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Steering exact penalty DCA for nonsmooth DC optimization problems with equality and inequality constraints

M. V. Dolgopolik

Abstract

We propose and study a version of the DCA (Difference-of-Convex functions Algorithm) using the $\ell_1$ penalty function for solving nonsmooth DC optimization problems with nonsmooth DC equality and inequality constraints. The method employs an adaptive penalty updating strategy to improve its performance. This strategy is based on the so-called steering exact penalty methodology and relies on solving some auxiliary convex subproblems to determine a suitable value of the penalty parameter. We present a detailed convergence analysis of the method and illustrate its practical performance by applying the method to two nonsmooth discrete optimal control problem.

Steering exact penalty DCA for nonsmooth DC optimization problems with equality and inequality constraints

Abstract

We propose and study a version of the DCA (Difference-of-Convex functions Algorithm) using the penalty function for solving nonsmooth DC optimization problems with nonsmooth DC equality and inequality constraints. The method employs an adaptive penalty updating strategy to improve its performance. This strategy is based on the so-called steering exact penalty methodology and relies on solving some auxiliary convex subproblems to determine a suitable value of the penalty parameter. We present a detailed convergence analysis of the method and illustrate its practical performance by applying the method to two nonsmooth discrete optimal control problem.

Paper Structure

This paper contains 10 sections, 11 theorems, 96 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Critical points of DC optimisation problems
  3. Steering exact penalty DCA
  4. A description of the algorithm
  5. A discussion of the method
  6. Correctness of the method
  7. Convergence of the infeasibility measure
  8. Convergence to critical points
  9. Sufficient conditions for the boundedness of the penalty parameter
  10. Numerical experiments

Key Result

Lemma \oldthetheorem

A feasible point $x_*$ is critical for the problem $(\mathcal{P})$ if and only if there exist $c_* > 0$, $v_k \in \partial h_k(x_*)$, $k \in \{ 0 \} \cup \mathcal{I} \cup \mathcal{E}$, and $w_j \in \partial g_j(x_*)$, $j \in \mathcal{E}$, such that for any $c \ge c_*$ the point $x_*$ is a global min on the set $A$.

Figures (1)

  • Figure 1: The "optimal" tachogram.

Theorems & Definitions (34)

  • Definition \oldthetheorem
  • Remark 2.1
  • Lemma \oldthetheorem
  • proof
  • Definition \oldthetheorem
  • Remark 2.2
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • ...and 24 more