Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A note on the convergence of deterministic gradient sampling in nonsmooth optimization

Bennet Gebken

TL;DR

This article proposes a bisection method for weakly lower semismooth functions which is able to compute new subgradients that improve a given approximation in case a direction with insufficient descent was computed.

Abstract

Approximation of subdifferentials is one of the main tasks when computing descent directions for nonsmooth optimization problems. In this article, we propose a bisection method for weakly lower semismooth functions which is able to compute new subgradients that improve a given approximation in case a direction with insufficient descent was computed. Combined with a recently proposed deterministic gradient sampling approach, this yields a deterministic and provably convergent way to approximate subdifferentials for computing descent directions.

A note on the convergence of deterministic gradient sampling in nonsmooth optimization

TL;DR

This article proposes a bisection method for weakly lower semismooth functions which is able to compute new subgradients that improve a given approximation in case a direction with insufficient descent was computed.

Abstract

Approximation of subdifferentials is one of the main tasks when computing descent directions for nonsmooth optimization problems. In this article, we propose a bisection method for weakly lower semismooth functions which is able to compute new subgradients that improve a given approximation in case a direction with insufficient descent was computed. Combined with a recently proposed deterministic gradient sampling approach, this yields a deterministic and provably convergent way to approximate subdifferentials for computing descent directions.
Paper Structure (10 sections, 3 theorems, 34 equations, 2 figures, 1 table, 2 algorithms)

This paper contains 10 sections, 3 theorems, 34 equations, 2 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Computing descent directions for nonsmooth functions
  3. Random gradient sampling
  4. Deterministic gradient sampling
  5. Improved bisection method
  6. Derivation of the improved method
  7. Proof of termination
  8. Example
  9. Conclusion and future work
  10. A drawback of random sampling

Key Result

Lemma 1

If Algorithm algo:new_bisection does not terminate, then

Figures (2)

  • Figure 1: The graphs of (a) $\varphi = h$, (b) $\partial \varphi = \partial h$ and (c) $f$ in Example \ref{['example:counterexample']}. The red lines show the values of $t_j = 1 - 2^{-j}$ for $j \in \mathbb{N}$.
  • Figure 4: (a) The graph of $f$ for $n = 2$ in Example \ref{['example:GS_bad']}. The red lines indicate the points in which $f$ is not differentiable. (b) The boundary of the unit sphere $B_1(0)$ (dashed) and the sets $D^1$ (green) and $D^2$ (blue).

Theorems & Definitions (9)

  • Example 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Example 2
  • Example 3