A New Notion of Tykhonov Well-Posedness for Optimization Problems

TL;DR

This paper introduces a generalized notion of Tykhonov well-posedness for optimization problems, focusing on specific directions rather than the entire space, with theoretical characterizations and practical implications.

Contribution

It proposes a new directional framework for Tykhonov well-posedness, extending classical concepts and enabling better convergence analysis in optimization.

Findings

Characterizations of directional Tykhonov well-posedness via level set diameters

Relationships between level sets and admissible functions

Examples showing problems well-posed in new sense but not classically

Abstract

Building upon the minimal time function, we propose and study a novel notion of Tykhonov well-posedness with respect to a set of directions for optimization problems. This concept generalizes the classical Tykhonov well-posedness by focusing on existence, stability and convergence along specific directions, rather than over the entire space. We first establish several characterizations of Tykhonov well-posedness with respect to a set of directions, formulated in terms of the diameter of level sets and admissible functions. We then investigate relationships between these level sets and admissible functions. To highlight the advantages of the proposed framework, we present several illustrative examples. In particular, we show that by selecting a suitable set of directions, optimization problems that are not well-posed in the classical sense may still be Tykhonov well-posed with respect to those directions. This viewpoint not only broadens the theoretical landscape of well-posedness but also has practical implications, as it allows numerical methods to be effectively adapted so that the generated sequences converge reliably to minimizers.