v2007.11.26 - Convex Optimization

226 CHAPTER 4. SEMIDEFINITE PROGRAMMINGhave the generic convex optimization problemminimize g(X)Xsubject to X ∈ C(545)where constraints are abstract here in the membership of variable X tofeasible set C . Inequality constraint functions of a convex optimizationproblem are convex while equality constraint functions are conventionallyaffine, but not necessarily so. Affine equality constraint functions (necessarilyconvex), as opposed to the larger set of all convex equality constraintfunctions having convex level sets, make convex optimization tractable.Similarly, the problemmaximize g(X)Xsubject to X ∈ C(546)is convex were g a real concave function. As conversion to convex form is notalways possible, there is much ongoing research to determine which problemclasses have convex expression or relaxation. [27] [44] [102] [206] [262] [100]4.1 Conic problemStill, we are surprised to see the relatively small number ofsubmissions to semidefinite programming (SDP) solvers, as thisis an area of significant current interest to the optimizationcommunity. We speculate that semidefinite programming issimply experiencing the fate of most new areas: Users have yet tounderstand how to pose their problems as semidefinite programs,and the lack of support for SDP solvers in popular modellinglanguages likely discourages submissions.−SIAM News, 2002. [79, p.9]Consider a conic problem (p) and its dual (d): [219,3.3.1] [176,2.1](confer p.140)(p)minimize c T xxsubject to x ∈ KAx = bmaximize b T yy,ssubject to s ∈ K ∗A T y + s = c(d) (264)