ruiz2012-73">[73]</a> An example of the interactive method utilizing trade-off information is the <a href="Zionts-Wallenius_method" class="mw-redirect" title="Zionts-Wallenius method">Zionts-Wallenius method</a>,<a href="#cite_note-Zionts1976-74">[74]</a> where the decision maker is shown several objective trade-offs at each iteration, and (s)he is expected to say whether (s)he likes, dislikes, or is indifferent with respect to each trade-off. In reference point-based methods (see e.g.,<a href="#cite_note-Wierzbicki1986-75">[75]</a><a href="#cite_note-WierzbickiMakowski2000-76">[76]</a>), the decision maker is expected at each iteration to specify a reference point consisting of desired values for each objective and a corresponding Pareto optimal solution(s) is then computed and shown to them for analysis. In classification-based interactive methods, the decision maker is assumed to give preferences in the form of classifying objectives at the current Pareto optimal solution into different classes, indicating how the values of the objectives should be changed to get a more preferred solution. Then, the classification information is considered when new (more preferred) Pareto optimal solution(s) are computed. In the satisficing trade-off method (STOM),<a href="#cite_note-Nakayama1984-77">[77]</a> three classes are used: objectives whose values 1) should be improved, 2) can be relaxed, and 3) are acceptable as such. In the NIMBUS method,<a href="#cite_note-Miettinen1995-78">[78]</a><a href="#cite_note-Miettinen2006-79">[79]</a> two additional classes are also used: objectives whose values 4) should be improved until a given bound and 5) can be relaxed until a given bound.

<div class="mw-heading mw-heading2"><h2 id="Hybrid_methods">Hybrid methods</h2></div>

Different <a href="Hybrid_algorithm" title="Hybrid algorithm">hybrid</a> methods exist, but here we consider hybridizing MCDM (<a href="Multi-criteria_decision-making" class="mw-redirect" title="Multi-criteria decision-making">multi-criteria decision-making</a>) and EMO (evolutionary multi-objective optimization). A hybrid algorithm in multi-objective optimization combines algorithms/approaches from these two fields (see e.g.,<a href="#cite_note-Miettinen2008-71">[71]</a>). Hybrid algorithms of EMO and MCDM are mainly used to overcome shortcomings by utilizing strengths. Several types of hybrid algorithms have been proposed in the literature, e.g., incorporating MCDM approaches into EMO algorithms as a local search operator, leading a DM to the most preferred solution(s), etc. A local search operator is mainly used to enhance the rate of convergence of EMO algorithms.

The roots for hybrid multi-objective optimization can be traced to the first Dagstuhl seminar organized in November 2004 (see <a rel="nofollow" class="external text" href="http://www.dagstuhl.de/en/program/calendar/semhp/?semnr=04461">here</a>). Here, some of the best minds in EMO (Professor Kalyanmoy Deb, Professor Jürgen Branke, etc.) and MCDM (Professor Kaisa Miettinen, Professor Ralph E. Steuer, etc.) realized the potential in combining ideas and approaches of MCDM and EMO fields to prepare hybrids of them. Subsequently, many more Dagstuhl seminars have been arranged to foster collaboration. Recently, hybrid multi-objective optimization has become an important theme in several international conferences in the area of EMO and MCDM (see e.g.,<a href="#cite_note-Sindhya2011-80">[80]</a><a href="#cite_note-Sindhya2008-81">[81]</a>).

