v2007.09.13 - Convex Optimization

4.1. CONIC PROBLEM 227where K is a closed convex cone, K ∗ is its dual, matrix A is fixed, and theremaining quantities are vectors.When K is a polyhedral cone (2.12.1), then each conic problem becomesa linear program [64]. More generally, each optimization problem is convexwhen K is a closed convex cone. Unlike the optimal objective value, asolution to each problem is not necessarily unique; in other words, the optimalsolution set {x ⋆ } or {y ⋆ , s ⋆ } is convex and may comprise more than a singlepoint although the corresponding optimal objective value is unique when thefeasible set is nonempty.When K is the self-dual cone of positive semidefinite matrices in thesubspace of symmetric matrices, then each conic problem is called asemidefinite program (SDP); [203,6.4] primal problem (P) having matrixvariable X ∈ S n while corresponding dual (D) has matrix slack variableS ∈ S n and vector variable y = [y i ]∈ R m : [8] [9,2] [296,1.3.8](P)minimizeX∈ S n 〈C , X〉subject to X ≽ 0A svec X = bmaximizey∈R m , S∈S n 〈b, y〉subject to S ≽ 0svec −1 (A T y) + S = C(D)(546)This is the prototypical semidefinite program and its dual, where matrixC ∈ S n and vector b∈R m are fixed, as is⎡A =∆ ⎣⎤svec(A 1 ) T. ⎦ ∈ R m×n(n+1)/2 (547)svec(A m ) Twhere A i ∈ S n , i=1... m , are given. Thus⎡ ⎤〈A 1 , X〉A svec X = ⎣ . ⎦〈A m , X〉(548)∑svec −1 (A T y) = m y i A iThe vector inner-product for matrices is defined in the Euclidean/Frobeniussense in the isomorphic vector space R n(n+1)/2 ; id est,i=1〈C , X〉 ∆ = tr(C T X) = svec(C) T svec X (31)