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Robust optimization

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Robust optimization is a field of <a href="Mathematical_optimization" title="Mathematical optimization">mathematical optimization</a> theory that deals with optimization problems in which a certain measure of robustness is sought against <a href="Uncertainty" title="Uncertainty">uncertainty</a> that can be represented as deterministic variability in the value of the parameters of the problem itself and/or its solution. It is related to, but often distinguished from, probabilistic optimization methods such as chance-constrained optimization.<a href="#cite_note-1">[1]</a><a href="#cite_note-2">[2]</a>

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

The origins of robust optimization date back to the establishment of modern <a href="Decision_theory" title="Decision theory">decision theory</a> in the 1950s and the use of worst case analysis and <a href="Wald's_maximin_model" title="Wald's maximin model">Wald's maximin model</a> as a tool for the treatment of severe uncertainty. It became a discipline of its own in the 1970s with parallel developments in several scientific and technological fields. Over the years, it has been applied in <a href="Statistics" title="Statistics">statistics</a>, but also in <a href="Operations_research" title="Operations research">operations research</a>,<a href="#cite_note-3">[3]</a> <a href="Electrical_engineering" title="Electrical engineering">electrical engineering</a>,<a href="#cite_note-4">[4]</a><a href="#cite_note-VPP_Robust_2015-5">[5]</a><a href="#cite_note-RO2015-6">[6]</a> <a href="Control_theory" title="Control theory">control theory</a>,<a href="#cite_note-7">[7]</a> <a href="Finance" title="Finance">finance</a>,<a href="#cite_note-8">[8]</a> <a href="Investment_management" title="Investment management">portfolio management</a><a href="#cite_note-9">[9]</a> <a href="Logistics" title="Logistics">logistics</a>,<a href="#cite_note-10">[10]</a> <a href="Manufacturing_engineering" title="Manufacturing engineering">manufacturing engineering</a>,<a href="#cite_note-11">[11]</a> <a href="Chemical_engineering" title="Chemical engineering">chemical engineering</a>,<a href="#cite_note-12">[12]</a> <a href="Medicine" title="Medicine">medicine</a>,<a href="#cite_note-13">[13]</a> and <a href="Computer_science" title="Computer science">computer science</a>. In <a href="Engineering" title="Engineering">engineering</a> problems, these formulations often take the name of "Robust Design Optimization", RDO or "Reliability Based Design Optimization", RBDO.

<div class="mw-heading mw-heading2"><h2 id="Example_1">Example 1</h2></div>

Consider the following <a href="Linear_programming" title="Linear programming">linear programming</a> problem

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<annotation encoding="application/x-tex">{\displaystyle \max _{x,y}\ \{3x+2y\}\ \ \mathrm {subject\ to} \ \ x,y\geq 0;cx+dy\leq 10,\forall (c,d)\in P}</annotation>

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</math><img src="./3483df1e84aee5e5e566de16f79cfd10e4abaae9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:61.186ex; height:4.176ex;" alt="{\displaystyle \max _{x,y}\ \{3x+2y\}\ \ \mathrm {subject\ to} \ \ x,y\geq 0;cx+dy\leq 10,\forall (c,d)\in P}" loading="lazy"></dd></dl>

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

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

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</math><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> is a given subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation>

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</math><img src="./e150115ab9f63023215109595b76686a1ff890fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}" loading="lazy">.

What makes this a 'robust optimization' problem is the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall (c,d)\in P}">

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<annotation encoding="application/x-tex">{\displaystyle \forall (c,d)\in P}</annotation>

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</math><img src="./15824976c97fe1d92cecb52d7fa2c86990734c39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.945ex; height:2.843ex;" alt="{\displaystyle \forall (c,d)\in P}" loading="lazy"> clause in the constraints. Its implication is that for a pair <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)}">

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<annotation encoding="application/x-tex">{\displaystyle (x,y)}</annotation>

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</math><img src="./41cf50e4a314ca8e2c30964baa8d26e5be7a9386.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle (x,y)}" loading="lazy"> to be admissible, the constraint <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle cx+dy\leq 10}">

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<annotation encoding="application/x-tex">{\displaystyle cx+dy\leq 10}</annotation>

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</math><img src="./14429ff621f686a435c522cf297ac38fafb686c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.972ex; height:2.509ex;" alt="{\displaystyle cx+dy\leq 10}" loading="lazy"> must be satisfied by the worst <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (c,d)\in P}">

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<annotation encoding="application/x-tex">{\displaystyle (c,d)\in P}</annotation>

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</math><img src="./be089db9e14557504eec168a694b99917c47aea6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.652ex; height:2.843ex;" alt="{\displaystyle (c,d)\in P}" loading="lazy"> pertaining to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)}">

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<annotation encoding="application/x-tex">{\displaystyle (x,y)}</annotation>

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</math><img src="./41cf50e4a314ca8e2c30964baa8d26e5be7a9386.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle (x,y)}" loading="lazy">, namely the pair <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (c,d)\in P}">

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<annotation encoding="application/x-tex">{\displaystyle (c,d)\in P}</annotation>

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</math><img src="./be089db9e14557504eec168a694b99917c47aea6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.652ex; height:2.843ex;" alt="{\displaystyle (c,d)\in P}" loading="lazy"> that maximizes the value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle cx+dy}">

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<annotation encoding="application/x-tex">{\displaystyle cx+dy}</annotation>

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</math><img src="./2d1fb0c6a48aa0be12f78154225cf2641ad1dcf5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.548ex; height:2.509ex;" alt="{\displaystyle cx+dy}" loading="lazy"> for the given value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)}">

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<annotation encoding="application/x-tex">{\displaystyle (x,y)}</annotation>

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</math><img src="./41cf50e4a314ca8e2c30964baa8d26e5be7a9386.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle (x,y)}" loading="lazy">.

