SherbyNet - /page/wikibook/entry.mu

Micron Document

<!DOCTYPE html>

<title>Bees algorithm</title>

</head>

Bees algorithm

</h1>

</header>

/* start https://en.wikipedia.org/ */

.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}

/* end https://en.wikipedia.org/ */

</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="Artificial_bee_colony_algorithm" title="Artificial bee colony algorithm">Artificial bee colony algorithm</a>.</div>

In <a href="Computer_science" title="Computer science">computer science</a> and <a href="Operations_research" title="Operations research">operations research</a>, the bees algorithm is a population-based <a href="Search_algorithm" title="Search algorithm">search algorithm</a> which was developed by Pham, Ghanbarzadeh et al. in 2005.<a href="#cite_note-Pham_&_al,_2005-1">[1]</a> It mimics the food foraging behaviour of honey bee colonies. In its basic version the algorithm performs a kind of neighbourhood search combined with global search, and can be used for both <a href="Combinatorial_optimization" title="Combinatorial optimization">combinatorial optimization</a> and <a href="Continuous_optimization" title="Continuous optimization">continuous optimization</a>. The only condition for the application of the bees algorithm is that some measure of distance between the solutions is defined. The effectiveness and specific abilities of the bees algorithm have been proven in a number of studies.<a href="#cite_note-Pham_&_Castellani,_2009-2">[2]</a><a href="#cite_note-3">[3]</a><a href="#cite_note-4">[4]</a><a href="#cite_note-Nasrinpour_&_Massah_&_Teshnehlab_2017-5">[5]</a><a href="#cite_note-6">[6]</a>

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

A colony of <a href="Honey_bees" class="mw-redirect" title="Honey bees">honey bees</a> can extend itself over long distances (over 14 km)<a href="#cite_note-Tereshko_&_Loengarov,_2005-7">[7]</a> and in multiple directions simultaneously to harvest nectar or pollen from multiple food sources (flower patches).

A small fraction of the colony constantly searches the environment looking for new flower patches. These scout bees move randomly in the area surrounding the hive, evaluating the profitability (net energy yield) of the food sources encountered.<a href="#cite_note-Tereshko_&_Loengarov,_2005-7">[7]</a> When they return to the hive, the scouts deposit the food harvested. Those individuals that found a highly profitable food source go to an area in the hive called the “dance floor”, and perform a ritual known as the <a href="Waggle_dance" title="Waggle dance">waggle dance</a>.<a href="#cite_note-8">[8]</a>

Through the waggle dance a scout bee communicates the location of its discovery to idle onlookers, which join in the exploitation of the flower patch. Since the length of the dance is proportional to the scout’s rating of the food source, more foragers get recruited to harvest the best rated flower patches. After dancing, the scout returns to the food source it discovered to collect more food.

As long as they are evaluated as profitable, rich food sources will be advertised by the scouts when they return to the hive. Recruited foragers may waggle dance as well, increasing the recruitment for highly rewarding flower patches. Thanks to this autocatalytic process, the bee colony is able to quickly switch the focus of the foraging effort on the most profitable flower patches.<a href="#cite_note-Tereshko_&_Loengarov,_2005-7">[7]</a>

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

The bees algorithm<a href="#cite_note-Pham_&_Castellani,_2009-2">[2]</a><a href="#cite_note-Pham_&_Ghanbarzadeh_et._a._2006-9">[9]</a> mimics the foraging strategy of honey bees to look for the best solution to an optimisation problem. Each candidate solution is thought of as a food source (flower), and a population (colony) of n agents (bees) is used to search the solution space. Each time an artificial bee visits a flower (lands on a solution), it evaluates its profitability (fitness).

The bees algorithm consists of an initialisation procedure and a main search cycle which is iterated for a given number T of times, or until a solution of acceptable fitness is found. Each search cycle is composed of five procedures: recruitment, local search, neighbourhood shrinking, site abandonment, and global search.

<pre>Pseudocode for the standard bees algorithm<a href="#cite_note-Pham_&_Castellani,_2009-2">[2]</a>

1 for i = 1, ..., ns

i scout[i] = Initialise_scout()

ii flower_patch[i] = Initialise_flower_patch(scout[i])

2 do until stopping_condition = TRUE

i Recruitment()

ii for i = 1, ..., na

1 flower_patch[i] = Local_search(flower_patch[i])

2 flower_patch[i] = Site_abandonment(flower_patch[i])

3 flower_patch[i] = Neighbourhood_shrinking(flower_patch[i])

iii for i = nb, ..., ns

1 flower_patch[i] = Global_search(flower_patch[i])}

</pre>

In the initialisation routine ns scout bees are randomly placed in the search space, and evaluate the fitness of the solutions where they land. For each solution, a neighbourhood (called flower patch) is delimited.

