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

C. G. Gebhardt, I. Romero

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 $Γ$-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.

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

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 -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.
Paper Structure (5 sections, 6 theorems, 52 equations)

This paper contains 5 sections, 6 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Equivalent formulations
  3. A perturbative $L$-regularization - continuous setting
  4. A perturbative $L$-regularization - discrete setting
  5. Acknowledgments

Key Result

Proposition 1

Let $[u]\in H/K$ be the solution of $\mathrm{P0}$. The representative $u\in[u]$ satisfying $\Pi^\parallel(u)=0$ is the unique element of the equivalence class $[u]$ with minimal $L$-norm. This representative belongs to $K^\perp$.

Theorems & Definitions (15)

  • Proposition
  • proof
  • Proposition
  • proof
  • Remark
  • Proposition
  • proof
  • Proposition
  • proof
  • Definition : Admissible finite element spaces
  • ...and 5 more