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

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postscript
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### 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
### August 16, 2001.

### 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.
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