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 ten problems from across mathematics.

翻译：隐变量近点（LVPP）算法是求解具有逐点不等式约束的无限维变分问题的框架。该算法是布雷格曼近点算法的鞍点重构形式。在连续层面上，两种表述具有等价性，但鞍点表述更易于离散化处理，因其在隐函数空间与可行集之间引入了结构保持变换。相较于可行集，在此隐函数空间中实施不等式约束更为便捷：离散化过程可采用适当基函数的一般线性组合，非线性求解器则可进行通用加法更新。LVPP算法针对障碍问题、接触问题、断裂问题、塑性问题及其他领域产生了具有观测网格无关性的数值方法，其中诸多案例属首次实现。该框架还可扩展至更复杂的约束条件，为蒙日-安培方程中的凸性约束实施提供了方法，并能处理准变分不等式——此类问题的底层约束隐式依赖于未知解。本文将以通用形式阐述LVPP算法，并将其应用于数学领域的十个具体问题。