Reduced first-level representations via the reformulation-linearization technique: results, counterexamples, and computations

Abstract In this paper, we consider the reformulation-linearization technique (RLT) of Sherali and Adams (SIAM J. Discrete Math. 3 (3) (1990) 411–430, Discrete Appl. Math. 52 (1994) 83–106) and explore the generation of reduced first-level representations for 0–1 mixed-integer programs that tend to retain the strength of the full first-level linear programming relaxation. The motivation for this study is provided by the computational success of the first-level RLT representation (in full or partial form) experienced by several researchers working on various classes of problems. We show that there exists a first-level representation having only about half the RLT constraints that yields the same lower bound value via its relaxation. Accordingly, we attempt to a priori predict the form of this representation and identify many special cases for which this prediction is accurate. However, using various counter examples, we show that this prediction as well as several variants of it are not accurate in general, even for the case of a single binary variable. In addition, since the full first-level relaxation produces the convex hull representation for the case of a single binary variable, we investigate whether this is the case with respect to the reduced first-level relaxation as well, showing similarly that it holds true only for some special cases. Some empirical results on the relative merit and prediction capability of the reduced, versus the full, first-level representation are also provided.

[1]  Warren P. Adams,et al.  Improved Linear Programming-based Lower Bounds for the Quadratic Assignment Proglem , 1993, Quadratic Assignment and Related Problems.

[2]  Hanif D. Sherali,et al.  A Quadratic Partial Assignment and Packing Model and Algorithm for the Airline Gate Assignment Problem , 1993, Quadratic Assignment and Related Problems.

[3]  Warren P. Adams,et al.  A Tight Linearization and an Algorithm for Zero-One Quadratic Programming Problems , 1986 .

[4]  Egon Balas,et al.  A restricted Lagrangean approach to the traveling salesman problem , 1981, Math. Program..

[5]  Maurício Resende.  A Branch and Bound Algorithm for the Quadratic Assignment Problem using a Lower Bound Based on Linear Programming , 1996 .

[6]  Hanif D. Sherali,et al.  Mixed-integer bilinear programming problems , 1993, Math. Program..

[7]  Laurence A. Wolsey,et al.  Integer and Combinatorial Optimization , 1988 .

[8]  Hanif D. Sherali,et al.  A Hierarchy of Relaxations and Convex Hull Characterizations for Mixed-integer Zero-one Programming Problems , 1994, Discret. Appl. Math..

[9]  Egon Balas,et al.  A lift-and-project cutting plane algorithm for mixed 0–1 programs , 1993, Math. Program..

[10]  Laurence A. Wolsey,et al.  Integer and Combinatorial Optimization , 1988, Wiley interscience series in discrete mathematics and optimization.

[11]  Warren P. Adams,et al.  A hierarchy of relaxation between the continuous and convex hull representations , 1990 .

[12]  Hanif D. Sherali,et al.  Tighter Representations for Set Partitioning Problems , 1996, Discret. Appl. Math..

[13]  Alexander Schrijver,et al.  Cones of Matrices and Set-Functions and 0-1 Optimization , 1991, SIAM J. Optim..

[14]  Hanif D. Sherali,et al.  A Hierarchy of Relaxations Between the Continuous and Convex Hull Representations for Zero-One Programming Problems , 1990, SIAM J. Discret. Math..

[15]  Martin W. P. Savelsbergh,et al.  An Updated Mixed Integer Programming Library: MIPLIB 3.0 , 1998 .

[16]  Joseph F. Pekny,et al.  Dynamic Matrix Factorization Methods for Using Formulations Derived From Higher Order Lifting Techniques in the Solution of the Quadratic Assignment Problem , 1996 .

[17]  Hanif D. Sherali,et al.  A Decomposition Algorithm for a Discrete Location-Allocation Problem , 1984, Oper. Res..

[18]  E. Balas,et al.  Mixed 0-1 Programming by Lift-and-Project in a Branch-and-Cut Framework , 1996 .