No-gap second-order conditions for minimization problems in spaces of measures

¹ Institute of Mathematics, Brandenburgische Technische Universität Cottbus–Senftenberg, 03046 Cottbus, Germany
² Institut für Numerische Mathematik, Johannes Kepler Universität Linz, 4040 Linz, Austria

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 18 March 2025
Accepted: 2 February 2026

Abstract

Over the last years, minimization problems over spaces of measures have received increased interest due to their relevance in the context of inverse problems, optimal control and machine learning. A fundamental role in their numerical analysis is played by the assumption that the optimal dual state admits finitely many global extrema and satisfies a second-order sufficient optimality condition in each one of them. In this work, we show the full equivalence of these structural assumptions to a no-gap second-order condition involving the second subderivative of the Radon norm as well as to a local quadratic growth property of the objective functional with respect to the bounded Lipschitz norm.