Parametrization and convergence of a primal-dual block-coordinate approach to linearly-constrained nonsmooth optimization (2408.16424v1)
Abstract: This note is concerned with the problem of minimizing a separable, convex, composite (smooth and nonsmooth) function subject to linear constraints. We study a randomized block-coordinate interpretation of the Chambolle-Pock primal-dual algorithm, based on inexact proximal gradient steps. A specificity of the considered algorithm is its robustness, as it converges even in the absence of strong duality or when the linear program is inconsistent. Using matrix preconditiong, we derive tight sublinear convergence rates with and without duality assumptions and for both the convex and the strongly convex settings. Our developments are extensions and particularizations of original algorithms proposed by Malitsky (2019) and Luke and Malitsky (2018). Numerical experiments are provided for an optimal transport problem of service pricing.
- A first-order primal-dual algorithm for convex problems with applications to imaging. Journal of Mathematical Imaging and Vision 40, 120–145. URL: https://doi.org/10.1007/s10851-010-0251-1, doi:10.1007/s10851-010-0251-1.
- Accelerated, parallel, and proximal coordinate descent. SIAM Journal on Optimization 25, 1997–2023. URL: https://doi.org/10.1137/130949993, doi:10.1137/130949993, arXiv:https://doi.org/10.1137/130949993.
- Block-coordinate primal-dual method for nonsmooth minimization over linear constraints, in: Giselsson, P., Rantzer, A. (Eds.), Large-Scale and Distributed Optimization. Springer International Publishing, Cham, pp. 121--147. URL: https://doi.org/10.1007/978-3-319-97478-1_6, doi:10.1007/978-3-319-97478-1_6.
- The primal-dual hybrid gradient method reduces to a primal method for linearly constrained optimization problems. URL: https://arxiv.org/abs/1706.02602, arXiv:1706.02602.
- Iteration complexity of randomized block-coordinate descent methods for minimizing a composite function. Mathematical Programming 144, 1--38. URL: https://doi.org/10.1007/s10107-012-0614-z.
- Parallel coordinate descent methods for big data optimization. Mathematical Programming 156, 433--484. URL: https://doi.org/10.1007/s10107-015-0901-6.
- Über allgemeine lineare mittelbildungen. Prace Matematyczno-Fizyczne 22, 113--119. URL: http://eudml.org/doc/215310.
- On accelerated proximal gradient methods for convex-concave optimization. Submitted to SIAM J. Optim.