<div class="mw-heading mw-heading2"><h2 id="Visualization_of_the_Pareto_front">Visualization of the Pareto front</h2></div>

Visualization of the Pareto front is one of the a posteriori preference techniques of multi-objective optimization. The a posteriori preference techniques provide an important class of multi-objective optimization techniques.<a href="#cite_note-Miettinen1999-2">[2]</a> Usually, the a posteriori preference techniques include four steps: (1) computer approximates the Pareto front, i.e., the Pareto optimal set in the objective space; (2) the decision maker studies the Pareto front approximation; (3) the decision maker identifies the preferred point at the Pareto front; (4) computer provides the Pareto optimal decision, whose output coincides with the objective point identified by the decision maker. From the point of view of the decision maker, the second step of the a posteriori preference techniques is the most complicated. There are two main approaches to informing the decision maker. First, a number of points of the Pareto front can be provided in the form of a list (interesting discussion and references are given in<a href="#cite_note-BensonSayin1997-82">[82]</a>) or using heatmaps.<a href="#cite_note-Pryke,_Mostaghim,_Nazemi-83">[83]</a>

<div class="mw-heading mw-heading3"><h3 id="Visualization_in_bi-objective_problems:_tradeoff_curve">Visualization in bi-objective problems: tradeoff curve</h3></div>

In the case of bi-objective problems, informing the decision maker concerning the Pareto front is usually carried out by its visualization: the Pareto front, often named the tradeoff curve in this case, can be drawn at the objective plane. The tradeoff curve gives full information on objective values and on objective tradeoffs, which inform how improving one objective is related to deteriorating the second one while moving along the tradeoff curve. The decision maker takes this information into account while specifying the preferred Pareto optimal objective point. The idea to approximate and visualize the Pareto front was introduced for linear bi-objective decision problems by S. Gass and T. Saaty.<a href="#cite_note-GassSaaty1955-84">[84]</a> This idea was developed and applied in environmental problems by J.L. Cohon.<a href="#cite_note-Cohon2004-85">[85]</a> A review of methods for approximating the Pareto front for various decision problems with a small number of objectives (mainly, two) is provided in.<a href="#cite_note-RuzikaWiecek2005-86">[86]</a>

<div class="mw-heading mw-heading3"><h3 id="Visualization_in_high-order_multi-objective_optimization_problems">Visualization in high-order multi-objective optimization problems</h3></div>

There are two generic ideas for visualizing the Pareto front in high-order multi-objective decision problems (problems with more than two objectives). One of them, which is applicable in the case of a relatively small number of objective points that represent the Pareto front, is based on using the visualization techniques developed in statistics (various diagrams, etc.; see the corresponding subsection below). The second idea proposes the display of bi-objective cross-sections (slices) of the Pareto front. It was introduced by W.S. Meisel in 1973<a href="#cite_note-87">[87]</a> who argued that such slices inform the decision maker on objective tradeoffs. The figures that display a series of bi-objective slices of the Pareto front for three-objective problems are known as the decision maps. They give a clear picture of tradeoffs between the three criteria. The disadvantages of such an approach are related to the following two facts. First, the computational procedures for constructing the Pareto front's bi-objective slices are unstable since the Pareto front is usually not stable. Secondly, it is applicable in the case of only three objectives. In the 1980s, the idea of W.S. Meisel was implemented in a different form—in the form of the <a href="Interactive_Decision_Maps" title="Interactive Decision Maps">Interactive Decision Maps</a> (IDM) technique.<a href="#cite_note-LotovBushenkov2004-88">[88]</a> More recently, N. Wesner<a href="#cite_note-89">[89]</a> proposed using a combination of a Venn diagram and multiple scatterplots of the objective space to explore the Pareto frontier and select optimal solutions.

<ul><li><a href="Concurrent_computing" title="Concurrent computing">Concurrent programming</a></li>

<li><a href="Decision-making_software" title="Decision-making software">Decision-making software</a></li>

<li><a href="Goal_programming" title="Goal programming">Goal programming</a></li>

<li><a href="Interactive_Decision_Maps" title="Interactive Decision Maps">Interactive Decision Maps</a></li>

<li><a href="MCDM" class="mw-redirect" title="MCDM">Multiple-criteria decision-making</a></li>

<li><a href="Multi-objective_linear_programming" title="Multi-objective linear programming">Multi-objective linear programming</a></li>

<li><a href="Multidisciplinary_design_optimization" title="Multidisciplinary design optimization">Multi-disciplinary design optimization</a></li>

