# A vector linear programming approach for certain global optimization problems

## TL;DR

This paper introduces a vector linear programming approach with modified Benson-type algorithms to efficiently solve certain global optimization problems involving quasi-concave functions and linear constraints, extending previous theoretical results.

## Contribution

It presents new algorithms based on MOLP techniques that improve the treatment of quasi-concave and related optimization problems, generalizing and enhancing prior research.

## Key findings

- Algorithms outperform existing methods in numerical tests.
- The approach effectively handles a broad class of global optimization problems.
- Results compare favorably with BARON software.

## Abstract

Global optimization problems with a quasi-concave objective function and linear constraints are studied. We point out that various other classes of global optimization problems can be expressed in this way. We present two algorithms, which can be seen as slight modifications of Benson-type algorithms for multiple objective linear programs (MOLP). The modification of the MOLP algorithms results in a more efficient treatment of the studied optimization problems. This paper generalizes results of Schulz and Mittal on quasi-concave problems and Shao and Ehrgott on multiplicative linear programs. Furthermore, it improves results of L\"ohne and Wagner on minimizing the difference $f=g-h$ of two convex functions $g$, $h$ where either $g$ or $h$ is polyhedral. Numerical examples are given and the results are compared with the global optimization software BARON.