Exact SDP relaxations for classes of nonlinear semidefinite programming problems

This paper addresses the issue of which nonlinear semidefinite linear programming problems possess exact semidefinite linear programming (SDP) relaxations under a constraint qualification. We establish 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 convex quadratic functions and separable convex polynomials. We also derive numerically checkable conditions, completely characterizing minimizers of these classes of problems.