A regularized method for quadratic optimization problems with finite-dimensional degeneracy

TL;DR

This paper introduces a perturbative regularization and finite element discretization method for quadratic optimization problems with finite-dimensional degeneracy, ensuring convergence of solutions and extending previous approaches for Neumann problems.

Contribution

It generalizes a regularization approach for quadratic optimization with degeneracy, providing a convergent finite element approximation framework.

Findings

Families of functionals $ ext{Gamma}$-converge to the original problem

Minimizers of regularized problems converge to the true solution

Method offers a sparsity-preserving, numerically efficient alternative

Abstract

We propose and analyze a perturbative regularization method to approximate quadratic optimization problems with finite-dimensional degeneracy. The original problem is first approximated by a regularized problem depending on a small positive parameter, and then discretized using the finite element method. The resulting families of continuous and discrete functionals $\Gamma$-converge to the functional of the original problem and the corresponding minimizers converge as well. Our method generalizes the approach proposed in Kaleem et al. (2026) for numerically approximating pure Neumann problems, which represents the cornerstone of a sparsity-preserving, numerically efficient alternative to the methods developed in Bochev and Lehoucq (2005), Ivanov et al. (2019) and Roccia et al. (2024). References: A. Kaleem, C. Gebhardt, and I. Romero. On the pure traction problem of linear elasticity: a regularized formulation and its robust approximation. arXiv preprint arXiv:2602.04359, 2026. P. Bochev and R. Lehoucq. On the finite element solution of the pure Neumann problem. SIAM Review, 47:50-66, 2005. M. Ivanov, I. Kremer, and M. Urev. Solving the pure Neumann problem by a finite element method. Numerical Analysis and Applications, 12:359-371, 2019. B. Roccia, C. Alturria, F. Mazzone, and C. Gebhardt. On the homogeneous torsion problem for heterogeneous and orthotropic cross-sections: theoretical and numerical aspects. Applied Numerical Mathematics, 201:579-607, 2024.