Solving a Class of Semideenite Programs via Nonlinear Programming

In this paper, we introduce a transformation that converts a class of linear and nonlinear semideenite programming(SDP) problemsinto nonlinear optimizationproblems. For those problemsof interest, the transformation replaces matrix-valued constraints by vector-valued ones, hence reducing the number of constraints by an order of magnitude. 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 gradient formulas for the objective function of the resulting nonlinear optimization problem and show that both function and gradient evaluations have aaordable complexities that eeectively exploit the sparsity of the problem data. This transformation, together with the eecient gradient formulas, enables the solution of very large-scale SDP problems by gradient-based nonlinear optimization techniques. In particular, we propose a rst-order log-barrier method designed for solving a class of large-scale linear SDP problems. This algorithm operates entirely within the space of the transformed problem while still maintaining close ties with both the primal and the dual of the original SDP problem. Global convergence of the algorithm is established under mild and reasonable assumptions. Key words. transformation { semideenite program { semideenite relaxation { nonlinear programming { rst-order methods { log-barrier algorithms { interior-point methods