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--Amp\`ere 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.

翻译：隐变量邻近点（LVPP）算法是求解带逐点不等式约束的无穷维变分问题的框架。该算法是Bregman邻近点算法的鞍点重构形式。在连续层面上，两种表述是等价的，但鞍点表述更易于离散化，因为它在隐函数空间与可行集之间引入了结构保持变换。在隐空间中进行操作比在可行集中执行不等式约束更为便捷，因为离散化可采用合适基函数的一般线性组合，且非线性求解器可涉及一般的加性更新。LVPP为障碍问题、接触、断裂、塑性及其他问题产生了具有观测网格无关性的数值方法；在许多情况下，这是首次实现。该框架还可扩展至更复杂的约束，为Monge--Ampère方程中的凸性约束提供了实施手段，并能处理拟变分不等式——其中底层约束隐式依赖于未知解。本文以一般形式描述LVPP算法，并将其应用于数学领域的十二个问题。