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Proximal Algorithms for a class of abstract convex functions

Ewa Bednarczuk, Dirk Lorenz, The Hung Tran

Abstract

In this paper we analyze a class of nonconvex optimization problem from the viewpoint of abstract convexity. Using the respective generalizations of the subgradient we propose an abstract notion proximal operator and derive a number of algorithms, namely an abstract proximal point method, an abstract forward-backward method and an abstract projected subgradient method. Global convergence results for all algorithms are discussed and numerical examples are given

Proximal Algorithms for a class of abstract convex functions

Abstract

In this paper we analyze a class of nonconvex optimization problem from the viewpoint of abstract convexity. Using the respective generalizations of the subgradient we propose an abstract notion proximal operator and derive a number of algorithms, namely an abstract proximal point method, an abstract forward-backward method and an abstract projected subgradient method. Global convergence results for all algorithms are discussed and numerical examples are given
Paper Structure (15 sections, 17 theorems, 177 equations, 3 figures, 4 algorithms)

This paper contains 15 sections, 17 theorems, 177 equations, 3 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Background and state of the art
  3. Notation
  4. Outline
  5. $\Phi_{lsc}^\mathbb{R}$-Convexity and $\Phi_{lsc}^\mathbb{R}$-Subdifferentials
  6. $\Phi_{lsc}^\mathbb{R}$-Proximal Operator
  7. $\Phi_{lsc}^\mathbb{R}$-Proximal Point Algorithm
  8. Auxiliary Convergence Results
  9. $\Phi_{lsc}^\mathbb{R}$-Proximal Point Algorithm
  10. $\Phi_{lsc}^\mathbb{R}$-Forward Backward Algorithm
  11. Projected Subgradient for $\Phi_{lsc}^\mathbb{R}$-convex function
  12. Numerical Examples
  13. $\Phi_{lsc}^\mathbb{R}$-Proximal Point Algorithm:
  14. $\Phi_{lsc}^\mathbb{R}$-Projected Subgradient Algorithm:
  15. Funding

Key Result

Proposition 2.2

Let $X$ be a Hilbert space, $f:X\to (-\infty,+\infty]$ be proper. We have ${\Phi_{lsc}^{\geq} \subset \Phi_{lsc}^\mathbb{R} }$. If $f$ is $\Phi_{lsc}^\mathbb{R}$-convex and there exists $\phi\in\Phi_{lsc}^\mathbb{R}$ with $a_\phi <0$ and then $\lim_{\Vert x\Vert \to +\infty} f(x) = +\infty$ and there exists $\psi\in\Phi_{lsc}^{\geq}$ such that which implies $f$ is $\Phi_{lsc}^{\geq}$-convex.

Figures (3)

  • Figure 1: $\Phi_{lsc}^\mathbb{R}$-Subgradient of $f$ at different points
  • Figure 2: From left to right: distance between the current iteration and the minimizer; distance to the optimal value; distance between two successive iterates.
  • Figure 8: $\Phi_{lsc}^\mathbb{R}$-(PSG) for Example \ref{['ex: negative hessian function']}. From left to right: distance between the current iterate and the solution; function value at each iteration.

Theorems & Definitions (50)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Remark 2.4
  • Example 2.5
  • Example 2.6
  • Example 2.7
  • Proposition 2.8
  • proof
  • ...and 40 more