In the recruitment procedure, the scouts that visited the nb≤ns fittest solutions (best sites) perform the waggle dance. That is, they recruit foragers to search further the neighbourhoods of the most promising solutions. The scouts that located the very best ne≤nb solutions (elite sites) recruit nre foragers each, whilst the remaining nb-ne scouts recruit nrb≤nre foragers each. Thus, the number of foragers recruited depends on the profitability of the food source.

In the local search procedure, the recruited foragers are randomly scattered within the flower patches enclosing the solutions visited by the scouts (local exploitation). If any of the foragers in a flower patch lands on a solution of higher fitness than the solution visited by the scout, that forager becomes the new scout. If no forager finds a solution of higher fitness, the size of the flower patch is shrunk (neighbourhood shrinking procedure). Usually, flower patches are initially defined over a large area, and their size is gradually shrunk by the neighbourhood shrinking procedure. As a result, the scope of the local exploration is progressively focused on the area immediately close to the local fitness best. If no improvement in fitness is recorded in a given flower patch for a pre-set number of search cycles, the local maximum of fitness is considered found, the patch is abandoned (site abandonment), and a new scout is randomly generated.

As in biological bee colonies,<a href="#cite_note-Tereshko_&_Loengarov,_2005-7">[7]</a> a small number of scouts keeps exploring the solution space looking for new regions of high fitness (global search). The global search procedure re-initialises the last ns-nb flower patches with randomly generated solutions.

At the end of one search cycle, the scout population is again composed of ns scouts: nr scouts produced by the local search procedure (some of which may have been re-initialised by the site abandonment procedure), and ns-nb scouts generated by the global search procedure. The total artificial bee colony size is n=ne•nre+(nb-ne)•nrb+ns (elite sites foragers + remaining best sites foragers + scouts) bees.

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

In addition to the basic bees algorithm,<a href="#cite_note-Pham_&_Ghanbarzadeh_et._a._2006-9">[9]</a> there are a number of improved or hybrid versions of the BA, each of which focuses on some shortcomings of the basic BA. These variants include (but are not limited to) fuzzy or enhanced BA (EBA),<a href="#cite_note-Pham_&_Darwish_008-10">[10]</a> grouped BA (GBA),<a href="#cite_note-Nasrinpour_&_Massah_&_Teshnehlab_2017-5">[5]</a> hybrid modified BA (MBA)<a href="#cite_note-Pham_&_Pham_&_Castellani_2011-11">[11]</a> and so on.

The pseudo-code for the grouped BA (GBA) <a href="#cite_note-Nasrinpour_&_Massah_&_Teshnehlab_2017-5">[5]</a> is as follows.

<div class="mw-highlight mw-highlight-lang-matlab mw-content-ltr" dir="ltr"><pre>function GBA

%% Set the problem parameters

maxIteration = ..; % number of iterations (e.g. 1000-5000)

maxParameters = ..; % number of input variables

min = [..] ; % an array of the size maxParameters to indicate the minimum value of each input parameter

max = [..] ; % an array of the size maxParameters to indicate the maximum value of each input parameter

%% Set the grouped bees algorithm (GBA) parameters

R_ngh = ..; % patch radius of the neighborhood search for bees in the first group (e.g. 0.001 - 1)

n = ..; % number of scout bees (e.g. 4-30)

nGroups = ..; % number of groups, excluding the random group

%% GBA's automatic parameter settings

k = 3 * n / ((nGroups+1)^3 - 1); % GBA's parameter to set the number of scout bees in each group

groups = zeros(1,nGroups); % An array to keep the number of scout bees for each group

recruited_bees = zeros(1,nGroups); % An array to keep the number of recruited bees for each group

a = (((max - min) ./ 2) - R_ngh) ./ (nGroups^2 - 1); % GBA's parameter for setting neighborhood radiuses

b = R_ngh - a; % GBA's parameter for setting neighborhood radiuses

for i=1:nGroups % For each group

groups(i) = floor(k*i^2); % determine the number of scout bees in each group

if groups(i) == 0

groups(i) = 1; % there has to be at least one scout bee per each group

end

recruited_bees = (nGroups+1-i)^2; % set the number of recruited bees for each group

ngh(i) = a * i*i + b; % set the radius patch for each group

end

group_random = n - sum(groups); % assign the remainder bees (if any) to random search

group_random = max(group_random,0); % make sure it is not a negative number

%% initialize the population matrix

population = zeros(n,maxParameters+1); % A population of n bees including all input variables and their fitness

for i=1:n

population(i,1:maxParameters)= generate_random_solution(maxParameters,min, max); % random initialization of maxParameters variables between max and min

population(i,maxParameters+1) = evalulate_fitness(population(i,:)); % fitness evaluation of each solution and saving it at the last index of the population matrix

end

sorted_population = sortrows(population); % sort the population based on their fitnesses

