On a New Constraint Reduction Heuristic Using Improved Bisection Method for Mixed Integer Linear Programming

Hande Günay Akdemir

doi:10.28924/2291-8639-19-2021-65

Abstract

In this study, we develop a surrogate relaxation-based procedure to reduce mixed-integer linear programming (MILP) problem sizes. This technique starts with one surrogate constraint which is a nonnegative linear combination of multiple constraints of the problem. At this initial step, we calculate optimal Lagrangian multipliers from LP relaxation of the problem and use them as initial surrogate multipliers. We incorporate the improved bisection method (IBM) (B. Gavish, F. Glover, and H. Pirkul, Surrogate Constraints in Integer Programming, J. Inform. Optim. Sci. 12(2) (1991), 219-228.) into our algorithm. This simple heuristic algorithm is designed to iteratively generate a new surrogate cut that is to guarantee to satisfy the most violated two constraints of the corresponding iteration. The performance of the heuristic is tested using both some problems from the OR libraries and randomly generated ones.

Highlights

The objective here is to attempt to reduce the number of constraints of mixed-integer linear programming (MILP) type of problems
We develop a surrogate relaxation-based procedure to reduce mixed-integer linear programming (MILP) problem sizes
This technique starts with one surrogate constraint which is a nonnegative linear combination of multiple constraints of the problem

Summary

Introduction

The objective here is to attempt to reduce the number of constraints of MILP type of problems. Mixed integer linear programming; surrogate relaxation; surrogate constraints; Lagrangian multipliers; improved bisection method; heuristics; reduction. Min cT x s.t. μT (Ax − b) ≤ 0 x∈X where surrogate multipliers vector μ ≥ 0 Both techniques enlarge the feasible region and provide a lower bound on the optimal objective value of Problem (P ). Lorena and Narciso [31, 39] proposed six heuristics based on both the surrogate and Lagrangian relaxations and a subgrandient search algorithm for large scale generalized assignment problems. Applications of a classical combinatorial optimization problem called the set covering were given in [1,11] The former used the Surrogate constraint normalization technique to create appropriate weights for surrogate constraint relaxations, and the latter compared heuristics based on Lagrangian and surrogate relaxations for the Maximal Covering Location Problem.

Computing the optimal surrogate multipliers by IBM

Constraint reduction heuristic

Computational results

Conclusions