A Conditional Logic Approach for Strengthening Mixed 0-1 Linear Programs

Naval Research Logistics

Hadavas

2009

Abstract:The reformulation-linearization technique (RLT) is a methodology for constructing tight linear programming relaxations of mixed discrete problems. A key construct is the multiplication of "product factors" of the discrete variables with problem constraints to form polynomial restrictions, which are subsequently linearized. For special problem forms, the structure of these linearized constraints tends to suggest that certain classes may be more beneficial than others. We examine the usefulness of subsets of constraints for a family of 0-1 quadratic multidimensional knapsack programs and perform extensive computational tests on a classical special case known as the 0-1 quadratic knapsack problem. We consider RLT forms both with and without these inequalities, and their comparisons with linearizations derived from published methods. Interestingly, the computational results depend in part upon the commercial software used.

Section: Concise Linear Representationsmentioning

confidence: 56%

Concise RLT forms of binary programs: A computational study of the quadratic knapsack problem

Forrester

Naval Research Logistics

Hadavas

2009

“…The first such method generates concise programs while the second promotes tight linear programming relaxations. Our study begins by enhancing the formulations in [11] using a conditional logic argument of [19,26] to adjust certain constraint coefficients, and a rewrite that alters the form of the objective function using a variable substitution based on binary identities. Both these enhancements are designed to strengthen the relaxation value.…”

Section: Discussionmentioning

confidence: 99%

“…The bounds L j and U j computed in (3) can directly impact the optimal objective function value to the continuous relaxation of Problem G. We desire to increase the values of the lower bounds L j and decrease the values of the upper bounds U j to potentially tighten the continuous relaxation. To do so, we employ a conditional logic argument introduced in [26] and expanded in [19].…”

Section: Enhancement 1: Strengthening L J and U Jmentioning

confidence: 99%

Comparisons and enhancement strategies for linearizing mixed 0-1 quadratic programs

Forrester

Glover

2004

Discrete Optimization

We present a linearization strategy for mixed 0-1 quadratic programs that produces small formulations with tight relaxations. It combines constructs from a classical method of Glover and a more recent reformulation-linearization technique (RLT). By using binary identities to rewrite the objective, a variant of the first method results in a concise formulation with the level-1 RLT strength. This variant is achieved as a modified surrogate dual of a Lagrangian subproblem to the RLT. Special structures can be exploited to obtain reductions in problem size, without forfeiting strength. Preliminary computational experience demonstrates the potential of the new representations.

Use ofLagrange Interpolating Polynomials in theRLT

Wiley Encyclopedia of Operations Research and Management Science

2011

This article examines the use of Lagrange interpolating polynomials (LIPs) in constructing polyhedral outer‐approximations and convex hull representations of mixed‐discrete sets. The LIPs provide the theoretical foundation for extending the reformulation‐linearization technique (RLT), used for automatically computing tight relaxations of mixed‐binary programs, to handle discrete variables that can realize values within finite sets. The LIPs operate by generalizing the idempotent property of binary variables x that x 2 = x , thereby promoting a hierarchy of progressively tighter forms that culminates in an explicit algebraic characterization of the convex hull of feasible solutions at the highest level. The methods of proof, which employ Kronecker products of transposes of Vandermonde matrices, both subsume and provide insight into the mixed‐binary RLT.