A One-Dimensional Local Tuning Algorithm for Solving GO Problems with Partially Defined Constraints

TL;DR

This paper introduces a one-dimensional local tuning algorithm for solving Lipschitz constrained global optimization problems with partially defined constraints, demonstrating improved performance over traditional methods.

Contribution

The paper presents a novel geometric local tuning method that adaptively estimates local Lipschitz constants without extra parameters, enhancing optimization efficiency.

Findings

The new method outperforms penalty and fixed Lipschitz approaches in numerical tests.

Adaptive local tuning improves convergence speed and solution accuracy.

The approach effectively handles partially defined constraints in one-dimensional GO problems.

Abstract

Lipschitz one-dimensional constrained global optimization (GO) problems where both the objective function and constraints can be multiextremal and non-differentiable are considered in this paper. Problems, where the constraints are verified in an a priori given order fixed by the nature of the problem are studied. Moreover, if a constraint is not satisfied at a point, then the remaining constraints and the objective function can be undefined at this point. The constrained problem is reduced to a discontinuous unconstrained problem by the index scheme without introducing additional parameters or variables. A new geometric method using adaptive estimates of local Lipschitz constants is introduced. The estimates are calculated by using the local tuning technique proposed recently. Numerical experiments show quite a satisfactory performance of the new method in comparison with the penalty approach and a method using a priori given Lipschitz constants.