If the parameter space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

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

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</math><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> is finite (consisting of finitely many elements), then this robust optimization problem itself is a <a href="Linear_programming" title="Linear programming">linear programming</a> problem: for each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (c,d)\in P}">

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<annotation encoding="application/x-tex">{\displaystyle (c,d)\in P}</annotation>

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</math><img src="./be089db9e14557504eec168a694b99917c47aea6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.652ex; height:2.843ex;" alt="{\displaystyle (c,d)\in P}" loading="lazy"> there is a linear constraint <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle cx+dy\leq 10}">

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<annotation encoding="application/x-tex">{\displaystyle cx+dy\leq 10}</annotation>

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</math><img src="./14429ff621f686a435c522cf297ac38fafb686c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.972ex; height:2.509ex;" alt="{\displaystyle cx+dy\leq 10}" loading="lazy">.

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

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

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</math><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> is not a finite set, then this problem is a linear <a href="Semi-infinite_programming" title="Semi-infinite programming">semi-infinite programming</a> problem, namely a linear programming problem with finitely many (2) decision variables and infinitely many constraints.

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

There are a number of classification criteria for robust optimization problems/models. In particular, one can distinguish between problems dealing with local and global models of robustness; and between probabilistic and non-probabilistic models of robustness. Modern robust optimization deals primarily with non-probabilistic models of robustness that are <a href="Worst_case" class="mw-redirect" title="Worst case">worst case</a> oriented and as such usually deploy <a href="Wald's_maximin_model" title="Wald's maximin model">Wald's maximin models</a>.

<div class="mw-heading mw-heading3"><h3 id="Local_robustness">Local robustness</h3></div>

There are cases where robustness is sought against small perturbations in a nominal value of a parameter. A very popular model of local robustness is the <a href="Stability_radius" title="Stability radius">radius of stability</a> model:

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<annotation encoding="application/x-tex">{\displaystyle {\hat {\rho }}(x,{\hat {u}}):=\max _{\rho \geq 0}\ \{\rho :u\in S(x),\forall u\in B(\rho ,{\hat {u}})\}}</annotation>

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</math><img src="./79dd71923c9b718237b59c081e2a7356685adbab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:43.825ex; height:4.343ex;" alt="{\displaystyle {\hat {\rho }}(x,{\hat {u}}):=\max _{\rho \geq 0}\ \{\rho :u\in S(x),\forall u\in B(\rho ,{\hat {u}})\}}" loading="lazy"></dd></dl>

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

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

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</math><img src="./28b0daee3e5310b67eb2222b45bea6236d002c69.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.176ex;" alt="{\displaystyle {\hat {u}}}" loading="lazy"> denotes the nominal value of the parameter, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B(\rho ,{\hat {u}})}">

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<annotation encoding="application/x-tex">{\displaystyle B(\rho ,{\hat {u}})}</annotation>

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</math><img src="./008d940f43bad5c5500d51b088389dc10f2882dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.139ex; height:2.843ex;" alt="{\displaystyle B(\rho ,{\hat {u}})}" loading="lazy"> denotes a ball of radius <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

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

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</math><img src="./1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"> centered at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {u}}}">

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

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</math><img src="./28b0daee3e5310b67eb2222b45bea6236d002c69.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.176ex;" alt="{\displaystyle {\hat {u}}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(x)}">

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

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</math><img src="./1886b5a535ed8f168a7c3a83afc8ca440edcdc6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.638ex; height:2.843ex;" alt="{\displaystyle S(x)}" loading="lazy"> denotes the set of values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

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

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</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"> that satisfy given stability/performance conditions associated with decision <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="./87f9e315fd7e2ba406057a97300593c4802b53e4.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 x}" loading="lazy">.

In words, the robustness (radius of stability) of decision <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="./87f9e315fd7e2ba406057a97300593c4802b53e4.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 x}" loading="lazy"> is the radius of the largest ball centered at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {u}}}">

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

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</math><img src="./28b0daee3e5310b67eb2222b45bea6236d002c69.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.176ex;" alt="{\displaystyle {\hat {u}}}" loading="lazy"> all of whose elements satisfy the stability requirements imposed on <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="./87f9e315fd7e2ba406057a97300593c4802b53e4.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 x}" loading="lazy">. The picture is this:

where the rectangle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(x)}">

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

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</math><img src="./6d626d3a1e65c94535c811c73fa83389cfb76683.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.922ex; height:2.843ex;" alt="{\displaystyle U(x)}" loading="lazy"> represents the set of all the values <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

</mstyle>

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

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</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"> associated with decision <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="./87f9e315fd7e2ba406057a97300593c4802b53e4.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 x}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Global_robustness">Global robustness</h3></div>

Consider the simple abstract robust optimization problem

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<annotation encoding="application/x-tex">{\displaystyle \max _{x\in X}\ \{f(x):g(x,u)\leq b,\forall u\in U\}}</annotation>

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</math><img src="./7bfe61c68196c0aeb79f2e227595d9a09163663c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:32.967ex; height:4.176ex;" alt="{\displaystyle \max _{x\in X}\ \{f(x):g(x,u)\leq b,\forall u\in U\}}" loading="lazy"></dd></dl>

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

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

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</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> denotes the set of all possible values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

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

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</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"> under consideration.

