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The alternating simultaneous Halpern-Lions-Wittmann-Bauschke algorithm for finding the best approximation pair for two disjoint intersections of convex sets

Yair Censor, Rafiq Mansour, Daniel Reem

TL;DR

The paper addresses finding a best approximation pair for two disjoint intersections $A=\bigcap_{i}A_i$ and $B=\bigcap_{j}B_j$ of convex sets in a finite-dimensional Euclidean space. It introduces the alternating Simultaneous-HLWB (A-S-HLWB) algorithm, which uses projections onto the generating sets $\{A_i\}$ and $\{B_j\}$ rather than directly onto the intersections, via a pair of weighted, averaged projection operators with increasing sweep counts. Under conditions such as compactness and strict convexity of the generators (ensuring a unique best approximation pair), the algorithm’s odd and even subsequences converge to a and b in $A$ and $B$, respectively, and $(a,b)$ achieves $\mathrm{dist}(A,B)$. The analysis combines fixed-point theory for averaged nonexpansive operators with a generalized Dini-type uniform convergence result to establish convergence of the composed operators to $P_BP_A$ and $P_AP_B$. The work extends prior results for polyhedra to general intersections of convex sets and outlines directions for relaxing assumptions and exploring rates and infinite-dimensional extensions.

Abstract

Given two nonempty and disjoint intersections of closed and convex subsets, we look for a best approximation pair relative to them, i.e., a pair of points, one in each intersection, attaining the minimum distance between the disjoint intersections. We propose an iterative process based on projections onto the subsets which generate the intersections. The process is inspired by the Halpern-Lions-Wittmann-Bauschke algorithm and the classical alternating process of Cheney and Goldstein, and its advantage is that there is no need to project onto the intersections themselves, a task which can be rather demanding. We prove that under certain conditions the two interlaced subsequences converge to a best approximation pair. These conditions hold, in particular, when the space is Euclidean and the subsets which generate the intersections are compact and strictly convex. Our result extends the one of Aharoni, Censor and Jiang ["Finding a best approximation pair of points for two polyhedra'', Computational Optimization and Applications 71 (2018), 509--523] who considered the case of finite-dimensional polyhedra.

The alternating simultaneous Halpern-Lions-Wittmann-Bauschke algorithm for finding the best approximation pair for two disjoint intersections of convex sets

TL;DR

The paper addresses finding a best approximation pair for two disjoint intersections and of convex sets in a finite-dimensional Euclidean space. It introduces the alternating Simultaneous-HLWB (A-S-HLWB) algorithm, which uses projections onto the generating sets and rather than directly onto the intersections, via a pair of weighted, averaged projection operators with increasing sweep counts. Under conditions such as compactness and strict convexity of the generators (ensuring a unique best approximation pair), the algorithm’s odd and even subsequences converge to a and b in and , respectively, and achieves . The analysis combines fixed-point theory for averaged nonexpansive operators with a generalized Dini-type uniform convergence result to establish convergence of the composed operators to and . The work extends prior results for polyhedra to general intersections of convex sets and outlines directions for relaxing assumptions and exploring rates and infinite-dimensional extensions.

Abstract

Given two nonempty and disjoint intersections of closed and convex subsets, we look for a best approximation pair relative to them, i.e., a pair of points, one in each intersection, attaining the minimum distance between the disjoint intersections. We propose an iterative process based on projections onto the subsets which generate the intersections. The process is inspired by the Halpern-Lions-Wittmann-Bauschke algorithm and the classical alternating process of Cheney and Goldstein, and its advantage is that there is no need to project onto the intersections themselves, a task which can be rather demanding. We prove that under certain conditions the two interlaced subsequences converge to a best approximation pair. These conditions hold, in particular, when the space is Euclidean and the subsets which generate the intersections are compact and strictly convex. Our result extends the one of Aharoni, Censor and Jiang ["Finding a best approximation pair of points for two polyhedra'', Computational Optimization and Applications 71 (2018), 509--523] who considered the case of finite-dimensional polyhedra.
Paper Structure (9 sections, 23 theorems, 59 equations, 1 figure, 1 algorithm)

This paper contains 9 sections, 23 theorems, 59 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background:
  3. Paper layout:
  4. Preliminaries
  5. A sufficient condition for the existence of a unique best approximation pair
  6. The Simultaneous-HLWB projection operator
  7. The alternating simultaneous HLWB algorithm
  8. Convergence of the A-S-HLWB algorithm
  9. Concluding remarks and lines for further investigation

Key Result

Lemma 2

Ceg-book Given $m\in\mathbb{N},$ let $S_{i}:X\rightarrow X$, be NE for all $i\in\left\{ 1,2,\ldots,m\right\}$ and let $w\in\Delta_{m}$. Then:

Figures (1)

  • Figure 1: A case where there is a unique best approximation pair $(a,b)$ despite the fact that $A$ and $B$ are not strictly convex

Theorems & Definitions (48)

  • Definition 1
  • Lemma 2
  • Proposition 3
  • proof
  • Definition 4
  • Remark 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Proposition 9
  • ...and 38 more