Chapter 3. Duality in convex optimization

Semidefinite program: Let A 0 , A 1 , . . . , A n ∈ S m , c ∈ R n
and let x ∈ R n be the vector of unknowns.
The primal SDP is:
{
(P) min c T x | F (x) := −A 0 + ∑ n
}
A k x k ≽ 0
x k=1
The dual problem is (as we shall see):
(D) max
Z {A 0 • Z | A k • Z = c k , k = 1, .., n , Z ≽ 0}
Ex. Show: Both programs, (P) and (D) are convex
problems.
Rem. If the matrices A 0 , A 1 , . . . , A n are diagonal matrices then
(P) is simply a linear program with (linear) dual (D).
CO, Chapter 3 p 16/18