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SIAM Journal on Matrix Analysis and Applications

Fixing Variables in Semidefinite Relaxations

Christoph Helmberg

Published Jan 1, 2000• 42 citations
Publisher

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.

DOI
10.1137/s089547989631442x

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In computer vision, many problems can be formulated as binary quadratic programs (BQPs), which are in general NP hard. Finding a solution when the problem is of large size to be of practical interest typically requires relaxation. Semidefinite relaxation usually yields tight bounds, but its computational complexity is high. In this work, we present a semidefinite programming (SDP) formulation for BQPs, with two desirable properties. First, it produces similar bounds to the standard SDP formulation. Second, compared with the conventional SDP formulation, the proposed SDP formulation leads to a considerably more efficient and scalable dual optimization approach. We then propose two solvers, namely, quasi-Newton and smoothing Newton methods, for the simplified dual problem. Both of them are significantly more efficient than standard interior-point methods. Empirically the smoothing Newton solver is faster than the quasi-Newton solver for dense or medium-sized problems, while the quasi-Newton solver is preferable for large sparse/structured problems.
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A special reformulation-linearization technique based linear conic relaxation is proposed for discrete quadratic programming (DQP). We show that the proposed relaxation is tighter than the traditional positive semidefinite programming relaxation. More importantly, when the proposed relaxation problem has an optimal solution with rank one or two, optimal solutions to the original DQP problem can be explicitly generated. This rank-two property is further extended to binary quadratic optimization problems and linearly constrained DQP problems. Numerical results indicate that the proposed relaxation is capable of providing high quality and robust lower bounds for DQP.
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We consider problem (QP) of minimizing a quadratic function subject to linear or quadratic constraints. Variables are integer and bounded. This very general problem can model many classical problems in Combinatorial Optimization.A major difference between (QP) and integer linear programs lies in the fact that, in general, its continuous relaxation is an NP-hard optimization problem. To overcome this difficulty, the Convex Quadratic Reformulation approach transforms (QP) into a problem (QP'), equivalent to (QP), but whose continuous relaxation is a convex problem. To compute a global optimal solution to (QP) it becomes possible to solve (QP') by an implicit enumeration algorithm based on continuous convex optimization.We make an overview of recent developments in this approach. We show in particular how positive semidefinite relaxations can be used to build the most interesting equivalent problems (QP'). In the case of binary variables, we give a vision of classical linearization as a special case of Convex Quadratic Reformulation.Finally, we illustrate the generality and the computational efficiency of the Quadratic Convex Reformulation approach for several combinatorial optimization problems.