No-Gap Second-Order Conditions via a Directional Curvature Functional

TL;DR

This paper develops a unified framework for second-order optimality conditions in constrained optimization, applicable in both finite and infinite dimensions, including cases lacking classical regularity assumptions.

Contribution

It introduces a directional curvature functional to establish no-gap second-order conditions, extending applicability beyond classical assumptions like polyhedricity.

Findings

Derived no-gap second-order conditions using the curvature functional

Applicable to both finite and infinite-dimensional problems

Includes analysis of bang-bang optimal control problems

Abstract

This paper is concerned with necessary and sufficient second-order conditions for finite-dimensional and infinite-dimensional constrained optimization problems. Using a suitably defined directional curvature functional for the admissible set, we derive no-gap second-order optimality conditions in an abstract functional analytic setting. Our theory not only covers those cases where the classical assumptions of polyhedricity or second-order regularity are satisfied but also allows to study problems in the absence of these requirements. As a tangible example, we consider no-gap second-order conditions for bang-bang optimal control problems.