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Semi-infinite linear programming approaches to semidefinite programming problems. (English) Zbl 1028.65066

Pardalos, Panos (ed.) et al., Novel approaches to hard discrete optimization. Providence, RI: American Mathematical Society (AMS). Fields Inst. Commun. 123-142 (2003).
Summary: Interior point methods, the traditional methods for the semidefinite programming (SDP), are fairly limited in the size of problems they can handle. This paper deals with an linear programming (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 to G. Pataki [Math. Oper. Res. 23, No. 2, 339-358 (1998; Zbl 0977.90051)] 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 C. Helmberg and F. Rendl [SIAM J. Optim. 10, No. 4, 673-696 (2000; Zbl 0960.65074)]. 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.
For the entire collection see [Zbl 1013.00014].

MSC:

65K05 Numerical mathematical programming methods
90C10 Integer programming
90C51 Interior-point methods
90C05 Linear programming
90C06 Large-scale problems in mathematical programming
90C22 Semidefinite programming
90C27 Combinatorial optimization
90C34 Semi-infinite programming
90C57 Polyhedral combinatorics, branch-and-bound, branch-and-cut
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