%% Iterations of the grouped bees algorithm

for i=1:maxIteration % GBA's main loop

beeIndex = 0; % keep track of all bees (i.e, patches)

for g=1:nGroups % for each group of scout bees

for j = 1 : groups(g) % exploit each patch within each group

beeIndex = beeIndex + 1; % increase the counter per each patch

for i = 1 : recruited_bees(g) % for each recruited bees of the group

solution = bee_waggle_dance(sorted_population(beeIndex,1:maxParameters),ngh(g)); % search the neighborhood around selected patch/solution within the radius of ngh

fit = evaluate_fitness(solution); % evaluate the fitness of recently found solution

if fit < sorted_population(beeIndex,maxParameters+1) % A minimization problem: if a better location/patch/solution is found by the recuiter bee

sorted_population(beeIndex,1 : maxParameters+1) = [solution(1 : maxParameters),fit]; % copy new solution and its fitness to the sorted population matrix

end

for i= 1 : group_random % For the remaining random bees

solution(beeIndex,1:maxParameters)= generate_random_solution(maxParameters,min, max); % generate a new random solution at the index beeIndex

solution(beeIndex,maxParameters+1)= evaluate_fitness(solution); % evaluate its fitness

sorted_population(beeIndex,:) = [solution(1 : maxParameters),fit]; % copy the new random solution and its fitness to the sorted population matrix

end

sorted_population=sortrows(sorted_population); % sort the population based on their fitnesses

Best_solution_sofar=sorted_population(1,:);

disp('Best:');disp(Best_solution_sofar); % Display the best solution of current iteration

end % end of GBA's main loop

end % end of main function

%% Function Bee Waggle Dance

function new_solution=bee_waggle_dance(solution, ngh, maxParameters)

new_solution(1:maxParameters) = (solution-ngh)+(2*ngh.*rand(1, maxParameters));

end

</pre></div>

<ul><li><a href="Ant_colony_optimization_algorithms" title="Ant colony optimization algorithms">Ant colony optimization algorithms</a></li>

<li><a href="Artificial_bee_colony_algorithm" title="Artificial bee colony algorithm">Artificial bee colony algorithm</a></li>

<li><a href="Evolutionary_computation" title="Evolutionary computation">Evolutionary computation</a></li>

<li><a href="L%C3%A9vy_flight_foraging_hypothesis" title="Lévy flight foraging hypothesis">Lévy flight foraging hypothesis</a></li>

<li><a href="Manufacturing_Engineering_Centre" title="Manufacturing Engineering Centre">Manufacturing Engineering Centre</a></li>

<li><a href="Mathematical_optimization" title="Mathematical optimization">Mathematical optimization</a></li>

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

<li><a href="Particle_swarm_optimization" title="Particle swarm optimization">Particle swarm optimization</a></li>

<li><a href="Swarm_intelligence" title="Swarm intelligence">Swarm intelligence</a></li></ul>

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

/* start https://en.wikipedia.org/ */

.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}

/* end https://en.wikipedia.org/ */

</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;">

<li id="cite_note-Pham_&_al,_2005-1"><a href="#cite_ref-Pham_&_al,_2005_1-0">^</a> Pham DT, Ghanbarzadeh A, Koc E, Otri S, Rahim S and Zaidi M. The Bees Algorithm. Technical Note, Manufacturing Engineering Centre, Cardiff University, UK, 2005.

</li>

<li id="cite_note-Pham_&_Castellani,_2009-2">^ <a href="#cite_ref-Pham_&_Castellani,_2009_2-0">a</a> <a href="#cite_ref-Pham_&_Castellani,_2009_2-1">b</a> <a href="#cite_ref-Pham_&_Castellani,_2009_2-2">c</a> Pham, D.T., Castellani, M. (2009), <a rel="nofollow" class="external text" href="http://pic.sagepub.com/content/223/12/2919.short">The Bees Algorithm – Modelling Foraging Behaviour to Solve Continuous Optimisation Problems</a>. Proc. ImechE, Part C, 223(12), 2919-2938.