This is a global robust optimization problem in the sense that the robustness constraint <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x,u)\leq b,\forall u\in U}">

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<annotation encoding="application/x-tex">{\displaystyle g(x,u)\leq b,\forall u\in U}</annotation>

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</math><img src="./4da9602e1e7c7ed755fceb478a37944c4b8b6f23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.994ex; height:2.843ex;" alt="{\displaystyle g(x,u)\leq b,\forall u\in U}" loading="lazy"> represents all the possible values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

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</mrow>

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

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</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">.

The difficulty is that such a "global" constraint can be too demanding in that there is no <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="./3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"> that satisfies this constraint. But even if such an <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="./3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"> exists, the constraint can be too "conservative" in that it yields 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>

</semantics>

</math><img src="./3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"> that generates a very small payoff <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./202945cce41ecebb6f643f31d119c514bec7a074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" loading="lazy"> that is not representative of the performance of other decisions in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</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">. For instance, there could be an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x'\in X}">

<msup>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./81084d8383a416e5ca37b0ea531a56eebb02661c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.835ex; height:2.509ex;" alt="{\displaystyle x'\in X}" loading="lazy"> that only slightly violates the robustness constraint but yields a very large payoff <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x')}">

<msup>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./8f2a352e80e95a54cd90e4f21b64dc6d44581acc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.102ex; height:3.009ex;" alt="{\displaystyle f(x')}" loading="lazy">. In such cases it might be necessary to relax a bit the robustness constraint and/or modify the statement of the problem.

<div class="mw-heading mw-heading4"><h4 id="Example_2">Example 2</h4></div>

Consider the case where the objective is to satisfy a constraint <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x,u)\leq b,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(x,u)\leq b,}</annotation>

</semantics>

</math><img src="./aace924b403ad6bc70cdd9026abd8f063711176a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.361ex; height:2.843ex;" alt="{\displaystyle g(x,u)\leq b,}" loading="lazy">. where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"> denotes the decision variable and <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 a parameter whose set of possible values in <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="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy">. If there is no <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x,u)\leq b,\forall u\in U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(x,u)\leq b,\forall u\in U}</annotation>

</semantics>

</math><img src="./4da9602e1e7c7ed755fceb478a37944c4b8b6f23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.994ex; height:2.843ex;" alt="{\displaystyle g(x,u)\leq b,\forall u\in U}" loading="lazy">, then the following intuitive measure of robustness suggests itself:

</mrow>

</munder>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho (x):=\max _{Y\subseteq U}\ \{size(Y):g(x,u)\leq b,\forall u\in Y\}\ ,\ x\in X}</annotation>

</semantics>

</math><img src="./f32c2cdfbf6869d65e50827c3114403901bda117.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:52.619ex; height:4.176ex;" alt="{\displaystyle \rho (x):=\max _{Y\subseteq U}\ \{size(Y):g(x,u)\leq b,\forall u\in Y\}\ ,\ x\in X}" loading="lazy"></dd></dl>

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

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle size(Y)}</annotation>

</semantics>

</math><img src="./a6a1694d8c4741c31ef7abac5b60296478160fe2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.647ex; height:2.843ex;" alt="{\displaystyle size(Y)}" loading="lazy"> denotes an appropriate measure of the "size" of set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy">. For example, if <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="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> is a finite set, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle size(Y)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle size(Y)}</annotation>

</semantics>

</math><img src="./a6a1694d8c4741c31ef7abac5b60296478160fe2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.647ex; height:2.843ex;" alt="{\displaystyle size(Y)}" loading="lazy"> could be defined as the <a href="Cardinality" title="Cardinality">cardinality</a> of set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy">.

In words, the robustness of decision is the size of the largest subset of <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="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> for which the constraint <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x,u)\leq b}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(x,u)\leq b}</annotation>

</semantics>

</math><img src="./29dccdb0a11a44f54f5e988d2ba74d48e6acd786.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.715ex; height:2.843ex;" alt="{\displaystyle g(x,u)\leq b}" loading="lazy"> is satisfied for each <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"> in this set. An optimal decision is then a decision whose robustness is the largest.

This yields the following robust optimization problem:

</mrow>

</munder>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \max _{x\in X,Y\subseteq U}\ \{size(Y):g(x,u)\leq b,\forall u\in Y\}}</annotation>

</semantics>

</math><img src="./c61d875ab2897547e6db934f147982f5db5ca7f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:39.549ex; height:4.343ex;" alt="{\displaystyle \max _{x\in X,Y\subseteq U}\ \{size(Y):g(x,u)\leq b,\forall u\in Y\}}" loading="lazy"></dd></dl>

This intuitive notion of global robustness is not used often in practice because the robust optimization problems that it induces are usually (not always) very difficult to solve.

