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Norm-induced Cuts: Outer Approximation for Lipschitzian Constraint Functions

Adrian Göß, Alexander Martin, Sebastian Pokutta, Kartikey Sharma

TL;DR

This work tackles constrained Lipschitz optimization with a known Lipschitz constant L by proposing Norm-induced Cuts (NIC), a non-convex outer-approximation framework. NIC iteratively relaxes the constraint set and refines the feasible region using cuts whose radii are proportional to the constraint violation divided by L, requiring a subproblem oracle for global solutions. The authors prove correctness and convergence, analyze termination bounds for infeasible problems, and discuss complexity under ε-approximation, complemented by computational illustrations. The approach generalizes prior one-dimensional Lipschitz methods to multi-dimensional, vector-valued constraints under general norms and accommodates derivative-free settings and surrogate-based constraints, offering a flexible tool for global optimization with Lipschitz constraints. Practical impact lies in enabling existing solvers to handle broad classes of non-convex constraints via structured outer-approximations, with clear guidance on Lipschitz constant computation and subproblem solvability.

Abstract

In this paper, we consider a finite-dimensional optimization problem minimizing a continuous objective on a compact domain subject to a multi-dimensional constraint function. For the latter, we assume the availability of a global Lipschitz constant. In recent literature, methods based on non-convex outer approximation are proposed for tackling one-dimensional equality constraints that are Lipschitz with respect to the maximum norm. To the best of our knowledge, however, there does not exist a non-convex outer approximation method for a general problem class. We introduce a meta-level solution framework to solve such problems and tackle the underlying theoretical foundations. Considering the feasible domain without the constraint function as manageable, our method relaxes the multidimensional constraint and iteratively refines the feasible region by means of norm-induced cuts, relying on an oracle for the resulting subproblems. We show the method's correctness and investigate the problem complexity. In order to account for discussions about functionality, limits, and extensions, we present computational examples including illustrations.

Norm-induced Cuts: Outer Approximation for Lipschitzian Constraint Functions

TL;DR

This work tackles constrained Lipschitz optimization with a known Lipschitz constant L by proposing Norm-induced Cuts (NIC), a non-convex outer-approximation framework. NIC iteratively relaxes the constraint set and refines the feasible region using cuts whose radii are proportional to the constraint violation divided by L, requiring a subproblem oracle for global solutions. The authors prove correctness and convergence, analyze termination bounds for infeasible problems, and discuss complexity under ε-approximation, complemented by computational illustrations. The approach generalizes prior one-dimensional Lipschitz methods to multi-dimensional, vector-valued constraints under general norms and accommodates derivative-free settings and surrogate-based constraints, offering a flexible tool for global optimization with Lipschitz constraints. Practical impact lies in enabling existing solvers to handle broad classes of non-convex constraints via structured outer-approximations, with clear guidance on Lipschitz constant computation and subproblem solvability.

Abstract

In this paper, we consider a finite-dimensional optimization problem minimizing a continuous objective on a compact domain subject to a multi-dimensional constraint function. For the latter, we assume the availability of a global Lipschitz constant. In recent literature, methods based on non-convex outer approximation are proposed for tackling one-dimensional equality constraints that are Lipschitz with respect to the maximum norm. To the best of our knowledge, however, there does not exist a non-convex outer approximation method for a general problem class. We introduce a meta-level solution framework to solve such problems and tackle the underlying theoretical foundations. Considering the feasible domain without the constraint function as manageable, our method relaxes the multidimensional constraint and iteratively refines the feasible region by means of norm-induced cuts, relying on an oracle for the resulting subproblems. We show the method's correctness and investigate the problem complexity. In order to account for discussions about functionality, limits, and extensions, we present computational examples including illustrations.
Paper Structure (21 sections, 8 theorems, 59 equations, 2 figures, 3 tables, 1 algorithm)

This paper contains 21 sections, 8 theorems, 59 equations, 2 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contribution
  3. Related Work
  4. Lipschitz Optimization
  5. Surrogate-based Optimization
  6. Norm-induced Cut Method
  7. Computing Global Lipschitz Constants
  8. Continuously Differentiable Constraint Functions
  9. Bilevel Optimization
  10. Neural Networks
  11. General Functions
  12. Solving \ref{['kth-relaxed-prob']}
  13. Definitions
  14. Convergence and Optimality Results
  15. Termination and Problem Complexity
  16. ...and 6 more sections

Key Result

Lemma 3.2

Consider some $k \in \mathbb{N}_0$ and a solution $\mathbf{x}^k$ to kth-relaxed-prob such that $\mathbf{r}(\mathbf{x}^k) \nleq \textup{0}$. Then, $\mathbf{x}^k \notin Q_{k+1}$, i.e., the point $\mathbf{x}^k$ is infeasible for problem $\mathrm{(P}_{k+1}\mathrm{)}$.

Figures (2)

  • Figure 1: Illustration of \ref{['ex:nic']}.
  • Figure 2: Objective value per iteration of \ref{['alg:nic']} on different versions of \ref{['example_berthold']} when considering $\mathbf{r}$ or its separate components.

Theorems & Definitions (32)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Definition 2.6
  • Definition 2.7
  • Remark 3.1
  • Lemma 3.2
  • proof
  • ...and 22 more