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> Pham, D.T. and Castellani, M. (2013), <a rel="nofollow" class="external text" href="https://link.springer.com/article/10.1007/s00500-013-1104-9">Benchmarking and Comparison of Nature-Inspired Population-Based Continuous Optimisation Algorithms</a>, Soft Computing, 1-33.

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> Pham, D.T. and Castellani, M. (2015), <a rel="nofollow" class="external text" href="http://www.tandfonline.com/doi/abs/10.1080/23311916.2015.1091540">A comparative study of the bees algorithm as a tool for function optimisation</a>, Cogent Engineering 2(1), 1091540.

</li>

<li id="cite_note-Nasrinpour_&_Massah_&_Teshnehlab_2017-5">^ <a href="#cite_ref-Nasrinpour_&_Massah_&_Teshnehlab_2017_5-0">a</a> <a href="#cite_ref-Nasrinpour_&_Massah_&_Teshnehlab_2017_5-1">b</a> <a href="#cite_ref-Nasrinpour_&_Massah_&_Teshnehlab_2017_5-2">c</a> Nasrinpour, H. R., Massah Bavani, A., Teshnehlab, M., (2017), <a rel="nofollow" class="external text" href="http://www.mdpi.com/2073-431X/6/1/5">Grouped Bees Algorithm: A Grouped Version of the Bees Algorithm</a>, Computers 2017, 6(1), 5; (<style data-mw-deduplicate="TemplateStyles:r1238218222">

/* start https://en.wikipedia.org/ */

.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

/* end https://en.wikipedia.org/ */

</style><a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.3390%2Fcomputers6010005">10.3390/computers6010005</a>)

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> Baronti, Luca & Castellani, Marco & Pham, D.. (2020),<a rel="nofollow" class="external text" href="http://pure-oai.bham.ac.uk/ws/files/100752622/An_Analysis_of_the_Search_Mechanisms_of_the_Bees_Algorithm.pdf.">An Analysis of the Search Mechanisms of the Bees Algorithm.</a>, Swarm and Evolutionary Computation. 59. 100746. 10.1016/j.swevo.2020.100746

</li>

<li id="cite_note-Tereshko_&_Loengarov,_2005-7">^ <a href="#cite_ref-Tereshko_&_Loengarov,_2005_7-0">a</a> <a href="#cite_ref-Tereshko_&_Loengarov,_2005_7-1">b</a> <a href="#cite_ref-Tereshko_&_Loengarov,_2005_7-2">c</a> <a href="#cite_ref-Tereshko_&_Loengarov,_2005_7-3">d</a> Tereshko V., Loengarov A., (2005) <a rel="nofollow" class="external text" href="http://cis.uws.ac.uk/research/journal/V9/V9N3/bees.pdf">Collective Decision-Making in Honey Bee Foraging Dynamics</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20140201175130/http://cis.uws.ac.uk/research/journal/V9/V9N3/bees.pdf">Archived</a> 2014-02-01 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. Journal of Computing and Information Systems, 9(3), 1-7.

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> Von Frisch, K. (1967) The Dance Language and Orientation of Bees. Harvard University Press, Cambridge, Massachusetts.

</li>

<li id="cite_note-Pham_&_Ghanbarzadeh_et._a._2006-9">^ <a href="#cite_ref-Pham_&_Ghanbarzadeh_et._a._2006_9-0">a</a> <a href="#cite_ref-Pham_&_Ghanbarzadeh_et._a._2006_9-1">b</a> Pham D.T., Ghanbarzadeh A., Koc E., Otri S., Rahim S., Zaidi M., <a rel="nofollow" class="external text" href="https://web.archive.org/web/20231226051053/https://www.academia.edu/8358231/The_Bees_Algorithm_A_Novel_Tool_for_Complex_Optimisation_Problems">The Bees Algorithm, A Novel Tool for Complex Optimisation Problems</a>, Proc 2nd Int Virtual Conf on Intelligent Production Machines and Systems (IPROMS 2006), Oxford: Elsevier, pp. 454-459, 2006.

