Sparse optimization problems in fractional order Sobolev spaces

TL;DR

This paper studies sparse optimization in fractional Sobolev spaces with non-convex $L^p$-pseudonorms, establishing existence, deriving optimality conditions, and developing an algorithm with convergence analysis.

Contribution

It introduces a smoothing scheme for fractional Sobolev space optimization problems with non-convex sparsity terms, providing existence proofs, optimality conditions, and an iterative algorithm.

Findings

Existence of solutions in fractional Sobolev spaces is proven.

A smoothing scheme-based algorithm is developed for the optimization problem.

Weak limit points of the algorithm satisfy a stationarity condition.

Abstract

We consider optimization problems in the fractional order Sobolev spaces $H^s(\Omega)$, $s\in (0,1)$, with sparsity promoting objective functionals containing $L^p$-pseudonorms, $p\in (0,1)$. Existence of solutions is proven. By means of a smoothing scheme, we obtain first-order optimality conditions. An algorithm based on this smoothing scheme is developed. Weak limit points of iterates are shown to satisfy a stationarity system that is slightly weaker than that given by the necessary condition.