Semi-infinite linear programming approaches to semidefinite programming problems∗

Interior point methods, the traditional methods for the SDP , are fairly limited in the size of problems they can handle. This paper deals with an LP approach to overcome some of these shortcomings. We begin with a semi-infinite linear programming formulation of the SDP and discuss the issue of its discretization in some detail. We further show that a lemma due Pataki on the geometry of the SDP , implies that no more than O( √ k) (where k is the number of constraints in the SDP ) linear constraints are required. To generate these constraints we employ the spectral bundle approach due to Helmberg and Rendl. This scheme recasts any SDP with a bounded primal feasible set as an eigenvalue optimization problem. These are convex nonsmooth problems that can be tackled by bundle methods for nondifferentiable optimization. Finally we present the rationale for using the columns of the bundle P maintained by the spectral bundle approach, as our linear constraints. We present numerical experiments that demonstrate the efficiency of the LP approach on two combinatorial examples, namely the max cut and min bisection problems. The LP approach potentially allows one to approximately solve large scale semidefinite programs using state of the art linear solvers. This work was supported in part by NSF grant numbers CCR–9901822 and DMS9872019 Department of Mathematical Sciences, Rensselaer Polytechnic Institute, 110 8th Street, Troy, New York, 12180 ([email protected]). Department of Mathematical Sciences, Rensselaer Polytechnic Institute, 110 8th Street, Troy, New York, 12180 ([email protected]).