On definable multifunctions and {\L}ojasiewicz inequalities

TL;DR

This paper explores conditions under which { extL}ojasiewicz inequalities hold for definable multifunctions, demonstrating definability of the Hausdorff distance and deriving inequalities related to subgradients and tangent cones.

Contribution

It introduces new { extL}ojasiewicz inequalities for definable multifunctions, including those involving Clarke's subgradient and tangent cones, expanding the theoretical framework.

Findings

Hausdorff distance is definable with definable multifunctions

{ extL}ojasiewicz inequalities hold for subgradients and tangent cones

Establishes a { extL}ojasiewicz-type subgradient inequality

Abstract

We investigate several possibilities of obtaining a {\L}ojasiewicz inequality for definable multifunctions and give some examples of applications thereof. In particular, we prove that the Hausdorff distance and its extension to closed sets is definable when composed with definable multifunctions. This allows us to obtain {\L}ojasiewicz-type inequalities for definable multifunctions obtained from Clarke's subgradient or the tangent cone. The paper ends with a {\L}ojasiewicz-type subgradient inequality in the spirit of Bolte-Daniilidis-Lewis-Shiota or Ph\d{a}m.