Primal-dual proximal splitting and generalized conjugation in non-smooth non-convex optimization

Abstract

We demonstrate that difficult non-convex non-smooth optimization problems, such as Nash equilibrium problems and anisotropic as well as isotropic Potts segmentation model, 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 of 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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Cited by in corpus (5)

• Primal-dual proximal splitting and generalized conjugation in non-smooth non-convex optimization
• First-order primal-dual methods for nonsmooth nonconvex optimisation
• Mumford-Shah regularization in electrical impedance tomography with complete electrode model
• Primal-dual block-proximal splitting for a class of non-convex problems
• A nonsmooth primal-dual method with interwoven PDE constraint solver