<div class="mw-heading mw-heading4"><h4 id="Example_3">Example 3</h4></div>

Consider the robust optimization problem

</mrow>

</munder>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z(U):=\max _{x\in X}\ \{f(x):g(x,u)\leq b,\forall u\in U\}}</annotation>

</semantics>

</math><img src="./185a4223408bdfe52803c7bf6e09c912b5e93429.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:41.393ex; height:4.176ex;" alt="{\displaystyle z(U):=\max _{x\in X}\ \{f(x):g(x,u)\leq b,\forall u\in U\}}" loading="lazy"></dd></dl>

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

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> is a real-valued function on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\times U}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./af68e115f7485e3b3d3774f06f2efc0f78d9256a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.603ex; height:2.176ex;" alt="{\displaystyle X\times U}" loading="lazy">, and assume that there is no feasible solution to this problem because the robustness constraint <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x,u)\leq b,\forall u\in U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(x,u)\leq b,\forall u\in U}</annotation>

</semantics>

</math><img src="./4da9602e1e7c7ed755fceb478a37944c4b8b6f23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.994ex; height:2.843ex;" alt="{\displaystyle g(x,u)\leq b,\forall u\in U}" loading="lazy"> is too demanding.

To overcome this difficulty, let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {N}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b7551c7bed2cd2ee83e10536d157c94a5f8f72fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-left: -0.062ex; width:2.337ex; height:2.509ex;" alt="{\displaystyle {\mathcal {N}}}" loading="lazy"> be a relatively small subset of <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="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> representing "normal" values of <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"> and consider the following robust optimization problem:

</mrow>

</mrow>

</munder>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z({\mathcal {N}}):=\max _{x\in X}\ \{f(x):g(x,u)\leq b,\forall u\in {\mathcal {N}}\}}</annotation>

</semantics>

</math><img src="./05a6a12bf38246103b5dd9ac9db4c5021f434be6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:42.377ex; height:4.343ex;" alt="{\displaystyle z({\mathcal {N}}):=\max _{x\in X}\ \{f(x):g(x,u)\leq b,\forall u\in {\mathcal {N}}\}}" loading="lazy"></dd></dl>

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

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b7551c7bed2cd2ee83e10536d157c94a5f8f72fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-left: -0.062ex; width:2.337ex; height:2.509ex;" alt="{\displaystyle {\mathcal {N}}}" loading="lazy"> is much smaller than <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="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy">, its optimal solution may not perform well on a large portion of <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="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> and therefore may not be robust against the variability of <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"> over <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="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy">.

One way to fix this difficulty is to relax the constraint <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x,u)\leq b}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(x,u)\leq b}</annotation>

</semantics>

</math><img src="./29dccdb0a11a44f54f5e988d2ba74d48e6acd786.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.715ex; height:2.843ex;" alt="{\displaystyle g(x,u)\leq b}" loading="lazy"> for values of <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"> outside the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {N}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b7551c7bed2cd2ee83e10536d157c94a5f8f72fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-left: -0.062ex; width:2.337ex; height:2.509ex;" alt="{\displaystyle {\mathcal {N}}}" loading="lazy"> in a controlled manner so that larger violations are allowed as the distance of <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"> from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {N}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b7551c7bed2cd2ee83e10536d157c94a5f8f72fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-left: -0.062ex; width:2.337ex; height:2.509ex;" alt="{\displaystyle {\mathcal {N}}}" loading="lazy"> increases. For instance, consider the relaxed robustness constraint

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(x,u)\leq b+\beta \cdot dist(u,{\mathcal {N}})\ ,\ \forall u\in U}</annotation>

</semantics>

</math><img src="./881b9d8da5a7143fd8ef5fc68fbd93b9c730329d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.403ex; height:3.009ex;" alt="{\displaystyle g(x,u)\leq b+\beta \cdot dist(u,{\mathcal {N}})\ ,\ \forall u\in U}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta \geq 0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./78f6fcdda6c164f2e0c8df177d9917a5d0c94214.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.593ex; height:2.509ex;" alt="{\displaystyle \beta \geq 0}" loading="lazy"> is a control parameter and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dist(u,{\mathcal {N}})}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle dist(u,{\mathcal {N}})}</annotation>

</semantics>

</math><img src="./38aecd3cf7d858f14f4ac7d43505a0ff0c6383a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.396ex; height:3.009ex;" alt="{\displaystyle dist(u,{\mathcal {N}})}" loading="lazy"> denotes the distance of <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"> from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {N}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b7551c7bed2cd2ee83e10536d157c94a5f8f72fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-left: -0.062ex; width:2.337ex; height:2.509ex;" alt="{\displaystyle {\mathcal {N}}}" loading="lazy">. Thus, for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta =0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./60b5e78663eba7ba08e0dd4915251e6261f4f35c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.593ex; height:2.509ex;" alt="{\displaystyle \beta =0}" loading="lazy"> the relaxed robustness constraint reduces back to the original robustness constraint.

