The Latent Variable Proximal Point Method: A New Solver Paradigm for Variational Inequalities, Nonlinear PDEs, and Beyond
Abstract.
The Latent Variable Proximal Point (LVPP) method is a novel, geometry-encoding scheme in which the continuous level informs the algorithms, discretization techniques, and implementation. Mathematically speaking, it embeds the problem at hand into a sequence of related saddle-point problems by introducing a structure-preserving transformation between a latent Banach space and the feasible set. LVPP arises at the confluence of information geometry, optimization, and convex analysis through its use of proximal point methods, Legendre functions, and the isomorphisms induced by their gradients. The method yields algorithms with mesh-independent convergence behaviour for obstacle problems, contact, topology optimization, fracture, plasticity, and more; in many cases, for the first time.
