Convex optimization: Difference between revisions

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{{cite journal |last1=Agrawal |first1=Akshay |last2=Verschueren |first2=Robin |last3=Diamond |first3=Steven |last4=Boyd |first4=Stephen |year=2018 |title=A rewriting system for convex optimization problems |url=https://web.stanford.edu/~boyd/papers/pdf/cvxpy_rewriting.pdf |journal=Control and Decision |volume=5 |issue=1 |pages=42–60 |arxiv=1709.04494 |doi=10.1080/23307706.2017.1397554 |s2cid=67856259}}</ref>

[[File:Hierarchy compact convex.pngsvg|thumb|A hierarchy of convex optimization problems. (LP: [[linear programming]], QP: [[quadratic programming]], SOCP [[Second-order cone programming|second-order cone program]], SDP: [[semidefinite programming]], CP: [[conic optimization]].)]]

*[[Linear programming]] problems are the simplest convex programs. In LP, the objective and constraint functions are all linear.

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*[[Geometric programming]]

*[[Entropy maximization]] with appropriate constraints.

==Properties==

The following are useful properties of convex optimization problems:<ref name="rockafellar93">{{cite journal | author = Rockafellar, R. Tyrrell | title = Lagrange multipliers and optimality | journal = SIAM Review | volume = 35 | issue = 2 | year = 1993 | pages = 183–238 |url = http://web.williams.edu/Mathematics/sjmiller/public_html/105Sp10/handouts/Rockafellar_LagrangeMultAndOptimality.pdf | doi=10.1137/1035044| citeseerx = 10.1.1.161.7209}}</ref><ref name=":2" />{{Rp|___location=chpt.4}}

* every point that is [[local minimum]] is a also a [[global minimum]];

* the optimal set is convex;

* if the objective function is ''strictly'' convex, then the problem has at most one optimal point.