This yields the following (relaxed) robust optimization problem:

</mrow>

</mrow>

</munder>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z({\mathcal {N}},U):=\max _{x\in X}\ \{f(x):g(x,u)\leq b+\beta \cdot dist(u,{\mathcal {N}})\ ,\ \forall u\in U\}}</annotation>

</semantics>

</math><img src="./68bd7246703a684f44a347fe9d2135da4404c187.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:62.111ex; height:4.343ex;" alt="{\displaystyle z({\mathcal {N}},U):=\max _{x\in X}\ \{f(x):g(x,u)\leq b+\beta \cdot dist(u,{\mathcal {N}})\ ,\ \forall u\in U\}}" loading="lazy"></dd></dl>

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

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b2a11f3bf7903f2935c9c68a3532d63e84153118.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.948ex; height:2.176ex;" alt="{\displaystyle dist}" loading="lazy"> is defined in such a manner that

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle dist(u,{\mathcal {N}})\geq 0,\forall u\in U}</annotation>

</semantics>

</math><img src="./54eae6f77d6e72218f453f5b08137283535015e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.937ex; height:3.009ex;" alt="{\displaystyle dist(u,{\mathcal {N}})\geq 0,\forall u\in U}" loading="lazy"></dd></dl>

and

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle dist(u,{\mathcal {N}})=0,\forall u\in {\mathcal {N}}}</annotation>

</semantics>

</math><img src="./0973860cb33a4462db11d747ea6bc3d9d10faadd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.429ex; height:3.009ex;" alt="{\displaystyle dist(u,{\mathcal {N}})=0,\forall u\in {\mathcal {N}}}" loading="lazy"></dd></dl>

and therefore the optimal solution to the relaxed problem satisfies the original constraint <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x,u)\leq b}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(x,u)\leq b}</annotation>

</semantics>

</math><img src="./29dccdb0a11a44f54f5e988d2ba74d48e6acd786.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.715ex; height:2.843ex;" alt="{\displaystyle g(x,u)\leq b}" loading="lazy"> for all values of <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"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {N}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b7551c7bed2cd2ee83e10536d157c94a5f8f72fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-left: -0.062ex; width:2.337ex; height:2.509ex;" alt="{\displaystyle {\mathcal {N}}}" loading="lazy">. It also satisfies the relaxed constraint

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(x,u)\leq b+\beta \cdot dist(u,{\mathcal {N}})}</annotation>

</semantics>

</math><img src="./868b792e32a847b3f4e646e93e556d8031d46a07.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.962ex; height:3.009ex;" alt="{\displaystyle g(x,u)\leq b+\beta \cdot dist(u,{\mathcal {N}})}" loading="lazy"></dd></dl>

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

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b7551c7bed2cd2ee83e10536d157c94a5f8f72fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-left: -0.062ex; width:2.337ex; height:2.509ex;" alt="{\displaystyle {\mathcal {N}}}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Non-probabilistic_robust_optimization_models">Non-probabilistic robust optimization models</h3></div>

The dominating paradigm in this area of robust optimization is <a href="Wald's_maximin_model" title="Wald's maximin model">Wald's maximin model</a>, namely

</mrow>

</munder>

</mrow>

</munder>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \max _{x\in X}\min _{u\in U(x)}f(x,u)}</annotation>

</semantics>

</math><img src="./d2bdd49689d87217636f116d9002b0dddc9496a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.398ex; height:4.509ex;" alt="{\displaystyle \max _{x\in X}\min _{u\in U(x)}f(x,u)}" loading="lazy"></dd></dl>

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

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b8e49fca3e322708b32d21eaa8b095dc05f09538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.326ex; height:1.676ex;" alt="{\displaystyle \max }" loading="lazy"> represents the decision maker, the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \min }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./695d28931288a686335c3969dfd15bb76ea873db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.875ex; height:2.176ex;" alt="{\displaystyle \min }" loading="lazy"> represents Nature, namely <a href="Uncertainty" title="Uncertainty">uncertainty</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</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"> represents the decision space and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(x)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./6d626d3a1e65c94535c811c73fa83389cfb76683.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.922ex; height:2.843ex;" alt="{\displaystyle U(x)}" loading="lazy"> denotes the set of possible values of <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"> associated with decision <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.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 x}" loading="lazy">. This is the classic format of the generic model, and is often referred to as minimax or maximin optimization problem. The non-probabilistic (deterministic) model has been and is being extensively used for robust optimization especially in the field of signal processing.<a href="#cite_note-14">[14]</a><a href="#cite_note-15">[15]</a><a href="#cite_note-16">[16]</a>

The equivalent <a href="Mathematical_programming" class="mw-redirect" title="Mathematical programming">mathematical programming</a> (MP) of the classic format above is

</mrow>

</munder>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \max _{x\in X,v\in \mathbb {R} }\ \{v:v\leq f(x,u),\forall u\in U(x)\}}</annotation>

</semantics>

</math><img src="./12b44b056a1d451eefe82230b0cf7ea16420bb5d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:35.757ex; height:4.343ex;" alt="{\displaystyle \max _{x\in X,v\in \mathbb {R} }\ \{v:v\leq f(x,u),\forall u\in U(x)\}}" loading="lazy"></dd></dl>

Constraints can be incorporated explicitly in these models. The generic constrained classic format is

</mrow>

</munder>

</mrow>

</munder>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \max _{x\in X}\min _{u\in U(x)}\ \{f(x,u):g(x,u)\leq b,\forall u\in U(x)\}}</annotation>

</semantics>

</math><img src="./cdaa49b27a845b1339a0446c75b50c61a2fe5e5f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:44.373ex; height:4.509ex;" alt="{\displaystyle \max _{x\in X}\min _{u\in U(x)}\ \{f(x,u):g(x,u)\leq b,\forall u\in U(x)\}}" loading="lazy"></dd></dl>

The equivalent constrained MP format is defined as:

</mrow>

</munder>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \max _{x\in X,v\in \mathbb {R} }\ \{v:v\leq f(x,u),g(x,u)\leq b,\forall u\in U(x)\}}</annotation>