</li>

<li id="cite_note-Pham_&_Darwish_008-10"><a href="#cite_ref-Pham_&_Darwish_008_10-0">^</a> Pham D. T., Haj Darwish A., (2008), A. <a rel="nofollow" class="external text" href="https://www.researchgate.net/profile/Ahmed_Haj_Darwish/publication/321849894_Fuzzy_Selection_of_Local_Search_Sites_in_the_Bees_Algorithm/links/5a37c834aca272a6ec1e511a/Fuzzy-Selection-of-Local-Search-Sites-in-the-Bees-Algorithm.pdf">Fuzzy Selection of Local Search Sites in the Bees Algorithm</a>. Proceedings of Innovative Production Machines and Systems (IPROMS 2008)

</li>

<li id="cite_note-Pham_&_Pham_&_Castellani_2011-11"><a href="#cite_ref-Pham_&_Pham_&_Castellani_2011_11-0">^</a> Pham Q. T., Pham D. T., Castellani M., A modified Bees Algorithm and a statistics-based method for tuning its parameters. Proceedings of the Institution of Mechanical Engineers (ImechE), Part I: Journal of Systems and Control Eng., 2011 (<a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1177%2F0959651811422759">10.1177/0959651811422759</a>)

</li>

</ol></div>

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

<ul><li><a rel="nofollow" class="external text" href="http://beesalgorithmsite.altervista.org/">The bees algorithm website</a></li>

<li><a rel="nofollow" class="external text" href="http://news.bbc.co.uk/1/hi/wales/south_east/5285106.stm">Boffins put dancing bees to work – BBC News</a></li>

<li><a rel="nofollow" class="external text" href="https://sites.google.com/view/baaworkshop/home">The bees algorithm workshop</a></li></ul>

/* start https://en.wikipedia.org/ */

.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}

/* end https://en.wikipedia.org/ */

</style><style data-mw-deduplicate="TemplateStyles:r1236075235">

/* start https://en.wikipedia.org/ */

.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}

/* end https://en.wikipedia.org/ */

</style></div><div role="navigation" class="navbox" aria-labelledby="Swarming203" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><style data-mw-deduplicate="TemplateStyles:r1239400231">

/* start https://en.wikipedia.org/ */

.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}

/* end https://en.wikipedia.org/ */

</style><div id="Swarming203" style="font-size:114%;margin:0 4em"><a href="Swarm_behaviour" title="Swarm behaviour">Swarming</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Swarm_behaviour#Biological_swarming" title="Swarm behaviour">Biological swarming</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Agent-based_model_in_biology" title="Agent-based model in biology">Agent-based model in biology</a></li>

<li><a href="Collective_animal_behavior" title="Collective animal behavior">Collective animal behavior</a></li>

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

<li><a href="Flock_(birds)" title="Flock (birds)">Flock</a>

<ul><li><a href="Flocking_(behavior)" class="mw-redirect" title="Flocking (behavior)">flocking</a></li>

<ul><li><a href="Herd_behavior" title="Herd behavior">herd behavior</a></li></ul></li>

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

<li><a href="Mixed-species_foraging_flock" title="Mixed-species foraging flock">Mixed-species foraging flock</a></li>

<li><a href="Mobbing_(animal_behavior)" title="Mobbing (animal behavior)">Mobbing behavior</a>

<ul><li><a href="Feeding_frenzy" title="Feeding frenzy">feeding frenzy</a></li></ul></li>

<ul><li><a href="Pack_hunter" title="Pack hunter">pack hunter</a></li></ul></li>

<li><a href="Patterns_of_self-organization_in_ants" title="Patterns of self-organization in ants">Patterns of self-organization in ants</a>

<li><a href="Symmetry_breaking_of_escaping_ants" title="Symmetry breaking of escaping ants">symmetry breaking of escaping ants</a></li></ul></li>

<li><a href="Shoaling_and_schooling" title="Shoaling and schooling">Shoaling and schooling</a>

<li><a href="Swarm_behaviour" title="Swarm behaviour">Swarming behaviour</a></li>

<li><a href="Swarming_(honey_bee)" title="Swarming (honey bee)">Swarming (honey bee)</a></li>

<li><a href="Swarming_motility" title="Swarming motility">Swarming motility</a></li></ul>

</div></td><td class="noviewer navbox-image" rowspan="6" style="width:1px;padding:0 0 0 2px"><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Animal_migration" title="Animal migration">Animal migration</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Animal_migration" title="Animal migration">Animal migration</a>

<ul><li><a href="Altitudinal_migration" title="Altitudinal migration">altitudinal</a></li>

<li><a href="Animal_migration_tracking" title="Animal migration tracking">tracking</a>

<ul><li><a href="History_of_wildlife_tracking_technology" title="History of wildlife tracking technology">history</a></li></ul></li>

