Exact SDP relaxations for classes of nonlinear semidefinite programming problems

An exact semidefinite linear programming (SDP) relaxation of a nonlinear semidefinite programming problem is a highly desirable feature because a semidefinite linear programming problem can efficiently be solved. This paper addresses the basic issue of which nonlinear semidefinite programming problems possess exact SDP relaxations under a constraint qualification. We do this by establishing exact SDP relaxations for classes of nonlinear semidefinite programming problems with SOS-convex polynomials. These classes include SOS-convex semidefinite programming problems and fractional semidefinite programming problems with SOS-convex polynomials. The class of SOS-convex polynomials contains, in particular, convex quadratic functions and separable convex polynomials. Consequently, we also derive numerically checkable conditions, completely characterizing minimizers of these classes of semidefinite programming problems. We finally present a constraint qualification, which is, in a sense, the weakest condition guaranteeing these checkable optimality conditions. The SOS-convexity of polynomials is a sufficient condition for convexity and it can be checked by semidefinite programming whereas deciding convexity is generally NP-hard.