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A proximal minimization algorithm for structured nonconvex and nonsmooth problems

Radu Ioan Bot(radu.bot***at***univie.ac.at)
Ernö Robert Csetnek(ernoe.robert.csetnek***at***univie.ac.at)
Dang-Khoa Nguyen(dang-khoa.nguyen***at***univie.ac.at)

Abstract: We propose a proximal algorithm for minimizing objective functions consisting of three summands: the composition of a nonsmooth function with a linear operator, another nonsmooth function, each of the nonsmooth summands depending on an independent block variable, and a smooth function which couples the two block variables. The algorithm is a full splitting method, which means that the nonsmooth functions are processed via their proximal operators, the smooth function via gradient steps, and the linear operator via matrix times vector multiplication. We provide sufficient conditions for the boundedness of the generated sequence and prove that any cluster point of the latter is a KKT point of the minimization problem. In the setting of the Kurdyka-\L{}ojasiewicz property we show global convergence, and derive convergence rates for the iterates in terms of the \L{}ojasiewicz exponent.

Keywords: structured nonconvex and nonsmooth optimization, proximal algorithm, full splitting scheme, Kurdyka-\L{}ojasiewicz property, limiting subdifferential

Category 1: Convex and Nonsmooth Optimization (Nonsmooth Optimization )

Citation:

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Entry Submitted: 05/28/2018
Entry Accepted: 05/28/2018
Entry Last Modified: 05/28/2018

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