Complete Dual Characterizations of Optimality and Feasibility for Convex Semidefinite Programming

A convex semidefinite programming problem is a convex constrained optimization problem, where the constraints are linear matrix inequalities, for which the standard Lagrangian condition is sufficient for optimality. However, this condition requires constraint qualifications to completely characterize optimality. We present a necessary and sufficient condition for optimality without a constraint qualification. This is achieved by relaxing the standard Lagrangian condition to a new asymptotic form. A dual condition characterizing feasibility of a convex semidefinite program is also obtained. As applications of the optimality conditions we derive a version of Farkas’ lemma for systems involving matrix inequalities and obtain a strong duality result for the convex semidefinite programming problems. A numerical example is also given to illustrate the nature of the asymptotic optimality condition.