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

Glaydston C. Bento, Boris S. Mordukhovich, Tiago S. Mota, Antoine Soubeyran

TL;DR

This work analyzes a structured nonsmooth, nonconvex minimization problem $L(x,y)=f(x)+Q(x,y)+g(y)$ using an alternating minimization scheme with proximal terms. It advances the theory by establishing refined convergence rates under PLK conditions, including finite termination or superlinear convergence for low-exponent PLK, and develops PLK exponent calculus to determine the effective exponent of $L$ from its components. The authors show both rate results for the objective values $L(x_k,y_k)$ and convergence behavior of the iterate sequences, with rates depending on the PLK exponent $q$; they also provide illustrative examples and discuss extensions to proximal perturbations. Finally, the paper connects these results to applications in noncooperative games and behavioral science, where fast termination and accurate rate predictions improve the practical use of alternating minimization in complex, structured optimization problems.

Abstract

This paper is devoted to developing the alternating minimization algorithm for problems of structured nonconvex optimization proposed by Attouch, Bolté, Redont, and Soubeyran in 2010. Our main result provides significant improvements of the convergence rate of the algorithm, especially under the low exponent Polyak-Łojasiewicz-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.

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

TL;DR

This work analyzes a structured nonsmooth, nonconvex minimization problem using an alternating minimization scheme with proximal terms. It advances the theory by establishing refined convergence rates under PLK conditions, including finite termination or superlinear convergence for low-exponent PLK, and develops PLK exponent calculus to determine the effective exponent of from its components. The authors show both rate results for the objective values and convergence behavior of the iterate sequences, with rates depending on the PLK exponent ; they also provide illustrative examples and discuss extensions to proximal perturbations. Finally, the paper connects these results to applications in noncooperative games and behavioral science, where fast termination and accurate rate predictions improve the practical use of alternating minimization in complex, structured optimization problems.

Abstract

This paper is devoted to developing the alternating minimization algorithm for problems of structured nonconvex optimization proposed by Attouch, Bolté, Redont, and Soubeyran in 2010. Our main result provides significant improvements of the convergence rate of the algorithm, especially under the low exponent Polyak-Łojasiewicz-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.
Paper Structure (5 sections, 6 theorems, 76 equations, 2 figures)

This paper contains 5 sections, 6 theorems, 76 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Basic Assumptions and PLK Conditions
  3. Calculus of PLK Exponents
  4. Main Results
  5. Discussions on Applications to Noncooperative Games and Behavioral Science

Key Result

Lemma 2.1

Under assumptions $(\mathcal{H})$ and $(\mathcal{H}_1)$, the sequences $\{x_k\}$ and $\{y_k\}$ are well-defined. Moreover, the following hold: (i) For all natural numbers $k\in\mathbb{N}$, we have the inequality and therefore the sequence $\{L(x_k,y_k)\}$ is not increasing. (ii) We have the convergent series which implies, in particular, that (iii) For all $k\in\mathbb{N}$, define Then we hav

Figures (2)

  • Figure 1: $\|\nabla L(x,y)\|^3$ and $L(x,y)$.
  • Figure 2: $\| \nabla L(x,y)\|^2$

Theorems & Definitions (13)

  • Lemma 2.1
  • Definition 2.2
  • Example 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Example 3.4
  • Example 3.5
  • Example 3.6
  • Example 3.7
  • Lemma 4.1
  • ...and 3 more