Convergence Rates for the Alternating Minimization Algorithm in Structured Nonsmooth and Nonconvex Optimization

TL;DR

This paper improves the convergence rates of the alternating minimization algorithm for structured nonconvex optimization, achieving finite termination or superlinear convergence under certain conditions, with applications to game theory and behavioral science.

Contribution

It provides new convergence rate results for the algorithm, including finite termination and superlinear convergence under the PLK condition, enhancing understanding of its efficiency.

Findings

Significant convergence rate improvements under PLK condition

Finite termination or superlinear convergence achieved

Applications demonstrated in game theory and behavioral science

Abstract

This paper is devoted to developing the alternating minimization algorithm for problems of structured nonconvex optimization proposed by Attouch, Bolt\'e, Redont, and Soubeyran in 2010. Our main result provides significant improvements of the convergence rate of the algorithm, especially under the low exponent Polyak-{\L}ojasiewicz-Kurdyka condition when we establish either finite termination of this algorithm or its superlinear convergence rate instead of the previously known linear convergence. We also investigate the PLK exponent calculus and discuss applications to noncooperative games and behavioral science.