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Hybrid subgradient and simulated annealing method for hemivariational inequalities

Piotr Bartman-Szwarc, Adil M. Bagirov, Anna Ochal

TL;DR

Addresses numerical solution of hemivariational inequalities in contact mechanics, which are nonsmooth and nonconvex. Proposes a hybrid method that merges a local aggregate subgradient search with a global simulated annealing procedure to identify high-quality starting points for local minimization of the energy function $\mathcal{L}(\mathbf{u}) = \frac{1}{2}\langle A\mathbf{u}, \mathbf{u}\rangle + \langle \mathbf{b}, \mathbf{u}\rangle + J(\mathbf{u})$. The authors provide convergence analysis, including finite termination of the inner descent loop and a bound on iterations to reach an $(\eta,\delta)$-stationary point, and show how simulated annealing facilitates escaping local minima. Numerical experiments on a 2D beam contact problem with multi-layer foundations demonstrate that the global subgradient variant reliably attains near-global minima faster than several classical solvers, and the implementation is open-source in a Python package conmech.

Abstract

In this paper, we employ a global aggregate subgradient method for the numerical solution of hemivariational inequality problems arising in contact mechanics. The method integrates a global search procedure to identify starting points for a local minimization algorithm. The algorithm consists of two types of steps: null steps and serious steps. In each null step, only two subgradients are utilized: the aggregate subgradient and the subgradient computed at the current iteration point, which together determine the search direction. Furthermore, we compare the performance of the proposed method with selected solvers using a representative contact mechanics problem as a case study.

Hybrid subgradient and simulated annealing method for hemivariational inequalities

TL;DR

Addresses numerical solution of hemivariational inequalities in contact mechanics, which are nonsmooth and nonconvex. Proposes a hybrid method that merges a local aggregate subgradient search with a global simulated annealing procedure to identify high-quality starting points for local minimization of the energy function . The authors provide convergence analysis, including finite termination of the inner descent loop and a bound on iterations to reach an -stationary point, and show how simulated annealing facilitates escaping local minima. Numerical experiments on a 2D beam contact problem with multi-layer foundations demonstrate that the global subgradient variant reliably attains near-global minima faster than several classical solvers, and the implementation is open-source in a Python package conmech.

Abstract

In this paper, we employ a global aggregate subgradient method for the numerical solution of hemivariational inequality problems arising in contact mechanics. The method integrates a global search procedure to identify starting points for a local minimization algorithm. The algorithm consists of two types of steps: null steps and serious steps. In each null step, only two subgradients are utilized: the aggregate subgradient and the subgradient computed at the current iteration point, which together determine the search direction. Furthermore, we compare the performance of the proposed method with selected solvers using a representative contact mechanics problem as a case study.

Paper Structure

This paper contains 7 sections, 2 theorems, 36 equations, 4 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. A subgradient method for hemivariational inequalities
  3. Contact Mechanics Problem
  4. Numerical results
  5. Conclusion
  6. Acknowledgments.
  7. Disclosure of Interests.

Key Result

Proposition 1

Suppose that $\mathcal{L}\colon \mathbb{R}^n\to\mathbb{R}$ is a locally Lipschitz function, ${\mathbf{u}} \in \mathbb{R}^n$, $\eta > 0$ and the constant $C_1 < +\infty$ is such that If $c_1 \in (0,1)$ and $\delta \in (0,C_1)$, then the inner loop in Algorithm alg:combined terminates after finite many iterations $m > 0$, where

Figures (4)

  • Figure 1: Modeled beam under loading.
  • Figure 2: A nonmonotone contact law and the corresponding functional $j(u)$.
  • Figure 3: Deflection of the beam under loading.
  • Figure 4: Energy function values for selected simulations (left column) and the corresponding computation time (right column).

Theorems & Definitions (5)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • remark thmcounterremark