Stochastic Primal-Dual Coordinate Method for Nonlinear Convex Cone Programs

Block coordinate descent (BCD) methods and their variants have been widely used in coping with large-scale nonconstrained optimization problems in many fields such as imaging processing, machine learning, compress sensing and so on. For problem with coupling constraints, Nonlinear convex cone programs (NCCP) are important problems with many practical applications, but these problems are hard to solve by using existing block coordinate type methods. This paper introduces a stochastic primal-dual coordinate (SPDC) method for solving large-scale NCCP. In this method, we randomly choose a block of variables based on the uniform distribution. The linearization and Bregman-like function (core function) to that randomly selected block allow us to get simple parallel primal-dual decomposition for NCCP. The sequence generated by our algorithm is proved almost surely converge to an optimal solution of primal problem. Two types of convergence rate with different probability (almost surely and expected) are also obtained. The probability complexity bound is also derived in this paper

Sampling-Based Approaches for Multimarginal Optimal Transport Problems with Coulomb Cost

The multimarginal optimal transport problem with Coulomb cost arises in quantum physics and is vital in understanding strongly correlated quantum systems. Its intrinsic curse of dimensionality can be overcome with a Monge-like ansatz. A nonconvex quadratic programmming then emerges after employing discretization and $\ell_1$ penalty. To globally solve this nonconvex problem, we adopt a grid refinements-based framework, in which a local solver is heavily invoked and hence significantly determines the overall efficiency. The block structure of this nonconvex problem suggests taking block coordinate descent-type methods as the local solvers, while the existing ones can get seriously afflicted with the poor scalability induced by the associated sparse-dense matrix multiplications. In this work, borrowing the tools from optimal transport, we develop novel methods that favor highly scalable schemes for subproblems and are completely free of the full matrix multiplications after introducing entrywise sampling. Convergence and asymptotic properties are built on the theory of random matrices. The numerical results on several typical physical systems corroborate the effectiveness and better scalability of our approach, which also allows the first visualization for the approximate optimal transport maps between electrons in three-dimensional contexts.Comment: 31 pages, 6 figures, 3 table