How Stringent is the Linear Independence Kink Qualification in Abs-Smooth Optimization?

TL;DR

This paper investigates the linear independence kink qualification (LIKQ) in abs-smooth optimization, demonstrating its generic nature and implications for deriving optimality conditions.

Contribution

It proves that LIKQ is a generic assumption in abs-smooth optimization problems using differential topology tools.

Findings

LIKQ holds generically for all feasible points

Optimality conditions can be derived in polynomial time when LIKQ is satisfied

LIKQ generalizes LICQ from smooth to abs-smooth optimization

Abstract

Abs-smooth functions are given by compositions of smooth functions and the evaluation of the absolute value. The linear independence kink qualification (LIKQ) is a fundamental assumption in optimization problems governed by these abs-smooth functions, generalizing the well-known LICQ from smooth optimization. In particular, provided that LIKQ holds it is possible to derive optimality conditions for abs-smooth optimization problems that can be checked in polynomial time. Utilizing tools from differential topology, namely a version of the jet-transversality theorem, it is shown that assuming LIKQ for all feasible points of an abs-smooth optimization problem is a generic assumption.