Approximate solutions of convex semi-infinite optimization problems in finitely many iterations

TL;DR

This paper introduces two adaptive discretization algorithms for convex semi-infinite optimization that efficiently find approximate solutions with arbitrary precision in finitely many steps, requiring smaller discretizations than existing methods.

Contribution

The paper presents novel adaptive algorithms for convex semi-infinite optimization that guarantee finite termination at approximate solutions with arbitrary accuracy, using smaller discretizations and only requiring approximate solutions of subproblems.

Findings

Algorithms terminate after finitely many steps.

Require smaller discretizations than existing methods.

Achieve arbitrary precision in approximate solutions.

Abstract

We develop two adaptive discretization algorithms for convex semi-infinite optimization, which terminate after finitely many iterations at approximate solutions of arbitrary precision. In particular, they terminate at a feasible point of the considered optimization problem. Compared to the existing finitely feasible algorithms for general semi-infinite optimization problems, our algorithms work with considerably smaller discretizations and are thus computationally favorable. Also, our algorithms terminate at approximate solutions of arbitrary precision, while for general semi-infinite optimization problems the best possible approximate-solution precision can be arbitrarily bad. All occurring finite optimization subproblems in our algorithms have to be solved only approximately, and continuity is the only regularity assumption on our objective and constraint functions. Applications to parametric and non-parametric regression problems under shape constraints are discussed.