v2009.01.01 - Convex Optimization

4.1. CONIC PROBLEM 247
where K is a closed convex cone, K ∗ is its dual, matrix A is fixed, and the
remaining quantities are vectors.
When K is a polyhedral cone (2.12.1), then each conic problem becomes
a linear program [82]; the self-dual nonnegative orthant providing the
prototypical primal linear program and its dual. More generally, each
optimization problem is convex when K is a closed convex cone. Unlike
the optimal objective value, a solution to each problem is not necessarily
unique; in other words, the optimal solution set {x ⋆ } or {y ⋆ , s ⋆ } is convex and
may comprise more than a single point although the corresponding optimal
objective value is unique when the feasible set is nonempty.
4.1.1 a Semidefinite program
When K is the self-dual cone of positive semidefinite matrices in the subspace
of symmetric matrices, then each conic problem is called a semidefinite
program (SDP); [239,6.4] primal problem (P) having matrix variable
X ∈ S n while corresponding dual (D) has matrix slack variable S ∈ S n and
vector variable y = [y i ]∈ R m : [9] [10,2] [342,1.3.8]
(P)
minimize
X∈ S n 〈C , X〉
subject to X ≽ 0
A svec X = b
maximize
y∈R m , S∈S n 〈b, y〉
subject to S ≽ 0
svec −1 (A T y) + S = C
(D)
(584)
This is the prototypical semidefinite program and its dual, where matrix
C ∈ S n and vector b∈R m are fixed, as is
⎡
A =
∆ ⎣
where A i ∈ S n , i=1... m , are given. Thus
⎤
svec(A 1 ) T
. ⎦∈ R m×n(n+1)/2 (585)
svec(A m ) T
⎡
A svec X = ⎣
〈A 1 , X〉
.
〈A m , X〉
(586)
∑
svec −1 (A T y) = m y i A i
i=1
⎤
⎦