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Global minimisation of nonconvex functions by generalising the mirror descent method

Reinier Díaz Millán, Julien Ugon

TL;DR

The paper extends optimization for abstract convex functions by introducing two conceptual, Bregman-based algorithms: an abstract mirror descent and an abstract proximal-point method. It provides convergence proofs under conditions on the abstract linear function set $L$ (either a vector space or closed under addition) and a suitable $L$-convex potential $\phi$, leveraging an abstract Bregman divergence $D^{\lambda}_\phi$. A key contribution is showing global minimisation of nonconvex problems expressible as an supremum of abstract linear functions, supported by a numerical example on a nonconvex function in one dimension. The work also discusses the classical convex case as a corollary and outlines limitations tied to the subdifferentiability sum-rule and the need to solve subproblems, pointing to future work on structure-exploiting subproblem solvers.

Abstract

In this paper we introduce two conceptual algorithms for minimising abstract convex functions. Both algorithms rely on solving a proximal-type subproblem with an abstract Bregman distance based proximal term. We prove their convergence when the set of abstract linear functions forms a linear space. This latter assumption can be relaxed to only require the set of abstract linear functions to be closed under the sum, which is a classical assumption in abstract convexity. We provide numerical examples on the minimisation of nonconvex functions with the presented algorithms.

Global minimisation of nonconvex functions by generalising the mirror descent method

TL;DR

The paper extends optimization for abstract convex functions by introducing two conceptual, Bregman-based algorithms: an abstract mirror descent and an abstract proximal-point method. It provides convergence proofs under conditions on the abstract linear function set (either a vector space or closed under addition) and a suitable -convex potential , leveraging an abstract Bregman divergence . A key contribution is showing global minimisation of nonconvex problems expressible as an supremum of abstract linear functions, supported by a numerical example on a nonconvex function in one dimension. The work also discusses the classical convex case as a corollary and outlines limitations tied to the subdifferentiability sum-rule and the need to solve subproblems, pointing to future work on structure-exploiting subproblem solvers.

Abstract

In this paper we introduce two conceptual algorithms for minimising abstract convex functions. Both algorithms rely on solving a proximal-type subproblem with an abstract Bregman distance based proximal term. We prove their convergence when the set of abstract linear functions forms a linear space. This latter assumption can be relaxed to only require the set of abstract linear functions to be closed under the sum, which is a classical assumption in abstract convexity. We provide numerical examples on the minimisation of nonconvex functions with the presented algorithms.
Paper Structure (8 sections, 8 theorems, 36 equations, 3 figures, 1 table, 2 algorithms)

This paper contains 8 sections, 8 theorems, 36 equations, 3 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Abstract Bregman Divergence
  4. Abstract Mirror Descent Algorithm
  5. Implications for the classical convex case
  6. Numerical Experiments
  7. Proximal point method
  8. Conclusion

Key Result

Proposition 1

The $L$-convex hull of a set $C$ is $L$-convex. More specifically, it is the smallest $L$-convex set containing $C$, and $\operatorname{co}_L C = \operatorname{supp}(\sup_{l\in C} l,L)$.

Figures (3)

  • Figure 1: Plot of the test function for the first experiment
  • Figure 2: Function values at the iterates for the Mirror descent method.
  • Figure 3: Iterates for $x_0=-5$ and $c_n=2/n^{0.8}$.

Theorems & Definitions (23)

  • Definition 1: Abstract convex function rubinov:2000
  • Definition 2: Abstract Convex Hull of a set rubinov:2000
  • Proposition 1: rubinov:2000
  • Definition 3: Abstract subdifferential rubinov:2000
  • Definition 4: $L$-conjugate
  • Proposition 2: rubinov:2000
  • Theorem 1: Sum Rule diaz-millan.ea:2025
  • Definition 5: Abstract Bregman Divergence grasmair:2010
  • Lemma 1
  • proof
  • ...and 13 more