Strong Valid Inequalities Identification for Mixed Integer Programming Problems

TL;DR

This paper introduces an iterative algorithm to systematically identify strong valid inequalities for mixed-integer programming, demonstrated through a new TSP formulation that improves relaxation quality.

Contribution

It presents a novel method to generate and validate strong inequalities systematically, aiding modelers in strengthening mixed-integer programming formulations.

Findings

The algorithm successfully generates strong valid inequalities.

The new TSP formulation yields better relaxation bounds.

Computational results show improved solution quality.

Abstract

The characterization of strong valid inequalities for integer and mixed-integer programs is more of an artistic task than a systematic methodology, requiring inspiration that can sometimes be elusive. Frequently, this task is facilitated by somehow exploiting the structure of problems for devising strong valid inequalities. Subsequently, various mathematical techniques are utilized for proving that those inequalities, which are often easily shown to be valid, are indeed strong in the sense that they represent facets or other high dimensional faces. This paper develops a method to assist modelers in the challenge to devise strong valid inequalities. In each iteration, the proposed algorithm generates a valid inequality by solving a suitably constructed linear mixed integer program and applies some quality criteria in order to determine if it is a new strong valid inequality. To illustrate the proposed algorithm, a new Traveling Salesman Problem (TSP) formulation is developed based on a set of constraints already constructed in the context of the Hamiltonian Cycle Problem (HCP), and then the proposed algorithm is employed to derive a set of strong inequalities to tighten this TSP formulation. Finally, a comparison study between the relaxation of the new TSP formulation and that of a state-of-the-art TSP formulation is conducted. The computational study confirms the effectiveness of the devised inequalities due to the better quality of the relaxation provided by the new formulation.