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arxiv: 1907.04469 · v1 · pith:AVY46NDDnew · submitted 2019-07-10 · 🧮 math.NA · cs.NA

A family of multi-parameterized proximal point algorithms

Jianchao Bai , Ke Guo , Xiaokai Chang This is my paper

Pith reviewed 2026-05-24 23:56 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords proximal point algorithmmulti-parameterizedrelaxation stepconvex minimizationlinear constraintsvariational inequalityconvergence ratesparse minimization
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The pith

A multi-parameterized proximal point algorithm with relaxation converges globally at sublinear rate for convex minimization under linear constraints.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a family of proximal point algorithms that incorporate multiple parameters and a relaxation step to solve convex minimization problems subject to linear constraints. It establishes global convergence and a sublinear rate of convergence by reformulating the problem as a variational inequality. Preliminary tests on sparse minimization tasks from signal processing indicate better practical performance than some existing methods. A sympathetic reader would care because the extra parameters provide flexibility that could make the method more effective on real constrained problems while retaining provable convergence.

Core claim

The authors develop a multi-parameterized proximal point algorithm combined with a relaxation step for solving convex minimization subject to linear constraints. They prove its global convergence and sublinear convergence rate from the perspective of variational inequality. Preliminary experiments indicate superior performance on sparse minimization problems.

What carries the argument

The multi-parameterized proximal point algorithm combined with a relaxation step, which introduces tunable parameters to handle the proximal mapping and relaxation for the linearly constrained convex problem.

If this is right

  • The algorithm converges globally for the stated class of problems regardless of specific parameter choices within the allowed range.
  • A sublinear convergence rate holds when the problem is recast as a variational inequality.
  • The method applies directly to sparse minimization arising in signal processing.
  • Numerical performance exceeds that of some established methods on the tested instances.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Adaptive selection of the multiple parameters during iteration might improve practical speed without losing the theoretical guarantees.
  • The same structure could be tested on quadratic programs that retain linear constraints.
  • Connections to other operator-splitting schemes might allow hybrid algorithms that inherit the multi-parameter flexibility.

Load-bearing premise

The objective function is convex and the constraints are linear.

What would settle it

The algorithm failing to converge or lacking a sublinear rate on a simple convex objective with linear constraints would falsify the central claim.

Figures

Figures reproduced from arXiv: 1907.04469 by Jianchao Bai, Ke Guo, Xiaokai Chang.

Figure 1
Figure 1. Figure 1: Comparison results by its minimum energy reconstruc [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Convergence behaviors of the residuals LER and LIR by [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Comparison results of the problem (21) with [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
read the original abstract

In this paper, a multi-parameterized proximal point algorithm combining with a relaxation step is developed for solving convex minimization problem subject to linear constraints. We show its global convergence and sublinear convergence rate from the prospective of variational inequality. Preliminary numerical experiments on testing a sparse minimization problem from signal processing indicate that the proposed algorithm performs better than some well-established methods

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript develops a family of multi-parameterized proximal point algorithms that incorporate a relaxation step for solving convex minimization problems subject to linear equality constraints. Global convergence and a sublinear O(1/k) convergence rate are established via a variational inequality reformulation of the problem. Preliminary numerical experiments on a sparse minimization problem from signal processing are reported to show improved performance relative to some established methods.

Significance. If the stated convergence results hold under the given parameter restrictions, the contribution lies in extending the proximal point framework with multiple tunable parameters and relaxation while preserving standard monotonicity arguments from variational inequality theory. The approach is consistent with existing literature on Fejér-monotone schemes for monotone operators. The numerical section, though limited in scope, provides initial evidence of practical utility without contradicting the theory.

minor comments (2)
  1. [Numerical section] The numerical experiments are confined to a single sparse-recovery instance; additional test problems with varying dimensions or conditioning would better support the claim of improved performance over well-established methods.
  2. [Abstract] The abstract and introduction refer to 'some well-established methods' without naming them or specifying the performance metrics used; explicit identification would improve clarity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained in standard VI theory

full rationale

The paper presents a multi-parameter proximal point algorithm with relaxation for convex minimization subject to linear equality constraints. Global convergence and O(1/k) sublinear rate are claimed via the variational inequality reformulation, using monotonicity of the operator and a Lyapunov function that absorbs the extra parameters. This is a standard proof pattern for proximal methods and does not reduce to any self-definition, fitted parameter renamed as prediction, or load-bearing self-citation. The abstract and skeptic summary contain no equations that equate a claimed result to its own inputs by construction. Numerical results on one sparse-recovery instance are presented only as preliminary validation, not as part of the derivation chain. The argument is therefore independent of the paper's own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the convergence argument is stated to rest on standard variational inequality theory for convex problems.

pith-pipeline@v0.9.0 · 5571 in / 959 out tokens · 23452 ms · 2026-05-24T23:56:45.241005+00:00 · methodology

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Reference graph

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