Best Approximation from the Kuhn-Tucker Set of Composite Monotone Inclusions

TL;DR

This paper introduces strongly convergent algorithms for finding the best approximation to a reference point from the Kuhn-Tucker set in composite monotone inclusion problems, without extra operator assumptions.

Contribution

It presents a novel class of algorithms that do not require knowledge of operator norms or inversions, applicable to general Hilbertian composite monotone inclusions.

Findings

Algorithms achieve strong convergence to the best approximation.

Applicable to systems of coupled monotone inclusions.

No additional assumptions on operators are needed.

Abstract

Kuhn-Tucker points play a fundamental role in the analysis and the numerical solution of monotone inclusion problems, providing in particular both primal and dual solutions. We propose a class of strongly convergent algorithms for constructing the best approximation to a reference point from the set of Kuhn-Tucker points of a general Hilbertian composite monotone inclusion problem. Applications to systems of coupled monotone inclusions are presented. Our framework does not impose additional assumptions on the operators present in the formulation, and it does not require knowledge of the norm of the linear operators involved in the compositions or the inversion of linear operators.