<li><a href="Pareto_efficiency" title="Pareto efficiency">Pareto efficiency</a></li>

<li><a href="Utility_function" class="mw-redirect" title="Utility function">Utility function</a></li>

<li><a href="Vector_optimization" title="Vector optimization">Vector optimization</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-89"><a href="#cite_ref-89">^</a> <cite id="CITEREFWesner2017" class="citation cs2">Wesner, N. (2017), "Multiobjective Optimization via Visualization", Economics Bulletin, 37 (2): 1226–1233</cite>

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</ol></div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li>Emmerich, M.T.M., Deutz, A.H. A tutorial on multiobjective optimization: fundamentals and evolutionary methods. Nat Comput 17, 585–609 (2018). <a rel="nofollow" class="external free" href="https://doi.org/10.1007/s11047-018-9685-y">https://doi.org/10.1007/s11047-018-9685-y</a></li>

<li><a rel="nofollow" class="external text" href="http://mcdmsociety.org/">International Society on Multiple Criteria Decision Making</a></li>

<li><a rel="nofollow" class="external text" href="http://demonstrations.wolfram.com/EvolutionaryMultiobjectiveOptimization/">Evolutionary Multiobjective Optimization</a>, <a href="The_Wolfram_Demonstrations_Project" class="mw-redirect" title="The Wolfram Demonstrations Project">The Wolfram Demonstrations Project</a></li>

<li><a rel="nofollow" class="external text" href="http://www.openeering.com/sites/default/files/Multiobjective_Optimization_NSGAII_0.pdf">A Tutorial on Multiobjective Optimization and Genetic Algorithms</a>, <a href="Scilab" title="Scilab">Scilab</a> Professional Partner</li>

<li><a rel="nofollow" class="external text" href="http://www.mdpi.com/1996-1073/6/3/1439/pdf">Tomoiagă, Bogdan; Chindriş, Mircea; Sumper, Andreas; Sudria-Andreu, Antoni; Villafafila-Robles, Roberto. 2013. "Pareto Optimal Reconfiguration of Power Distribution Systems Using a Genetic Algorithm Based on NSGA-II." Energies 6, no. 3: 1439-1455.</a></li>

<li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20140901152751/http://www.lania.mx/%7Eccoello/EMOO/EMOObib.html">List of References on Evolutionary Multiobjective Optimization</a></li></ul>

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</style><div id="Major_subfields_of_optimization78" style="font-size:114%;margin:0 4em"><a href="Optimization_(mathematics)" class="mw-redirect" title="Optimization (mathematics)">Major subfields of optimization</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Convex_programming" class="mw-redirect" title="Convex programming">Convex programming</a></li>

<li><a href="Fractional_programming" title="Fractional programming">Fractional programming</a></li>

<li><a href="Integer_programming" title="Integer programming">Integer programming</a></li>

<li><a href="Quadratic_programming" title="Quadratic programming">Quadratic programming</a></li>

<li><a href="Nonlinear_programming" title="Nonlinear programming">Nonlinear programming</a></li>

<li><a href="Stochastic_programming" title="Stochastic programming">Stochastic programming</a></li>

<li><a href="Robust_optimization" title="Robust optimization">Robust optimization</a></li>

<li><a href="Combinatorial_optimization" title="Combinatorial optimization">Combinatorial optimization</a></li>

<li><a href="Infinite-dimensional_optimization" title="Infinite-dimensional optimization">Infinite-dimensional optimization</a></li>

<li><a href="Metaheuristic" title="Metaheuristic">Metaheuristics</a></li>

<li><a href="Constraint_satisfaction" title="Constraint satisfaction">Constraint satisfaction</a></li>

<li><a href="Multiobjective_optimization" class="mw-redirect" title="Multiobjective optimization">Multiobjective optimization</a></li>

<li><a href="Simulated_annealing" title="Simulated annealing">Simulated annealing</a></li></ul>

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