On the Global Solution of Linear Programs with Linear Complementarity Constraints

Abstract

This paper presents a parameter-free integer-programming based algorithm for the global resolution of a linear program with linear complementarity constraints (LPCC). The cornerstone of the algorithm is a minimax integer program formulation that characterizes and provides certificates for the three outcomes-infeasibility, unboundedness, or solvability-of an LPCC. An extreme point/ray generation scheme in the spirit of Benders decomposition is developed, from which valid inequalities in the form of satisfiability constraints are obtained. The feasibility problem of these inequalities and the carefully guided linear programming relaxations of the LPCC are the workhorse of the algorithm, which also employs a specialized procedure for the sparsification of the satifiability cuts. We establish the finite termination of the algorithm and report computational results using the algorithm for solving randomly generated LPCCs of reasonable sizes. The results establish that the algorithm can handle infeasible, unbounded, and solvable LPCCs effectively.

Lift-and-project for general two-term disjunctions

“…. , m. We have compared our results to those of Hu et al [12] using the test instances [16], which were obtained by a sophisticated Bender's decomposition method in which pure integer programs (IPs) were used to test whether a system of inequalities each of the form i∈I z i + j∈J (1 − z j ) ≥ 1 admits a feasible binary solution, and linear programs (LPs) were solved repeatedly, to find new extreme points, or rays of a Bender's reformulation, and also to detect infeasibility or unboundedness [12]. We tested our method on datasets with 50, and 300 complementarity constraints.…”

“…In Tables 3 and 4, columns lb and opt provide the value of the LP relaxation and that of the optimal solution, respectively. The columns LPs and IPs are taken from [12] and indicate the number of linear and integer programs solved, respectively. Finally, the last four columns provide information about our method: the number of nodes explored, the number of L&P cuts generated, the total number of cut-generation rounds, and the run-time of the computation in seconds.…”

“…One should also note that these instances are easy as the number of nodes needed by any method is below 50 in all but one cases. In particular, on the large instances with [12] and branch-and-cut with L&P cuts on Table 6: Comparion of pure B&B, branch-and-cut with L&P cuts, and branchand-cut with CGLP cuts on general LPCCs with B = 0, n = m = 300, r = 300. …”

“…α normalization, or β normalization (11) u, u 0 , v, v 0 ≥ 0 (12) In the objective function, (x,ŷ) is a basic solution of the linear relaxation of (5) withx i fractional, which we want to cut off. Constraints (7)- (10) express that the inequality αx ≥ β is valid for both of the polyhedra P 0 = {(x, y) ∈ P | − x i ≥ 0}, and P 1 = {(x, y) ∈ P | x i ≥ 1}.…”

“…A drawback of the original approach was the computational overhead caused by forming CGLP, even in a reduced space, and by the many pivots needed to solve it to optimality. Both of these problems were eliminated by Balas and Perregaard [8], and Perregaard [15] by establishing a direct correspondence between the bases of the linear relaxation of (5) and the basic solutions of CGLP (6)- (12), and by showing how to pivot in the simplex tableau of (5) (small tableau) in order to improve the objective function value of CGLP. They have also shown that by pivoting in the small tableau, the true optimum of CGLP can be found.…”