<li><a href="Coded_wire_tag" title="Coded wire tag">coded wire tag</a></li></ul></li>

<li><a href="Bird_migration" title="Bird migration">Bird migration</a>

<ul><li>flyways</li>

<li><a href="Reverse_migration_(birds)" title="Reverse migration (birds)">reverse migration</a></li></ul></li>

<li><a href="Cell_migration" title="Cell migration">Cell migration</a></li>

<li><a href="Fish_migration" title="Fish migration">Fish migration</a>

<ul><li><a href="Diel_vertical_migration" title="Diel vertical migration">diel vertical</a></li>

<li><a href="Lessepsian_migration" title="Lessepsian migration">Lessepsian</a></li>

<li><a href="Salmon_run" title="Salmon run">salmon run</a></li>

<li><a href="Sardine_run" title="Sardine run">sardine run</a></li></ul></li>

<li><a href="Homing_(biology)" title="Homing (biology)">Homing</a>

<ul><li><a href="Natal_homing" title="Natal homing">natal</a></li>

<li><a href="Philopatry" title="Philopatry">philopatry</a></li></ul></li>

<li><a href="Insect_migration" title="Insect migration">Insect migration</a>

<ul><li><a href="Lepidoptera_migration" title="Lepidoptera migration">butterflies</a>

<ul><li><a href="Monarch_butterfly_migration" title="Monarch butterfly migration">monarch</a></li></ul></li></ul></li>

<li><a href="Sea_turtle_migration" title="Sea turtle migration">Sea turtle migration</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Swarm_behaviour#Models" title="Swarm behaviour">Swarm algorithms</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Agent-based_model" title="Agent-based model">Agent-based models</a></li>

<li><a href="Ant_colony_optimization_algorithm" class="mw-redirect" title="Ant colony optimization algorithm">Ant colony optimization</a></li>

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

<li><a href="Crowd_simulation" title="Crowd simulation">Crowd simulation</a></li>

<li><a href="Particle_swarm_optimization" title="Particle swarm optimization">Particle swarm optimization</a></li>

<li><a href="Swarm_intelligence" title="Swarm intelligence">Swarm intelligence</a></li>

<li><a href="Swarm_(simulation)" title="Swarm (simulation)">Swarm (simulation)</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Collective_motion" title="Collective motion">Collective motion</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Active_matter" title="Active matter">Active matter</a></li>

<li><a href="Collective_motion" title="Collective motion">Collective motion</a></li>

<li><a href="Self-propelled_particles" title="Self-propelled particles">Self-propelled particles</a>

<ul><li><a href="Clustering_of_self-propelled_particles" title="Clustering of self-propelled particles">clustering</a></li></ul></li>

<li><a href="Vicsek_model" title="Vicsek model">Vicsek model</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Swarm_robotics" title="Swarm robotics">Swarm robotics</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Ant_robotics" title="Ant robotics">Ant robotics</a></li>

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

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

<li><a href="Swarm_robotics" title="Swarm robotics">Swarm robotics</a></li>

<li><a href="Symbrion" title="Symbrion">Symbrion</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related topics</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Allee_effect" title="Allee effect">Allee effect</a></li>

<li><a href="Animal_navigation" title="Animal navigation">Animal navigation</a></li>

<li><a href="Collective_intelligence" title="Collective intelligence">Collective intelligence</a></li>

<li><a href="Decentralised_system" title="Decentralised system">Decentralised system</a></li>

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

<li><a href="Group_size_measures" title="Group size measures">Group size measures</a></li>

<li><a href="Microbial_intelligence" title="Microbial intelligence">Microbial intelligence</a></li>

<li><a href="Mutualism_(biology)" title="Mutualism (biology)">Mutualism</a></li>

<li><a href="Predator_satiation" title="Predator satiation">Predator satiation</a></li>

<li><a href="Quorum_sensing" title="Quorum sensing">Quorum sensing</a></li>

<li><a href="Spatial_organization" title="Spatial organization">Spatial organization</a></li>

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

<li><a href="Swarming_(military)" title="Swarming (military)">Military swarming</a></li>

<li><a href="Task_allocation_and_partitioning_of_social_insects" class="mw-redirect" title="Task allocation and partitioning of social insects">Task allocation and partitioning of social insects</a></li></ul>

