v2010.10.26 - Convex Optimization

208 CHAPTER 2. CONVEX GEOMETRYwhich is simply a restatement of optimality conditions (460) for conic problem(454). Suitable f(z) is the quadratic objective from convex problem1minimizez 2 zT Bz + q T zsubject to z ≽ 0(467)which means B ∈ S n + should be (symmetric) positive semidefinite for solutionof (465) by this method. Then (465) has solution iff (467) does. 2.13.10.1.5 Exercise. Optimality for equality constrained conic problem.Consider a conic optimization problem like (454) having real differentiableconvex objective function f(x) : R n →Rminimize f(x)xsubject to Cx = dx ∈ K(468)minimized over convex cone K but, this time, constrained to affine setA = {x | Cx = d}. Show, by means of first-order optimality condition(352) or (452), that necessary and sufficient optimality conditions are:(confer (460))x ⋆ ∈ KCx ⋆ = d∇f(x ⋆ ) + C T ν ⋆ ∈ K ∗(469)〈∇f(x ⋆ ) + C T ν ⋆ , x ⋆ 〉 = 0where ν ⋆ is any vector 2.83 satisfying these conditions.2.13.11 Proper nonsimplicial K , dual, X fat full-rankSince conically dependent columns can always be removed from X toconstruct K or to determine K ∗ [Wıκımization], then assume we are givena set of N conically independent generators (2.10) of an arbitrary properpolyhedral cone K in R n arranged columnar in X ∈ R n×N such that N > n(fat) and rankX = n . Having found formula (420) to determine the dual ofa simplicial cone, the easiest way to find a vertex-description of proper dual2.83 an optimal dual variable, these optimality conditions are equivalent to the KKTconditions [61,5.5.3].