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On a globally convergent semismooth* Newton method in nonsmooth nonconvex optimization

H. Gfrerer

TL;DR

This work addresses nonsmooth nonconvex optimization by proposing GSSN, a globalized SCD semismooth$^{*}$ Newton method that solves $0\in\partial\varphi(x)$ with $\varphi=f+g$. The method blends proximal-gradient globalization via the forward-backward envelope with a locally superlinear SCD semismooth$^{*}$ Newton step based on SC-derivative subspaces, enabling efficient Newton directions computed from small linear systems or CG subproblems. Theoretical guarantees include global convergence to stationary points and local superlinear convergence under mild SCD regularity assumptions, even when the Newton direction is computed approximately. Numerical experiments on Signorini contact problems and nonconvex sparse recovery illustrate strong performance and scalability, with competitive comparisons against ZeroFPR, SSNAL, and FISTA, and practical strategies to handle degeneracy and nonconvexity.

Abstract

In this paper we present GSSN, a globalized SCD semismooth* Newton method for solving nonsmooth nonconvex optimization problems. The global convergence properties of the method are ensured by the proximal gradient method, whereas locally superlinear convergence is established via the SCD semismooth* Newton method under quite weak assumptions. The Newton direction is based on the SC (subspace containing) derivative of the subdifferential mapping and can be computed by the (approximate) solution of an equality-constrained quadratic program. Special attention is given to the efficient numerical implementation of the overall method.

On a globally convergent semismooth* Newton method in nonsmooth nonconvex optimization

TL;DR

This work addresses nonsmooth nonconvex optimization by proposing GSSN, a globalized SCD semismooth Newton method that solves with . The method blends proximal-gradient globalization via the forward-backward envelope with a locally superlinear SCD semismooth Newton step based on SC-derivative subspaces, enabling efficient Newton directions computed from small linear systems or CG subproblems. Theoretical guarantees include global convergence to stationary points and local superlinear convergence under mild SCD regularity assumptions, even when the Newton direction is computed approximately. Numerical experiments on Signorini contact problems and nonconvex sparse recovery illustrate strong performance and scalability, with competitive comparisons against ZeroFPR, SSNAL, and FISTA, and practical strategies to handle degeneracy and nonconvexity.

Abstract

In this paper we present GSSN, a globalized SCD semismooth* Newton method for solving nonsmooth nonconvex optimization problems. The global convergence properties of the method are ensured by the proximal gradient method, whereas locally superlinear convergence is established via the SCD semismooth* Newton method under quite weak assumptions. The Newton direction is based on the SC (subspace containing) derivative of the subdifferential mapping and can be computed by the (approximate) solution of an equality-constrained quadratic program. Special attention is given to the efficient numerical implementation of the overall method.
Paper Structure (13 sections, 27 theorems, 150 equations, 1 figure, 6 tables, 3 algorithms)

This paper contains 13 sections, 27 theorems, 150 equations, 1 figure, 6 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Variational analysis
  4. On the forward-backward envelope
  5. On SCD mappings
  6. On the SCD semismooth$^{*}$ Newton method
  7. On the SCD semismooth$^{*}$ Newton step in optimization
  8. Globalizing the SCD semismooth$^{*}$ Newton method
  9. Combining BasGSSN with the SCD semismooth$^{*}$ Newton method (GSSN)
  10. Numerical experiments
  11. Signorini problem with Tresca friction
  12. (Nonconvex) sparse approximation
  13. Conclusion

Key Result

Lemma 2.1

Let $\nabla f$ be Lipschitz continuous with constant $L$ on the convex set $D$. Then for any $x,y\in D$ one has

Figures (1)

  • Figure 1: relative error $\frac{\|{x}^{(k)}-\bar{x}\|_\infty}{\|\bar{x}\|_\infty}$ and relative residuals $\frac{{r}^{(k)}}{{r}^{(0)}}$ for discretization levels $lev=7$ (left) and $lev=10$ (right)

Theorems & Definitions (60)

  • Lemma 2.1: see, e.g., Be17
  • Lemma 2.2: cf. MaTh22
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Proposition 2.5: cf. GfrOut22a
  • Definition 2.6
  • Proposition 2.7: cf.GfrOut22a
  • Lemma 2.8
  • Definition 2.9: tilt-stable minimizers
  • ...and 50 more