Solving Semidefinite Programs via Nonlinear Programming Part I: Transformations and Derivatives∗

In this paper, we introduce a transformation that converts a class of linear and nonlinear semidefinite programming (SDP) problems into nonlinear optimization problems over “orthants” of the form <++ × <N , where n is the size of the matrices involved in the problem and N is a nonnegative integer dependent upon the specific problem. For example, in the case of the SDP relaxation of a MAXCUT problem, n is the number of vertices in the underlying graph and N is zero. The class of transformable problems includes instances of SDP relaxations of combinatorial optimization problems with binary variables, as well as other important SDP problems. We also derive formulas for the first and second derivatives of the objective function of the resulting nonlinear optimization problem, hence enabling applications of existing nonlinear optimization techniques to the solution of large-scale SDP problems.