Editing Mathematical optimization (section)

{{short description|Study of mathematical algorithms for optimization problems}}
{{redirect|Mathematical programming|the peer-reviewed journal|Mathematical Programming}}
{{redirect-multi|2|Optimization|Optimum}}
[[File:Max paraboloid.svg|right|thumb|Graph of a
surface given by ''z'' = f(''x'', ''y'') = −(''x''² + ''y''²) + 4. The global [[maximum (mathematics)|maximum]] at (''x, y, z'') = (0, 0, 4) is indicated by a blue dot.]]

[[File:Nelder-Mead Simionescu.gif|thumb|Nelder-Mead minimum search of [[Test functions for optimization|Simionescu's function]]. Simplex vertices are ordered by their values, with 1 having the lowest (<math>f(x)</math> best) value.|alt=]]

'''Mathematical optimization''' (alternatively spelled ''optimisation'') or '''mathematical programming''' is the selection of a best element, with regard to some criteria, from some set of available alternatives.<ref>"[http://glossary.computing.society.informs.org/index.php?page=nature.html The Nature of Mathematical Programming] {{webarchive|url=https://web.archive.org/web/20140305080324/http://glossary.computing.society.informs.org/index.php?page=nature.html |date=2014-03-05 }}," ''Mathematical Programming Glossary'', INFORMS Computing Society.</ref><ref>{{Cite web |title=Mathematical Programming: An Overview |url=https://web.mit.edu/15.053/www/AppliedMathematicalProgramming.pdf |access-date=26 April 2024}}</ref> It is generally divided into two subfields: [[discrete optimization]] and [[continuous optimization]]. Optimization problems arise in all quantitative disciplines from [[computer science]] and [[engineering]]<ref name="edo2021">{{Cite book|url=https://www.researchgate.net/publication/352413464|title=Engineering Design Optimization|last1=Martins|first1=Joaquim R. R. A.|last2=Ning|first2=Andrew|date=2021-10-01|publisher=Cambridge University Press|isbn=978-1108833417|language=en}}</ref> to [[operations research]] and [[economics]], and the development of solution methods has been of interest in [[mathematics]] for centuries.<ref>{{cite book |last1=Du |first1=D. Z. |last2=Pardalos |first2=P. M. |last3=Wu |first3=W. |year=2008 |chapter=History of Optimization |editor-link=Christodoulos Floudas |editor-last=Floudas |editor-first=C. |editor2-last=Pardalos |editor2-first=P. |title=Encyclopedia of Optimization |publisher=Springer |location=Boston |pages=1538–1542 }}</ref>

In the more general approach, an [[optimization problem]] consists of [[maxima and minima|maximizing or minimizing]] a [[Function of a real variable|real function]] by systematically choosing [[Argument of a function|input]] values from within an allowed set and computing the [[Value (mathematics)|value]] of the function. The generalization of optimization theory and techniques to other formulations constitutes a large area of [[applied mathematics]].