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Optimal subgradient algorithms with application to large-scale linear inverse problems

Masoud Ahookhosh (masoud.ahookhosh***at***univie.ac.at)

Abstract: This study addresses some algorithms for solving structured unconstrained convex optimization problems using first-order information where the underlying function includes high-dimensional data. The primary aim is to develop an implementable algorithmic framework for solving problems with multi-term composite objective functions involving linear mappings using the optimal subgradient algorithm, OSGA, proposed by {\sc Neumaier} in \cite{NeuO}. To this end, we propose some prox-functions for which the corresponding subproblem of OSGA is solved in a closed form. Considering various inverse problems arising in signal and image processing, machine learning, statistics, we report extensive numerical and comparisons with several state-of-the-art solvers proposing favourably performance of our algorithm. We also compare with the most widely used optimal first-order methods for some smooth and nonsmooth convex problems. Surprisingly, when some Nesterov-type optimal methods originally proposed for smooth problems are adapted for solving nonsmooth problems by simply passing a subgradient instead of the gradient, the results of these subgradient-based algorithms are competitive and totally interesting for solving nonsmooth problems.

Keywords: Structured convex optimization, Linear inverse problems, Nonsmooth optimization, Sparse optimization, First-order black-box oracle, Optimal complexity, Subgradient methods, High-dimensional data, Compressed sensing, Image restoration

Category 1: Convex and Nonsmooth Optimization

Category 2: Convex and Nonsmooth Optimization (Convex Optimization )

Category 3: Convex and Nonsmooth Optimization (Nonsmooth Optimization )

Citation: Faculty of Mathematics, University of Vienna

Download: [PDF]

Entry Submitted: 02/06/2014
Entry Accepted: 02/06/2014
Entry Last Modified: 05/26/2014

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