v2010.10.26 - Convex Optimization

4.1. CONIC PROBLEM 275(confer p.162) Consider a conic problem (p) and its dual (d): [293,3.3.1][239,2.1] [240](p)minimize c T xxsubject to x ∈ KAx = bmaximize b T yy,ssubject to s ∈ K ∗A T y + s = c(d) (302)where 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; the selfdual nonnegative orthant providing the prototypicalprimal linear program and its dual. [94,3-1] 4.2 More generally, eachoptimization problem is convex when K is a closed convex cone. Unlike theoptimal objective value, a solution to each convex problem is not necessarilyunique; in other words, the optimal solution set {x ⋆ } or {y ⋆ , s ⋆ } is convex andmay comprise more than a single point although the corresponding optimalobjective value is unique when the feasible set is nonempty.4.1.1 a Semidefinite programWhen K is the selfdual cone of positive semidefinite matrices S n + in thesubspace of symmetric matrices S n , then each conic problem is called asemidefinite program (SDP); [277,6.4] primal problem (P) having matrixvariable X ∈ S n while corresponding dual (D) has slack variable S ∈ S n andvector variable y = [y i ]∈ R m : [10] [11,2] [391,1.3.8](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)(649)This is the prototypical semidefinite program and its dual, where matrixC ∈ S n and vector b∈R m are fixed, as is⎡ ⎤svec(A 1 ) TA ⎣ . ⎦∈ R m×n(n+1)/2 (650)svec(A m ) Twhere A i ∈ S n , i=1... m , are given. Thus4.2 Dantzig explains reasoning behind a nonnegativity constraint: . . .negative quantitiesof activities are not possible. . . .a negative number of cases cannot be shipped.