Exact penalty functions in optimization with unbounded constraint sets

TL;DR

This work develops a unified theory of exact penalty functions for constrained optimization with unbounded constraint sets. It provides global necessary and sufficient conditions in the general case and specialized results for locally Lipschitz, semi-algebraic, and non-degenerate polynomial data, all formulated through the objective $f$ and a residual function $\psi$. The analysis leverages variational analysis and semi-algebraic geometry, delivering Hölder-type error bounds and Newton-polyhedra-based criteria that ensure exact penalization without boundedness assumptions. The results extend classical exact-penalty theory, offering multipliers and explicit conditions that apply to a broad class of problems encountered in practice.

Abstract

This paper identifies necessary and sufficient conditions for the exactness of penalty functions in optimization problems whose constraint sets are not necessarily bounded. The case where the data of problems is locally Lipschitz, semi-algebraic or non-degenerate polynomials is studied in detail. The conditions are given in terms of properties of the objective and residual functions of the problems in question. The obtained results generalize and improve some known results in the literature on exact penalty functions.

Theorems & Definitions (84)

• Definition 2.1
• Remark 2.1
• Definition 2.2
• Remark 2.2
• Definition 2.3
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5: Fermat rule