A General Purpose Exact Solution Method for Mixed Integer Concave Minimization Problems

TL;DR

This paper introduces a general exact solution method for mixed integer concave minimization problems using a bilevel programming approach with linearization, demonstrating superior performance on supply chain problems.

Contribution

The paper develops a novel, general exact algorithm for mixed integer concave minimization problems, applicable to various classes of problems with improved efficiency.

Findings

Outperforms existing methods by an order of magnitude in computational tests.

Successfully applied to concave knapsack and production-transportation problems.

Guarantees convergence to the exact global optimum.

Abstract

In this article, we discuss an exact algorithm for solving mixed integer concave minimization problems. A piecewise inner-approximation of the concave function is achieved using an auxiliary linear program that leads to a bilevel program, which provides a lower bound to the original problem. The bilevel program is reduced to a single level formulation with the help of Karush-Kuhn-Tucker (KKT) conditions. Incorporating the KKT conditions lead to complementary slackness conditions that are linearized using BigM, for which we identify a tight value for general problems. Multiple bilevel programs, when solved over iterations, guarantee convergence to the exact optimum of the original problem. Though the algorithm is general and can be applied to any optimization problem with concave function(s), in this paper, we solve two common classes of operations and supply chain problems; namely, the concave knapsack problem, and the concave production-transportation problem. The computational experiments indicate that our proposed approach outperforms the customized methods that have been used in the literature to solve the two classes of problems by an order of magnitude in most of the test cases.