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Primal-Dual Proximal Splitting and Generalized Conjugation in Non-smooth Non-convex Optimization

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CLASON Christian MAZURENKO Stanislav VALKONEN Tuomo

Rok publikování 2020
Druh Článek v odborném periodiku
Časopis / Zdroj Applied Mathematics and Optimization
Fakulta / Pracoviště MU

Přírodovědecká fakulta

Citace
www https://doi.org/10.1007/s00245-020-09676-1
Doi http://dx.doi.org/10.1007/s00245-020-09676-1
Klíčová slova Nonsmooth optimization; Primal-dual method; Non-convex-concave saddle-points; Generalized conjugate; Potts model; Nash equilibria
Popis We demonstrate that difficult non-convex non-smooth optimization problems, such as Nash equilibrium problems and anisotropic as well as isotropic Potts segmentation models, can be written in terms of generalized conjugates of convex functionals. These, in turn, can be formulated as saddle-point problems involving convex non-smooth functionals and a general smooth but non-bilinear coupling term. We then show through detailed convergence analysis that a conceptually straightforward extension of the primal-dual proximal splitting method of Chambolle and Pock is applicable to the solution of such problems. Under sufficient local strong convexity assumptions on the functionals-but still with a non-bilinear coupling term-we even demonstrate local linear convergence of the method. We illustrate these theoretical results numerically on the aforementioned example problems.

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