</semantics>

</math><img src="./d6a25df17b612e077442210891a7cd7b2fc0a8a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:47.506ex; height:4.343ex;" alt="{\displaystyle \max _{x\in X,v\in \mathbb {R} }\ \{v:v\leq f(x,u),g(x,u)\leq b,\forall u\in U(x)\}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Probabilistically_robust_optimization_models">Probabilistically robust optimization models</h3></div>

These models quantify the uncertainty in the "true" value of the parameter of interest by probability distribution functions. They have been traditionally classified as <a href="Stochastic_programming" title="Stochastic programming">stochastic programming</a> and <a href="Stochastic_optimization" title="Stochastic optimization">stochastic optimization</a> models. Recently, probabilistically robust optimization has gained popularity by the introduction of rigorous theories such as <a href="Scenario_optimization" title="Scenario optimization">scenario optimization</a> able to quantify the robustness level of solutions obtained by randomization. These methods are also relevant to data-driven optimization methods.

<div class="mw-heading mw-heading3"><h3 id="Robust_counterpart">Robust counterpart</h3></div>

The solution method to many robust program involves creating a deterministic equivalent, called the robust counterpart. The practical difficulty of a robust program depends on if its robust counterpart is computationally tractable.<a href="#cite_note-17">[17]</a><a href="#cite_note-18">[18]</a>

<ul><li><a href="Stability_radius" title="Stability radius">Stability radius</a></li>

<li><a href="Minimax" title="Minimax">Minimax</a></li>

<li><a href="Minimax_estimator" title="Minimax estimator">Minimax estimator</a></li>

<li><a href="Minimax_regret" class="mw-redirect" title="Minimax regret">Minimax regret</a></li>

<li><a href="Robust_statistics" title="Robust statistics">Robust statistics</a></li>

<li><a href="Robust_decision_making" class="mw-redirect" title="Robust decision making">Robust decision making</a></li>

<li><a href="Robust_fuzzy_programming" title="Robust fuzzy programming">Robust fuzzy programming</a></li>

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

<li><a href="Stochastic_optimization" title="Stochastic optimization">Stochastic optimization</a></li>

<li><a href="Info-gap_decision_theory" title="Info-gap decision theory">Info-gap decision theory</a></li>

<li><a href="Taguchi_methods" title="Taguchi methods">Taguchi methods</a></li></ul>

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

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<li id="cite_note-9"><a href="#cite_ref-9">^</a> Md. Asadujjaman and Kais Zaman, "Robust Portfolio Optimization under Data Uncertainty" 15th National Statistical Conference, December 2014, Dhaka, Bangladesh.

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<li id="cite_note-10"><a href="#cite_ref-10">^</a> <cite id="CITEREFYuLi,_Han-Lin2000" class="citation journal cs1">Yu, Chian-Son; Li, Han-Lin (2000). "A robust optimization model for stochastic logistic problems". International Journal of Production Economics. 64 (1–3): 385–397. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0925-5273%2899%2900074-2">10.1016/S0925-5273(99)00074-2</a>.</cite>

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<li id="cite_note-11"><a href="#cite_ref-11">^</a> <cite id="CITEREFStrano2006" class="citation journal cs1">Strano, M (2006). "Optimization under uncertainty of sheet-metal-forming processes by the finite element method". Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture. 220 (8): 1305–1315. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1243%2F09544054JEM480">10.1243/09544054JEM480</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:108843522">108843522</a>.</cite>

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<li id="cite_note-12"><a href="#cite_ref-12">^</a> <cite id="CITEREFBernardoSaraiva,_Pedro_M.1998" class="citation journal cs1">Bernardo, Fernando P.; Saraiva, Pedro M. (1998). "Robust optimization framework for process parameter and tolerance design". AIChE Journal. 44 (9): 2007–2017. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1998AIChE..44.2007B">1998AIChE..44.2007B</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Faic.690440908">10.1002/aic.690440908</a>. <a href="Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/10316%2F8195">10316/8195</a>.</cite>

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<li id="cite_note-13"><a href="#cite_ref-13">^</a> <cite id="CITEREFChuZinchenko,_YuriyHenderson,_Shane_GSharpe,_Michael_B2005" class="citation journal cs1">Chu, Millie; Zinchenko, Yuriy; Henderson, Shane G; Sharpe, Michael B (2005). "Robust optimization for intensity modulated radiation therapy treatment planning under uncertainty". Physics in Medicine and Biology. 50 (23): 5463–5477. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2005PMB....50.5463C">2005PMB....50.5463C</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F0031-9155%2F50%2F23%2F003">10.1088/0031-9155/50/23/003</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/16306645">16306645</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:15713904">15713904</a>.</cite>

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<li id="cite_note-14"><a href="#cite_ref-14">^</a> <cite id="CITEREFVerduPoor1984" class="citation journal cs1">Verdu, S.; Poor, H. V. (1984). "On Minimax Robustness: A general approach and applications". IEEE Transactions on Information Theory. 30 (2): 328–340. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.132.837">10.1.1.132.837</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2Ftit.1984.1056876">10.1109/tit.1984.1056876</a>.</cite>

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<li id="cite_note-15"><a href="#cite_ref-15">^</a> <cite id="CITEREFKassamPoor1985" class="citation journal cs1">Kassam, S. A.; Poor, H. V. (1985). "Robust Techniques for Signal Processing: A Survey". Proceedings of the IEEE. 73 (3): 433–481. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2Fproc.1985.13167">10.1109/proc.1985.13167</a>. <a href="Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/2142%2F74118">2142/74118</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:30443041">30443041</a>.</cite>

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<li id="cite_note-16"><a href="#cite_ref-16">^</a> M. Danish Nisar. <a rel="nofollow" class="external text" href="https://www.shaker.eu/shop/978-3-8440-0332-1">"Minimax Robustness in Signal Processing for Communications"</a>, Shaker Verlag, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-8440-0332-1</bdi>, August 2011.

