  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.
