## Semi-infinite linear programming approaches to semidefinite programming (SDP) problems

### Authors:

Kartik Krishnan and John E. Mitchell
Department of Mathematical Sciences
Rensselaer Polytechnic Institute
Troy, NY 12180 USA
kartis@rpi.edu and mitchj@rpi.edu

### Abstract:

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 $O(\sqrt{k})$ (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. Moreover one can incorporate these linear programs in a branch and cut approach for solving large scale integer programs.

### Citation details:

Fields Institute Communications Series, Volume 37, Novel approaches to hard discrete optimization problems, edited by P. Pardalos and H. Wolkowicz, AMS, pages 123-142, 2003.