The latent variable proximal point algorithm for variational problems with inequality constraints
Authors
- Dokken, Jørgen S.
- Farrell, Patrick E.
- Keith, Brendan
- Papadopoulos, Ioannis
ORCID: 0000-0003-3522-8761 - Surowiec, Thomas M.
ORCID: 0000-0003-2473-4984
2020 Mathematics Subject Classification
- 35J86 5J96 35R35 49M15 65K15
Keywords
- Variational inequality, latent variable proximal point, PDE constrained optimization, pointwise constraints
DOI
Abstract
The latent variable proximal point (LVPP) algorithm is a framework for solving infinite-dimensional variational problems with pointwise inequality constraints. The algorithm is a saddle point reformulation of the Bregman proximal point algorithm. At the continuous level, the two formulations are equivalent, but the saddle point formulation is more amenable to discretization because it introduces a structure-preserving transformation between a latent function space and the feasible set. Working in this latent space is much more convenient for enforcing inequality constraints than the feasible set, as discretizations can employ general linear combinations of suitable basis functions, and nonlinear solvers can involve general additive updates. LVPP yields numerical methods with observed mesh-independence for obstacle problems, contact, fracture, plasticity, and others besides; in many cases, for the first time. The framework also extends to more complex constraints, providing means to enforce convexity in the Monge?Ampère equation and handling quasi-variational inequalities, where the underlying constraint depends implicitly on the unknown solution. In this paper, we describe the LVPP algorithm in a general form and apply it to twelve problems from across mathematics.
Appeared in
- Comput. Methods Appl. Mech. Engrg., 445 (2025) , Article 118181, (published online on 01.10.2025), DOI 10.1016/j.cma.2025.118181 .
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