Lower bounds for nonlinear assignment problems using many body interactions

Abstract This paper concerns lower bounding techniques for the general α-adic assignment problem. The nonlinear objective function is linearized by the introduction of additional variables and constraints, thus yielding a mixed integer linear programming formulation of the problem. The concept of many body interactions is introduced to strengthen this formulation and incorporated in a modified formulation obtained by lifting the original representation to a higher dimensional space. This process involves two steps — (i) addition of new variables and constraints and (ii) incorporation of the new variables in the objective function. If this lifting process is repeated β times on an α-adic assignment problem along with the incorporation of higher order interactions, it results in the mixed-integer formulation of an equivalent (α + β)-adic assignment problem. The incorporation of many body interactions in the higher dimensional formulation improves its degeneracy properties and is also critical to the derivation of decomposition methods for the solution of these large scale mathematical programs in the higher dimensional space. It is shown that a lower bound to the optimal solution of the corresponding linear programming relaxation can be obtained by dualizing a subset of constraints in this formulation and solving O ( N 2( α + β −1) ) linear assignment problems, whose coefficients depend on the dual values. Moreover, it is proved that the optimal solution to the LP relaxation is obtained if we use the optimal duals for the solution of the linear assignment problems. This concept of many body interactions could be applied in designing algorithms for the solution of formulations obtained by lifting general MILP's. We illustrate all these concepts on the quadratic assignment problems With these decomposition bounds, we have found the provably optimal solutions of two unsolved QAP's of size 32 and have also improved upon existing lower bounds for other QAP's.

[1]  Nicos Christofides,et al.  A Graph Theoretic Analysis of Bounds for the Quadratic Assignment Problem , 1981 .

[2]  E. Lawler.  The Quadratic Assignment Problem , 1963 .

[3]  Hanif D. Sherali,et al.  On the Use of Exact and Heuristic Cutting Plane Methods for the Quadratic Assignment Problem , 1982 .

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

[5]  Rainer E. Burkard,et al.  On the Biquadratic Assignment Problem , 1993, Quadratic Assignment and Related Problems.

[6]  H. Sherali,et al.  Benders' partitioning scheme applied to a new formulation of the quadratic assignment problem , 1980 .

[7]  Arthur M. Geoffrion,et al.  Scheduling Parallel Production Lines with Changeover Costs: Practical Application of a Quadratic Assignment/LP Approach , 1976, Oper. Res..

[8]  Zvi Drezner,et al.  Computing Lower Bounds for the Quadratic Assignment Problem with an Interior Point Algorithm for Linear Programming , 1995, Oper. Res..

[9]  John R. Beaumont,et al.  Studies on Graphs and Discrete Programming , 1982 .

[10]  P. Pardalos,et al.  State of the art in global optimization: computational methods and applications , 1996 .

[11]  Panos M. Pardalos,et al.  A parallel algorithm for the quadratic assignment problem , 1989, Proceedings of the 1989 ACM/IEEE Conference on Supercomputing (Supercomputing '89).

[12]  P. Gilmore.  Optimal and Suboptimal Algorithms for the Quadratic Assignment Problem , 1962 .

[13]  T. Koopmans,et al.  Assignment Problems and the Location of Economic Activities , 1957 .

[14]  Catherine Roucairol,et al.  A parallel branch and bound algorithm for the quadratic assignment problem , 1987, Discret. Appl. Math..

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

[16]  Franz Rendl,et al.  QAPLIB – A Quadratic Assignment Problem Library , 1997, J. Glob. Optim..

[17]  Catherine Roucairol,et al.  Difficulties of Exact Methods for Solving the Quadratic Assignment Problem , 1993, Quadratic Assignment and Related Problems.

[18]  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 .

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

[20]  Leon Steinberg,et al.  The Backboard Wiring Problem: A Placement Algorithm , 1961 .

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

[22]  Egon Balas,et al.  A parallel shortest augmenting path algorithm for the assignment problem , 1991, JACM.

[23]  Alan M. Frieze,et al.  On the quadratic assignment problem , 1983, Discret. Appl. Math..