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A feasible smoothing accelerated projected gradient method for nonsmooth convex optimization

Akatsuki Nishioka, Yoshihiro Kanno

TL;DR

The paper presents a feasible, smoothing-based accelerated first-order method for nonsmooth convex optimization by ensuring all variables remain feasible at every step. By replacing Nesterov-style extrapolation with convex combinations and employing a Lyapunov-based analysis, the authors prove an $O(k^{-1}\log k)$ convergence rate for the smoothing-accelerated projected gradient (S-APG) method. The smoothing framework uses an adaptive parameter $\mu_k$ and a log-sum-exp approximation to handle nonsmooth objectives, with a quantified relationship between smoothed and original objectives via $\beta$. A robust truss compliance optimization example demonstrates that S-APG can outperform smoothing projected gradient and subgradient methods in large-scale, structure-optimization problems, highlighting the method’s practical impact for feasible-variable nonsmooth convex problems.

Abstract

Smoothing accelerated gradient methods achieve faster convergence rates than that of the subgradient method for some nonsmooth convex optimization problems. However, Nesterov's extrapolation may require gradients at infeasible points, and thus they cannot be applied to some structural optimization problems. We introduce a variant of smoothing accelerated projected gradient methods where every variable is feasible. The $O(k^{-1}\log k)$ convergence rate is obtained using the Lyapunov function. We conduct a numerical experiment on the robust compliance optimization of a truss structure.

A feasible smoothing accelerated projected gradient method for nonsmooth convex optimization

TL;DR

The paper presents a feasible, smoothing-based accelerated first-order method for nonsmooth convex optimization by ensuring all variables remain feasible at every step. By replacing Nesterov-style extrapolation with convex combinations and employing a Lyapunov-based analysis, the authors prove an convergence rate for the smoothing-accelerated projected gradient (S-APG) method. The smoothing framework uses an adaptive parameter and a log-sum-exp approximation to handle nonsmooth objectives, with a quantified relationship between smoothed and original objectives via . A robust truss compliance optimization example demonstrates that S-APG can outperform smoothing projected gradient and subgradient methods in large-scale, structure-optimization problems, highlighting the method’s practical impact for feasible-variable nonsmooth convex problems.

Abstract

Smoothing accelerated gradient methods achieve faster convergence rates than that of the subgradient method for some nonsmooth convex optimization problems. However, Nesterov's extrapolation may require gradients at infeasible points, and thus they cannot be applied to some structural optimization problems. We introduce a variant of smoothing accelerated projected gradient methods where every variable is feasible. The convergence rate is obtained using the Lyapunov function. We conduct a numerical experiment on the robust compliance optimization of a truss structure.
Paper Structure (5 sections, 3 theorems, 27 equations, 2 figures, 1 algorithm)

This paper contains 5 sections, 3 theorems, 27 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem setting
  3. Motivating example
  4. Proposed method
  5. Numerical result

Key Result

Proposition 1

Let $A(\bm{x})=Q^\mathrm{T} K(\bm{x})^{-1}Q\in\mathbb{S}^n$ and $S$ be the feasible set of p_robust, which is compact. The smooth approximation $f_{\mu}$ defined by approx is convex and $\nabla f_{\mu}$ is $(L'+L/\mu)$-Lipschitz continuous for some $L,L'>0$ over $S$. Moreover, it satisfies $0 \le f_

Figures (2)

  • Figure 1: Comparison of the designs after 4000 iterations
  • Figure 6: Difference of the objective value and the optimal value at each iteration

Theorems & Definitions (8)

  • Proposition 1
  • proof
  • Lemma 1
  • proof
  • Remark 1
  • Theorem 1
  • proof
  • Remark 2