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Erel Segal (talk | contribs) |
m Removed phrase about "nearly as straightforward," as it is an opinion not really suitable for Wikipedia. It should probably be replaced with citations to successful applications of convex optimization that would demonstrate its straightforwardness or widespread use (if it is widespread in use, which I doubt). |
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{{short description|Subfield of mathematical optimization}}
'''Convex optimization''' is a subfield of [[mathematical optimization]] that studies the problem of minimizing [[convex function]]s over [[convex set]]s (or, equivalently, maximizing [[concave functions]] over convex sets). Many classes of convex optimization problems admit polynomial-time algorithms,<ref name="Nesterov 1994">{{harvnb|Nesterov|Nemirovskii|1994}}</ref> whereas mathematical optimization is in general [[NP-hard]].<ref>
{{cite journal | last1 = Murty | first1 = Katta | last2 = Kabadi | first2 = Santosh | title = Some NP-complete problems in quadratic and nonlinear programming | journal = Mathematical Programming | volume = 39 | issue = 2 | pages = 117–129 | year = 1987 | doi = 10.1007/BF02592948| hdl = 2027.42/6740 | s2cid = 30500771 | hdl-access = free}}</ref><ref>Sahni, S. "Computationally related problems," in SIAM Journal on Computing, 3, 262--279, 1974.</ref><ref>[https://link.springer.com/article/10.1007/BF00120662 Quadratic programming with one negative eigenvalue is NP-hard], Panos M. Pardalos and Stephen A. Vavasis in ''Journal of Global Optimization'', Volume 1, Number 1, 1991, pg.15-22.</ref>
==Definition==
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The goal of the problem is to find some <math>\mathbf{x^\ast} \in C</math> attaining
:<math>\inf \{ f(\mathbf{x}) : \mathbf{x} \in C \}</math>.
In general, there are three options regarding the existence of a solution:<ref name=":2">{{Cite book |last1=Boyd |first1=Stephen |url=https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf |title=Convex Optimization |last2=Vandenberghe |first2=Lieven |publisher=[[Cambridge University Press]] |year=2004 |isbn=978-0-521-83378-3 |access-date=12 Apr 2021}}</ref>{{Rp|___location=chpt.4}}
* If such a point ''x''* exists, it is referred to as an ''optimal point'' or ''solution''; the set of all optimal points is called the ''optimal set''; and the problem is called ''solvable''.
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