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<li id="cite_note-17"><a href="#cite_ref-17">^</a> Ben-Tal A., El Ghaoui, L. and Nemirovski, A. (2009). Robust Optimization. Princeton Series in Applied Mathematics, Princeton University Press, 9-16.

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<li id="cite_note-18"><a href="#cite_ref-18">^</a> <a href="Sven_Leyffer" title="Sven Leyffer">Leyffer S.</a>, Menickelly M., Munson T., Vanaret C. and Wild S. M (2020). A survey of nonlinear robust optimization. INFOR: Information Systems and Operational Research, Taylor \& Francis.

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li>H.J. Greenberg. Mathematical Programming Glossary. World Wide Web, <a rel="nofollow" class="external free" href="http://glossary.computing.society.informs.org/">http://glossary.computing.society.informs.org/</a>, 1996-2006. Edited by the INFORMS Computing Society.</li>

<li><cite id="CITEREFBen-TalNemirovski1998" class="citation journal cs1">Ben-Tal, A.; Nemirovski, A. (1998). "Robust Convex Optimization". <a href="Mathematics_of_Operations_Research" title="Mathematics of Operations Research">Mathematics of Operations Research</a>. 23 (4): 769–805. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.135.798">10.1.1.135.798</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1287%2Fmoor.23.4.769">10.1287/moor.23.4.769</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:15905691">15905691</a>.</cite></li>

<li><cite id="CITEREFBen-TalNemirovski1999" class="citation journal cs1">Ben-Tal, A.; Nemirovski, A. (1999). "Robust solutions to uncertain linear programs". <a href="Operations_Research_Letters" title="Operations Research Letters">Operations Research Letters</a>. 25: 1–13. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.424.861">10.1.1.424.861</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fs0167-6377%2899%2900016-4">10.1016/s0167-6377(99)00016-4</a>.</cite></li>

<li><cite id="CITEREFBen-TalArkadi_Nemirovski2002" class="citation journal cs1">Ben-Tal, A.; Arkadi Nemirovski, A. (2002). "Robust optimization—methodology and applications". Mathematical Programming, Series B. 92 (3): 453–480. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.298.7965">10.1.1.298.7965</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs101070100286">10.1007/s101070100286</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:1429482">1429482</a>.</cite></li>

<li>Ben-Tal A., El Ghaoui, L. and Nemirovski, A. (2006). Mathematical Programming, Special issue on Robust Optimization, Volume 107(1-2).</li>

<li>Ben-Tal A., El Ghaoui, L. and Nemirovski, A. (2009). Robust Optimization. Princeton Series in Applied Mathematics, Princeton University Press.</li>

<li><cite id="CITEREFBertsimasSim2003" class="citation journal cs1">Bertsimas, D.; Sim, M. (2003). "Robust Discrete Optimization and Network Flows". Mathematical Programming. 98 (1–3): 49–71. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.392.4470">10.1.1.392.4470</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10107-003-0396-4">10.1007/s10107-003-0396-4</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:1279073">1279073</a>.</cite></li>

<li><cite id="CITEREFBertsimasSim2006" class="citation journal cs1">Bertsimas, D.; Sim, M. (2006). "Tractable Approximations to Robust Conic Optimization Problems Dimitris Bertsimas". Mathematical Programming. 107 (1): 5–36. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.207.8378">10.1.1.207.8378</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10107-005-0677-1">10.1007/s10107-005-0677-1</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:900938">900938</a>.</cite></li>

<li><cite id="CITEREFChenSim2009" class="citation journal cs1">Chen, W.; Sim, M. (2009). "Goal Driven Optimization". Operations Research. 57 (2): 342–357. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1287%2Fopre.1080.0570">10.1287/opre.1080.0570</a>.</cite></li>

<li><cite id="CITEREFChenSimSunZhang2008" class="citation journal cs1">Chen, X.; Sim, M.; Sun, P.; Zhang, J. (2008). "A Linear-Decision Based Approximation Approach to Stochastic Programming". Operations Research. 56 (2): 344–357. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1287%2Fopre.1070.0457">10.1287/opre.1070.0457</a>.</cite></li>

<li><cite id="CITEREFChenSimSun2007" class="citation journal cs1">Chen, X.; Sim, M.; Sun, P. (2007). "A Robust Optimization Perspective on Stochastic Programming". Operations Research. 55 (6): 1058–1071. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1287%2Fopre.1070.0441">10.1287/opre.1070.0441</a>.</cite></li>

<li><cite id="CITEREFDembo1991" class="citation journal cs1">Dembo, R (1991). "Scenario optimization". Annals of Operations Research. 30 (1): 63–80. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fbf02204809">10.1007/bf02204809</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:44126126">44126126</a>.</cite></li>

