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Newton and interior-point methods for (constrained) nonconvex-nonconcave minmax optimization with stability and instability guarantees

Raphael Chinchilla, Guosong Yang, Joao P. Hespanha

TL;DR

This work tackles the challenge of finding local solutions to nonconvex-nonconcave minmax problems by developing Newton-type updates whose steps are derived from local quadratic approximations. By carefully modifying the Hessian and employing Local Quadratic Approximation Conditions, the authors guarantee that equilibria attractors correspond to local minmax points, while non-minmax equilibria are unstable. The framework extends to constrained problems via primal-dual interior-point methods, with analogous stability guarantees and inertia-based sufficiency conditions. Numerical experiments, including unconstrained benchmarks and a constrained homicidal chauffeur MPC example, demonstrate that the proposed approach reliably converges to local minmax points and scales efficiently when leveraging Hessian sparsity. The methods offer practical tools for robust optimization and adversarial settings, with potential extensions to nonlocal convergence analyses and merit-function schemes.

Abstract

We address the problem of finding a local solution to a nonconvex-nonconcave minmax optimization using Newton type methods, including interior-point ones. We modify the Hessian matrix of these methods such that, at each step, the modified Newton update direction can be seen as the solution to a quadratic program that locally approximates the minmax problem. Moreover, we show that by selecting the modification in an appropriate way, the only stable equilibrium points of the algorithm's iterations are local minmax points. As a consequence, the algorithm can only converge towards an equilibrium point if such point is a local minmax, and it will escape if the point is not a local minmax. Using numerical examples, we show that the computation time of our algorithm scales roughly linearly with the number of nonzero elements in the Hessian.

Newton and interior-point methods for (constrained) nonconvex-nonconcave minmax optimization with stability and instability guarantees

TL;DR

This work tackles the challenge of finding local solutions to nonconvex-nonconcave minmax problems by developing Newton-type updates whose steps are derived from local quadratic approximations. By carefully modifying the Hessian and employing Local Quadratic Approximation Conditions, the authors guarantee that equilibria attractors correspond to local minmax points, while non-minmax equilibria are unstable. The framework extends to constrained problems via primal-dual interior-point methods, with analogous stability guarantees and inertia-based sufficiency conditions. Numerical experiments, including unconstrained benchmarks and a constrained homicidal chauffeur MPC example, demonstrate that the proposed approach reliably converges to local minmax points and scales efficiently when leveraging Hessian sparsity. The methods offer practical tools for robust optimization and adversarial settings, with potential extensions to nonlocal convergence analyses and merit-function schemes.

Abstract

We address the problem of finding a local solution to a nonconvex-nonconcave minmax optimization using Newton type methods, including interior-point ones. We modify the Hessian matrix of these methods such that, at each step, the modified Newton update direction can be seen as the solution to a quadratic program that locally approximates the minmax problem. Moreover, we show that by selecting the modification in an appropriate way, the only stable equilibrium points of the algorithm's iterations are local minmax points. As a consequence, the algorithm can only converge towards an equilibrium point if such point is a local minmax, and it will escape if the point is not a local minmax. Using numerical examples, we show that the computation time of our algorithm scales roughly linearly with the number of nonzero elements in the Hessian.
Paper Structure (22 sections, 8 theorems, 114 equations, 3 figures, 1 table, 1 algorithm)

This paper contains 22 sections, 8 theorems, 114 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation:
  3. Literature Review
  4. Minimization
  5. Unconstrained minimization
  6. Constrained minimization
  7. Primal-dual interior-point method
  8. Minmax optimization
  9. Unconstrained minmax
  10. Constrained minmax
  11. Primal-dual interior-point method
  12. Algorithmic development and numerical examples
  13. Benchmark example for unconstrained minmax
  14. The homicidal chauffeur example for constrained minmax
  15. The importance of guaranteeing instability
  16. ...and 7 more sections

Key Result

Theorem 1

Let $x$ be an equilibrium point in the sense that $\grad_{x}f(x)=\zerob$. Assume that $\grad_{xx}f(x)$ is invertible and that $\grad_{xx}f(\cdot)$ is differentiable in a neighborhood around $x$. Then for any function $\epsilon_x(\cdot)$ that is constant in a neighborhood around $x$ and satisfies $\g

Figures (3)

  • Figure 1: Trajectory for Homicidal Chauffeur problem with and without guaranteeing instability at equilibrium points that are not a local minmax.
  • Figure 2: Scaling of homicidal chauffeur with horizon length and sparsity pattern of the Hessian when using the sequential approach
  • Figure 3: Scaling of homicidal chauffeur with horizon length and sparsity pattern of the Hessian

Theorems & Definitions (21)

  • Example 1
  • Example 1: Continuation
  • Theorem 1: Stability and instability of modified Newton method for unconstrained minimization
  • proof : Proof of Theorem \ref{['th:stab-newton-minimization']}
  • Definition 1: LICQ and strict complementarity
  • Proposition 1: Second order sufficient conditions for constrained minimization
  • Theorem 2: Stability and instability of modified primal-dual interior-point method for constrained minimization
  • proof : Proof sketch
  • Proposition 2: Second order sufficient condition for unconstrained minmax
  • Example 2
  • ...and 11 more