DC Semidefinite Programming and Cone Constrained DC Optimization

In the first part of this paper we discuss possible extensions of the main ideas and results of constrained DC optimization to the case of nonlinear semidefinite programming problems (i.e. problems with matrix constraints). To this end, we analyse two different approaches to the definition of DC matrix-valued functions (namely, order-theoretic and componentwise), study some properties of convex and DC matrix-valued functions and demonstrate how to compute DC decompositions of some nonlinear semidefinite constraints appearing in applications. We also compute a DC decomposition of the maximal eigenvalue of a DC matrix-valued function, which can be used to reformulate DC semidefinite constraints as DC inequality constrains. In the second part of the paper, we develop a general theory of cone constrained DC optimization problems. Namely, we obtain local optimality conditions for such problems and study an extension of the DC algorithm (the convex-concave procedure) to the case of general cone constrained DC optimization problems. We analyse a global convergence of this method and present a detailed study of a version of the DCA utilising exact penalty functions. In particular, we provide two types of sufficient conditions for the convergence of this method to a feasible and critical point of a cone constrained DC optimization problem from an infeasible starting point