v2010.10.26 - Convex Optimization

274 CHAPTER 4. SEMIDEFINITE PROGRAMMINGBy the fundamental theorem of Convex Optimization, any locally optimalpoint of a convex problem is globally optimal. [61,4.2.2] [306,1] Givenconvex real objective function g and convex feasible set D ⊆domg , whichis the set of all variable values satisfying the problem constraints, we pose ageneric convex optimization problemminimize g(X)Xsubject to X ∈ D(647)where constraints are abstract here in membership of variable X to convexfeasible set D . Inequality constraint functions of a convex optimizationproblem are convex while equality constraint functions are conventionallyaffine, but not necessarily so. Affine equality constraint functions, as opposedto the superset of all convex equality constraint functions having convex levelsets (3.4.0.0.4), make convex optimization tractable.Similarly, the problemmaximize g(X)Xsubject to X ∈ D(648)is called convex were g a real concave function and feasible set D convex.As conversion to convex form is not always possible, there is much ongoingresearch to determine which problem classes have convex expression orrelaxation. [34] [59] [151] [277] [345] [149]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. [115, p.9]