</div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Optimization:_Algorithms,_methods,_and_heuristics381" style="padding:3px"><table class="nowraplinks hlist mw-collapsible uncollapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><div id="Optimization:_Algorithms,_methods,_and_heuristics381" style="font-size:114%;margin:0 4em"><a href="Mathematical_optimization" title="Mathematical optimization">Optimization</a>: <a href="Optimization_algorithm" class="mw-redirect" title="Optimization algorithm">Algorithms</a>, <a href="Iterative_method" title="Iterative method">methods</a>, and <a href="Heuristic_algorithm" class="mw-redirect" title="Heuristic algorithm">heuristics</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Unconstrained_nonlinear381" style="font-size:114%;margin:0 4em"><a href="Nonlinear_programming" title="Nonlinear programming">Unconstrained nonlinear</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Function_(mathematics)" title="Function (mathematics)">Functions</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Golden-section_search" title="Golden-section search">Golden-section search</a></li>

<li><a href="Powell's_method" title="Powell's method">Powell's method</a></li>

<li><a href="Line_search" title="Line search">Line search</a></li>

<li><a href="Nelder%E2%80%93Mead_method" title="Nelder–Mead method">Nelder–Mead method</a></li>

<li><a href="Successive_parabolic_interpolation" title="Successive parabolic interpolation">Successive parabolic interpolation</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Gradient" title="Gradient">Gradients</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Local_convergence" title="Local convergence">Convergence</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Trust_region" title="Trust region">Trust region</a></li>

<li><a href="Wolfe_conditions" title="Wolfe conditions">Wolfe conditions</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Quasi-Newton_method" title="Quasi-Newton method">Quasi–Newton</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Berndt%E2%80%93Hall%E2%80%93Hall%E2%80%93Hausman_algorithm" title="Berndt–Hall–Hall–Hausman algorithm">Berndt–Hall–Hall–Hausman</a></li>

<li><a href="Broyden%E2%80%93Fletcher%E2%80%93Goldfarb%E2%80%93Shanno_algorithm" title="Broyden–Fletcher–Goldfarb–Shanno algorithm">Broyden–Fletcher–Goldfarb–Shanno</a> and <a href="Limited-memory_BFGS" title="Limited-memory BFGS">L-BFGS</a></li>

<li><a href="Davidon%E2%80%93Fletcher%E2%80%93Powell_formula" title="Davidon–Fletcher–Powell formula">Davidon–Fletcher–Powell</a></li>

<li><a href="Symmetric_rank-one" title="Symmetric rank-one">Symmetric rank-one (SR1)</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Iterative_method" title="Iterative method">Other methods</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Nonlinear_conjugate_gradient_method" title="Nonlinear conjugate gradient method">Conjugate gradient</a></li>

<li><a href="Gauss%E2%80%93Newton_algorithm" title="Gauss–Newton algorithm">Gauss–Newton</a></li>

<li><a href="Gradient_descent" title="Gradient descent">Gradient</a></li>

<li><a href="Mirror_descent" title="Mirror descent">Mirror</a></li>

<li><a href="Levenberg%E2%80%93Marquardt_algorithm" title="Levenberg–Marquardt algorithm">Levenberg–Marquardt</a></li>

<li><a href="Powell's_dog_leg_method" title="Powell's dog leg method">Powell's dog leg method</a></li>

<li><a href="Truncated_Newton_method" title="Truncated Newton method">Truncated Newton</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Hessian_matrix" title="Hessian matrix">Hessians</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Newton's_method_in_optimization" title="Newton's method in optimization">Newton's method</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td><td class="noviewer navbox-image" rowspan="5" style="width:1px;padding:0 0 0 2px"><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Constrained_nonlinear381" style="font-size:114%;margin:0 4em"><a href="Nonlinear_programming" title="Nonlinear programming">Constrained nonlinear</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">General</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Barrier_function" title="Barrier function">Barrier methods</a></li>

<li><a href="Penalty_method" title="Penalty method">Penalty methods</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Differentiable</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Augmented_Lagrangian_method" title="Augmented Lagrangian method">Augmented Lagrangian methods</a></li>

<li><a href="Sequential_quadratic_programming" title="Sequential quadratic programming">Sequential quadratic programming</a></li>

<li><a href="Successive_linear_programming" title="Successive linear programming">Successive linear programming</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Convex_optimization381" style="font-size:114%;margin:0 4em"><a href="Convex_optimization" title="Convex optimization">Convex optimization</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Convex_minimization" class="mw-redirect" title="Convex minimization">Convex minimization</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cutting-plane_method" title="Cutting-plane method">Cutting-plane method</a></li>

<li><a href="Frank%E2%80%93Wolfe_algorithm" title="Frank–Wolfe algorithm">Reduced gradient (Frank–Wolfe)</a></li>

