An adaptive discretization method solving semi-infinite optimization problems with quadratic rate of convergence

TL;DR

This paper introduces a new adaptive discretization method for semi-infinite optimization problems that combines classical and bi-level techniques, achieving quadratic convergence and ensuring convergence to stationary points.

Contribution

The paper develops a novel adaptive discretization approach that merges classical and bi-level methods, providing quadratic convergence for semi-infinite problems.

Findings

The method exhibits quadratic rate of convergence.

A limit of the iterates is a stationary point.

The approach effectively combines classical and bi-level techniques.

Abstract

Semi-infinite programming can be used to model a large variety of complex optimization problems. The simple description of such problems comes at a price: semi-infinite problems are often harder to solve than finite nonlinear problems. In this paper we combine a classical adaptive discretization method developed by Blankenship and Falk and techniques regarding a semi-infinite optimization problem as a bi-level optimization problem. We develop a new adaptive discretization method which combines the advantages of both techniques and exhibits a quadratic rate of convergence. We further show that a limit of the iterates is a stationary point, if the iterates are stationary points of the approximate problems.