Subdifferentiation of nonconvex sparsity-promoting functionals on Lebesgue spaces

TL;DR

This paper derives exact formulas for generalized derivatives of nonconvex, non-Lipschitz sparsity-promoting functionals on Lebesgue spaces, aiding in optimality condition derivation for control problems.

Contribution

It provides explicit formulas for subdifferentials of nonconvex sparsity functionals, facilitating their use in optimal control analysis.

Findings

Derived formulas for Fréchet, limiting, and singular subdifferentials.

Enabled necessary optimality conditions for control problems with sparsity terms.

Addressed variational challenges of nonconvex, non-Lipschitz functionals.

Abstract

Sparsity-promoting terms are incorporated into the objective functions of optimal control problems in order to ensure that optimal controls vanish on large parts of the underlying domain. Typical candidates for those terms are integral functions on Lebesgue spaces based on the $\ell_p$-metric for $p\in[0,1)$ which are nonconvex as well as non-Lipschitz and, thus, variationally challenging. In this paper, we derive exact formulas for the Fr\'echet, limiting, and singular subdifferential of these functionals. These generalized derivatives can be used for the derivation of necessary optimality conditions for optimal control problems comprising such sparsity-promoting terms.