Global optimization of multivariable functions satisfying the Vanderbei condition

TL;DR

This paper introduces two non-uniform covering algorithms for global optimization of multivariable functions satisfying the Vanderbei condition, with proven convergence and tested on non-Lipschitz functions.

Contribution

The paper presents novel algorithms for global optimization under the Vanderbei condition, including convergence proofs and numerical testing on non-Lipschitz functions.

Findings

Algorithms converge to an ε-solution

Effective on non-Lipschitz functions

Demonstrated through numerical examples

Abstract

We propose two algorithms for solving global optimization problems on a hyperrectangle with an objective function satisfying the Vanderbei condition (this function is also called an $\varepsilon$-Lipschitz continuous function). The algorithms belong to the class of non-uniform cover-ings methods. For the algorithms we prove propositions about convergence to an $\varepsilon$-solution in terms of the objective function. We illustrate the performance of the algorithms using several test numerical examples with non-Lipschitz continuous objective functions.