Index Branch-and-Bound Algorithm for Global Optimization with Multiextremal Constraints

TL;DR

This paper introduces a derivative-free Branch-and-Bound algorithm for Lipschitz multiextremal constrained global optimization, reducing problem complexity via an index scheme and efficiently handling constraints to improve search speed.

Contribution

It proposes a novel index-based reduction and a derivative-free Branch-and-Bound method with selective constraint evaluation, enhancing efficiency in solving complex constrained optimization problems.

Findings

The method effectively determines infeasibility or bounds for the global solution.

It accelerates the search by evaluating fewer constraints per iteration.

Numerical experiments demonstrate its practical efficiency and convergence.

Abstract

In this paper, Lipschitz univariate constrained global optimization problems where both the objective function and constraints can be multiextremal are considered. The constrained problem is reduced to a discontinuous unconstrained problem by the index scheme without introducing additional parameters or variables. A Branch-and-Bound method that does not use derivatives for solving the reduced problem is proposed. The method either determines the infeasibility of the original problem or finds lower and upper bounds for the global solution. Not all the constraints are evaluated during every iteration of the algorithm, providing a significant acceleration of the search. Convergence conditions of the new method are established. Test problems and extensive numerical experiments are presented.