<li><a href="Subgradient_method" title="Subgradient method">Subgradient method</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Linear_programming" title="Linear programming">Linear</a> and <a href="Quadratic_programming" title="Quadratic programming">quadratic</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Linear_programming#Interior_point" title="Linear programming">Interior point</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Affine_scaling" title="Affine scaling">Affine scaling</a></li>

<li><a href="Ellipsoid_method" title="Ellipsoid method">Ellipsoid algorithm of Khachiyan</a></li>

<li><a href="Karmarkar's_algorithm" title="Karmarkar's algorithm">Projective algorithm of Karmarkar</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Matroid" title="Matroid">Basis-</a><a href="Exchange_algorithm" class="mw-redirect" title="Exchange algorithm">exchange</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Simplex_algorithm" title="Simplex algorithm">Simplex algorithm of Dantzig</a></li>

<li><a href="Revised_simplex_method" title="Revised simplex method">Revised simplex algorithm</a></li>

<li><a href="Criss-cross_algorithm" title="Criss-cross algorithm">Criss-cross algorithm</a></li>

<li><a href="Lemke's_algorithm" title="Lemke's algorithm">Principal pivoting algorithm of Lemke</a></li>

<li><a href="Active-set_method" title="Active-set method">Active-set method</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Combinatorial381" style="font-size:114%;margin:0 4em"><a href="Combinatorial_optimization" title="Combinatorial optimization">Combinatorial</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Paradigms</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Approximation_algorithm" title="Approximation algorithm">Approximation algorithm</a></li>

<li><a href="Dynamic_programming" title="Dynamic programming">Dynamic programming</a></li>

<li><a href="Greedy_algorithm" title="Greedy algorithm">Greedy algorithm</a></li>

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

<ul><li><a href="Branch_and_bound" title="Branch and bound">Branch and bound</a>/<a href="Branch_and_cut" title="Branch and cut">cut</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Graph_algorithm" class="mw-redirect" title="Graph algorithm">Graph algorithms</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Minimum_spanning_tree52" scope="row" class="navbox-group" style="width:1%"><a href="Minimum_spanning_tree" title="Minimum spanning tree">Minimum spanning tree</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bor%C5%AFvka's_algorithm" title="Borůvka's algorithm">Borůvka</a></li>

<li><a href="Kruskal's_algorithm" title="Kruskal's algorithm">Kruskal</a></li></ul>

</div></td></tr></tbody></table><div>

</div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Shortest_path39" scope="row" class="navbox-group" style="width:1%"><a href="Shortest_path_problem" title="Shortest path problem">Shortest path</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bellman%E2%80%93Ford_algorithm" title="Bellman–Ford algorithm">Bellman–Ford</a>

<li><a href="Dijkstra's_algorithm" title="Dijkstra's algorithm">Dijkstra</a></li>

<li><a href="Floyd%E2%80%93Warshall_algorithm" title="Floyd–Warshall algorithm">Floyd–Warshall</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Flow_network" title="Flow network">Network flows</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Dinic's_algorithm" title="Dinic's algorithm">Dinic</a></li>

<li><a href="Edmonds%E2%80%93Karp_algorithm" title="Edmonds–Karp algorithm">Edmonds–Karp</a></li>

<li><a href="Ford%E2%80%93Fulkerson_algorithm" title="Ford–Fulkerson algorithm">Ford–Fulkerson</a></li>

<li><a href="Push%E2%80%93relabel_maximum_flow_algorithm" title="Push–relabel maximum flow algorithm">Push–relabel maximum flow</a></li></ul>

<ul><li><a href="Evolutionary_algorithm" title="Evolutionary algorithm">Evolutionary algorithm</a></li>

<li><a href="Hill_climbing" title="Hill climbing">Hill climbing</a></li>

<li><a href="Local_search_(optimization)" title="Local search (optimization)">Local search</a></li>

<li><a href="Parallel_metaheuristic" title="Parallel metaheuristic">Parallel metaheuristics</a></li>

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

<li><a href="Spiral_optimization_algorithm" title="Spiral optimization algorithm">Spiral optimization algorithm</a></li>

<li><a href="Tabu_search" title="Tabu search">Tabu search</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div>

<ul><li><a href="Comparison_of_optimization_software" title="Comparison of optimization software">Software</a></li></ul>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-06-02" href="https://en.wikipedia.org/wiki/?title=Bees_algorithm&oldid=1293484801">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

</div>

</div></div>

</div>

</main>

</div>

</body></html>