Bad Semidefinite Programs: They All Look the Same

Duality theory plays a central role in semidefinite programming, since in optimization algorithms a dual solution serves as a certificate of optimality. However, in semidefinite duality pathological phenomena occur: nonattainment of the optimal value, positive duality gaps, and infeasibility of the dual, even when the primal is bounded. We say that the semidefinite system PSD = {x | ∑m i=1 xiAi B } is badly behaved, or lacks uniform LP-duality, if for some linear objective function c the value sup { 〈c, x〉 |x ∈ PSD } is finite, but the dual program has no solution attaining the same value. We give simple, and exact characterizations of badly behaved semidefinite systems. Surprisingly, it turns out that the system x1 (