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

Sam Burer, Renato Monteiro, Yin Zhang

In this paper, we introduce transformations that convert a large class of linear and/or nonlinear semidefinite programming (SDP) problems into nonlinear optimization problems over ``orthants'' of the form $\Re^n_{++} \times \Re^N$, 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 zero and $n$, the number of variables of the resulting nonlinear optimization problem, is the number of vertices in the underlying graph. The class of transformable problems includes most, if not all, 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 the effective application of existing nonlinear optimization techniques to the solution of large-scale SDP problems.

Technical Report TR99-17 Department of Computational and Applied Mathematics, Rice University, Houston, Texas 77005, USA