Fixing Variables in Semidefinite Relaxations

Authors:
Christoph Helmberg
Affiliations:
-
Venue:
SIAM Journal on Matrix Analysis and Applications
Year:
2000

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Abstract

The standard technique of reduced cost fixing from linear programming is not trivially extensible to semidefinite relaxations because the corresponding Lagrange multipliers are usually not available. We propose a general technique for computing reasonable Lagrange multipliers for constraints that are not part of the problem description. Its specialization to the semidefinite {-1,1} relaxation of quadratic 0-1 programming yields an efficient routine for fixing variables. The routine offers the possibility of exploiting problem structure. We extend the traditional bijective map between {0,1} and {-1,1} formulations to the constraints so that the dual variables remain the same and structural properties are preserved. Consequently, the fixing routine can be applied efficiently to optimal solutions of the semidefinite {0,1} relaxation of constrained quadratic 0-1 programming as well. We provide numerical results showing the efficacy of this approach.