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A Newton-type Method for Non-smooth Under-determined Systems of Equations

Titus Pinta

TL;DR

This work extends Newton-type methods to under-determined, non-smooth systems by introducing uniform Newton differentiability and a Newton-differential framework. The method linearizes around the current iterate using a Newton differential and projects onto the resulting affine solution set, with a geometric compatibility condition ensuring well-defined projections. The authors prove that the distance to the zero set decreases super-linearly under these conditions and provide corollaries for smooth and semi-smooth cases, accompanied by numerical experiments including a complementarity-constraint toy problem. The results suggest strong potential for engineering applications and offer avenues for future work, such as quasi-Newton variants and weaker regularity assumptions.

Abstract

We study a variant of Newton's algorithm applied to under-determined systems of non-smooth equations. The notion of regularity employed in our work is based on Newton differentiability, which generalizes semi-smoothness. The classic notion of Newton differentiability does not suffice for our purpose, due to the existence of multiple zeros and as such we extend it to uniform Newton differentiability. In this context, we can show that the distance between the iterates and the set of zeros of the system decreases super-linearly. For the special case of smooth equations, the assumptions of our algorithm are simplified. Finally, we provide some numerical examples to showcase the behavior of our proposed method. The key example is a toy model of complementarity constraint problems, showing that our method has great application potential across engineering fields.

A Newton-type Method for Non-smooth Under-determined Systems of Equations

TL;DR

This work extends Newton-type methods to under-determined, non-smooth systems by introducing uniform Newton differentiability and a Newton-differential framework. The method linearizes around the current iterate using a Newton differential and projects onto the resulting affine solution set, with a geometric compatibility condition ensuring well-defined projections. The authors prove that the distance to the zero set decreases super-linearly under these conditions and provide corollaries for smooth and semi-smooth cases, accompanied by numerical experiments including a complementarity-constraint toy problem. The results suggest strong potential for engineering applications and offer avenues for future work, such as quasi-Newton variants and weaker regularity assumptions.

Abstract

We study a variant of Newton's algorithm applied to under-determined systems of non-smooth equations. The notion of regularity employed in our work is based on Newton differentiability, which generalizes semi-smoothness. The classic notion of Newton differentiability does not suffice for our purpose, due to the existence of multiple zeros and as such we extend it to uniform Newton differentiability. In this context, we can show that the distance between the iterates and the set of zeros of the system decreases super-linearly. For the special case of smooth equations, the assumptions of our algorithm are simplified. Finally, we provide some numerical examples to showcase the behavior of our proposed method. The key example is a toy model of complementarity constraint problems, showing that our method has great application potential across engineering fields.

Paper Structure

This paper contains 9 sections, 10 theorems, 82 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Notation, Definitions and Basic Properties
  3. Newton Differentiability
  4. Examples
  5. Under-determined Problems
  6. Application: (Semi)-Smooth Systems
  7. Numerical Experiments
  8. Conclusions and Further Works
  9. Acknowledgments

Key Result

Proposition 1.5

If $A \in \mathbb{R}^{m {\times} n}$, then

Figures (4)

  • Figure 1: One Step of a Newton-type Method for Under-determined Problems
  • Figure 2: Objective value, $\|G(x^k)\|$ over $k$ for our Algorithm and PolTre17Solv, observing nearly identical behavior due to the smoothness of the problem
  • Figure 3: Objective value, $\|G(x^k)\|$ over $k$ for out Algorithm applied to the problems from Kub11Inte, observing quadratic convergence for all instances, except for the infeasible problem
  • Figure 4: Objective value, $\|G(x^k)\|$ over $k$ for our Algorithm applied to a toy model steaming from the non-smooth formulation of a complementarity constraint problem, observing just super-linear convergence and not quadratic convergence

Theorems & Definitions (39)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4: Pseudo-Inverse
  • Proposition 1.5
  • Definition 2.1: Newton Differentiability
  • Definition 2.2: Uniform Newton Differentiability
  • Remark 2.3
  • Remark 2.4
  • Proposition 2.5: Newton Differentiability of Smooth Functions
  • ...and 29 more