Convergence analysis of the semismooth Newton method for sparse control problems governed by semilinear elliptic equations

TL;DR

This paper proves superlinear convergence of the semismooth Newton method for sparse control problems governed by semilinear elliptic equations, under certain optimality and complementarity conditions, even with sparsity terms and box constraints.

Contribution

It establishes convergence results for the semismooth Newton method in the context of sparse control problems with complex constraints, extending previous analyses.

Findings

Superlinear convergence achieved under second order sufficient and strict complementarity conditions.

Applicable to problems with sparsity-promoting terms and box constraints.

Provides theoretical foundation for efficient numerical solutions of sparse control problems.

Abstract

We show that a second order sufficient condition for local optimality, along with a strict complementarity condition, is enough to get the superlinear convergence of the semismooth Newton method for an optimal control problem governed by a semilinear elliptic equation. The objective functional may include a sparsity promoting term and we allow for box control constraints.