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A monotone block coordinate descent method for solving absolute value equations

Tingting Luo, Jiayu Liu, Cairong Chen, Qun Wang

TL;DR

The paper tackles solving the absolute value equation $A x - |x| = b$, a problem linked to generalized AVEs and known to be challenging. It introduces a monotone block coordinate descent algorithm (BCDA) that minimizes the auxiliary function $f(x)=x^T A x - |x|^T x - 2 b^T x$ along pairs of coordinate directions, providing closed-form step updates and ensuring monotone descent. With $A-I \succ 0$, BCDA is proven to converge to the AVE solution via level-set compactness and descent arguments, improving upon prior nonmonotone methods. Numerical experiments show BCDA's monotone convergence and superior performance against MGSM in SPD and certain non-SPD scenarios, highlighting its practical efficacy for AVE solving in numerical optimization.

Abstract

In this paper, we proposed a monotone block coordinate descent method for solving absolute value equation (AVE). Under appropriate conditions, we analyzed the global convergence of the algorithm and conduct numerical experiments to demonstrate its feasibility and effectiveness.

A monotone block coordinate descent method for solving absolute value equations

TL;DR

The paper tackles solving the absolute value equation , a problem linked to generalized AVEs and known to be challenging. It introduces a monotone block coordinate descent algorithm (BCDA) that minimizes the auxiliary function along pairs of coordinate directions, providing closed-form step updates and ensuring monotone descent. With , BCDA is proven to converge to the AVE solution via level-set compactness and descent arguments, improving upon prior nonmonotone methods. Numerical experiments show BCDA's monotone convergence and superior performance against MGSM in SPD and certain non-SPD scenarios, highlighting its practical efficacy for AVE solving in numerical optimization.

Abstract

In this paper, we proposed a monotone block coordinate descent method for solving absolute value equation (AVE). Under appropriate conditions, we analyzed the global convergence of the algorithm and conduct numerical experiments to demonstrate its feasibility and effectiveness.

Paper Structure

This paper contains 5 sections, 6 theorems, 46 equations, 2 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Monotone block coordinate descent method
  4. Numerical results
  5. Conclusion

Key Result

Lemma 2.1

The function $\nabla f$ defined as in eq:diff is Lipschitz continuous on $\mathbb{R}^n$ with the Lipschitz constant $2(\|A\|+1)$.

Figures (2)

  • Figure 1: Iterative trajectory of the algorithm.
  • Figure 2: Plot of relative residuals for Example \ref{['exam:4.3']}.

Theorems & Definitions (15)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.1
  • proof
  • Example 3.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 5 more