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Chambolle-Pock and Tseng's methods: relationship and extension to the bilevel optimization

Yura Malitsky(yurii.malitskyi***at***uni-goettingen.de)

Abstract: In the first part of the paper we focus on two problems: (a) regularized least squares and (b) nonsmooth minimization over an affine subspace. For these problems we establish the connection between the primal-dual method of Chambolle-Pock and Tseng's proximal gradient method. For problem (a) it allows us to derive a nonergodic $O(1/k^2)$ convergence rate of the objective function for the primal-dual method. For problem (b) we show that the primal-dual method converges even when the duality does not hold or when the linear system is inconsistent. We also obtain new convergence rates which are more suitable in practice. In the second part, using the established connection, we develop a novel method for the bilevel composite optimization problem. We prove its convergence without the assumption of strong convexity which is typical for many existing methods.

Keywords: Saddle-point problems, first order algorithms, primal-dual algorithms, convergence rates, bilevel optimization, regularized least squares problem

Category 1: Convex and Nonsmooth Optimization (Convex Optimization )

Category 2: Convex and Nonsmooth Optimization (Nonsmooth Optimization )


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Entry Submitted: 06/27/2017
Entry Accepted: 06/27/2017
Entry Last Modified: 06/27/2017

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