<li>Dodson, B., Hammett, P., and Klerx, R. (2014) Probabilistic Design for Optimization and Robustness for Engineers John Wiley & Sons, Inc. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-118-79619-1</bdi></li>

<li><cite id="CITEREFGuptaRosenhead1968" class="citation journal cs1">Gupta, S.K.; Rosenhead, J. (1968). "Robustness in sequential investment decisions". Management Science. 15 (2): 18–29. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1287%2Fmnsc.15.2.B18">10.1287/mnsc.15.2.B18</a>.</cite></li>

<li>Kouvelis P. and Yu G. (1997). Robust Discrete Optimization and Its Applications, Kluwer.</li>

<li><cite id="CITEREFMutapcicBoyd2009" class="citation journal cs1">Mutapcic, Almir; Boyd, Stephen (2009). "Cutting-set methods for robust convex optimization with pessimizing oracles". Optimization Methods and Software. 24 (3): 381–406. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.416.4912">10.1.1.416.4912</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F10556780802712889">10.1080/10556780802712889</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:16443437">16443437</a>.</cite></li>

<li><cite id="CITEREFMulveyVanderbeiZenios1995" class="citation journal cs1">Mulvey, J.M.; Vanderbei, R.J.; Zenios, S.A. (1995). "Robust Optimization of Large-Scale Systems". Operations Research. 43 (2): 264–281. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1287%2Fopre.43.2.264">10.1287/opre.43.2.264</a>.</cite></li>

<li>Nejadseyfi, O., Geijselaers H.J.M, van den Boogaard A.H. (2018). "Robust optimization based on analytical evaluation of uncertainty propagation". Engineering Optimization 51 (9): 1581-1603. <a href="https://doi.org/10.1080/0305215X.2018.1536752" class="extiw external" title="doi:10.1080/0305215X.2018.1536752">doi:10.1080/0305215X.2018.1536752</a>.</li>

<li><cite id="CITEREFRosenblat1987" class="citation journal cs1">Rosenblat, M.J. (1987). "A robust approach to facility design". International Journal of Production Research. 25 (4): 479–486. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00207548708919855">10.1080/00207548708919855</a>.</cite></li>

<li><cite id="CITEREFRosenheadEltonGupta1972" class="citation journal cs1">Rosenhead, M.J; Elton, M; Gupta, S.K. (1972). "Robustness and Optimality as Criteria for Strategic Decisions". Operational Research Quarterly. 23 (4): 413–430. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F3007957">10.2307/3007957</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/3007957">3007957</a>.</cite></li>

<li>Rustem B. and Howe M. (2002). Algorithms for Worst-case Design and Applications to Risk Management, Princeton University Press.</li>

<li><cite id="CITEREFSniedovich2007" class="citation journal cs1">Sniedovich, M (2007). <a rel="nofollow" class="external text" href="https://doi.org/10.7494%2Fdmms.2007.1.2.111">"The art and science of modeling decision-making under severe uncertainty"</a>. Decision Making in Manufacturing and Services. 1 (1–2): 111–136. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.7494%2Fdmms.2007.1.2.111">10.7494/dmms.2007.1.2.111</a>.</cite></li>

<li><cite id="CITEREFSniedovich2008" class="citation journal cs1">Sniedovich, M (2008). "Wald's Maximin Model: a Treasure in Disguise!". Journal of Risk Finance. 9 (3): 287–291. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1108%2F15265940810875603">10.1108/15265940810875603</a>.</cite></li>

<li><cite id="CITEREFSniedovich2010" class="citation journal cs1">Sniedovich, M (2010). "A bird's view of info-gap decision theory". Journal of Risk Finance. 11 (3): 268–283. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1108%2F15265941011043648">10.1108/15265941011043648</a>.</cite></li>

<li><cite id="CITEREFWald1939" class="citation journal cs1">Wald, A (1939). <a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faoms%2F1177732144">"Contributions to the theory of statistical estimation and testing hypotheses"</a>. The Annals of Mathematical Statistics. 10 (4): 299–326. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faoms%2F1177732144">10.1214/aoms/1177732144</a>.</cite></li>

<li><cite id="CITEREFWald1945" class="citation journal cs1">Wald, A (1945). "Statistical decision functions which minimize the maximum risk". The Annals of Mathematics. 46 (2): 265–280. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1969022">10.2307/1969022</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1969022">1969022</a>.</cite></li>

<li>Wald, A. (1950). Statistical Decision Functions, John Wiley, NY.</li>

<li><cite id="CITEREFShabanzadehFattahi2015" class="citation book cs1">Shabanzadeh, Morteza; Fattahi, Mohammad (2015). "Generation Maintenance Scheduling via robust optimization". 2015 23rd Iranian Conference on Electrical Engineering. pp. 1504–1509. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FIranianCEE.2015.7146458">10.1109/IranianCEE.2015.7146458</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4799-1972-7</bdi>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:8774918">8774918</a>.</cite></li></ul>

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

<ul><li><a rel="nofollow" class="external text" href="https://www.robustopt.com">ROME: Robust Optimization Made Easy</a></li>

<li><a rel="nofollow" class="external text" href="http://robust.moshe-online.com:">Robust Decision-Making Under Severe Uncertainty</a></li>

<li><a rel="nofollow" class="external text" href="https://robustimizer.com/">Robustimizer: Robust optimization software</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="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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