A semi-implicit scheme based on Arrow-Hurwicz method for saddle point problems

TL;DR

This paper introduces a semi-implicit iterative scheme based on the Arrow-Hurwicz method for efficiently finding saddle points in convex-concave problems, demonstrating improved convergence and robustness through numerical experiments.

Contribution

The paper proposes a novel semi-implicit scheme that accelerates convergence compared to explicit methods for saddle point problems.

Findings

Semi-implicit scheme converges faster than explicit scheme.

Numerical experiments show robustness and efficiency.

Applicable to complex shape optimization problems.

Abstract

We search saddle points for a large class of convex-concave Lagrangian. A generalized explicit iterative scheme based on Arrow-Hurwicz method converges to a saddle point of the problem. We also propose in this work, a convergent semi-implicit scheme in order to accelerate the convergence of the iterative process. Numerical experiments are provided for a nontrivial numerical problem modeling an optimal shape problem of thin torsion rods. This semi-implicit scheme is figured out in practice robustly efficient in comparison with the explicit one.