On duality theory for non-convex semidefinite programming

In this paper, with the help of convex-like function, we discuss the duality theory for nonconvex semidefinite programming. Our contributions are: duality theory for the general nonconvex semidefinite programming when Slater’s condition holds; perfect duality for a special case of the nonconvex semidefinite programming for which Slater’s condition fails. We point out that the results of Fan (Appl. Math. Lett. 18:1068–1073, 2005) can be regarded as a special case of our result.