Linear Programming ( Lp ) Approaches to Semidefinite Programming ( Sdp ) Problems

Until recently, the study of interior point methods has dominated algorithmic research in semidefinite programming (SDP). From a theoretical point of view, these interior point methods offer everything one can hope for; they apply to all SDP’s, exploit second order information and offer polynomial time complexity. Still for practical applications with many constraints k, the number of arithmetic operations, per iteration is often too high. This motivates the search for other approaches, that are suitable for large k and exploit problem structure. Recently Helmberg and Rendl developed a scheme that casts SDP’s with a constant trace on the primal feasible set as eigenvalue optimisation problems. These are convex nonsmooth programming problems and can be solved by bundle methods. In this talk we propose a linear programming framework to solving SDP’s with this structure. Although SDP’s are semi infinite linear programs, we show that only a small number of constraints, namely those in the bundle maintained by the spectral bundle approach, bounded by the square root of the number of constraints in the SDP, and others polynomial in the problem size are typically required. The resulting LP’s can be solved rather quickly and provide reasonably accurate solutions. We also describe a cutting plane approach, where an SDP is solved as a sequence of LP’s. However to make the resulting method competitive with interior point methods for the SDP, several refinements are necessary. In particular the cutting plane algorithm uses an interior point algorithm to solve the LP relaxations approximately, since this results in the generation of better constraints than a simplex cutting plane algorithm. We present numerical examples demonstrating the efficiency of the two approaches on combinatorial examples. This is joint work with my advisor John Mitchell.