Inexact proximal $$\epsilon $$ϵ-subgradient methods for composite convex optimization problems

R. Díaz Millán () and M. Pentón Machado ()
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R. Díaz Millán: Federal Institute of Goiás
M. Pentón Machado: Universidade Federal da Bahia

Journal of Global Optimization, 2019, vol. 75, issue 4, No 6, 1029-1060

Abstract: Abstract We present two approximate versions of the proximal subgradient method for minimizing the sum of two convex functions (not necessarily differentiable). At each iteration, the algorithms require inexact evaluations of the proximal operator, as well as, approximate subgradients of the functions (namely: the$$\epsilon $$ϵ-subgradients). The methods use different error criteria for approximating the proximal operators. We provide an analysis of the convergence and rate of convergence properties of these methods, considering various stepsize rules, including both, diminishing and constant stepsizes. For the case where one of the functions is smooth, we propose an inexact accelerated version of the proximal gradient method, and prove that the optimal convergence rate for the function values can be achieved. Moreover, we provide some numerical experiments comparing our algorithm with similar recent ones.

Keywords: Splitting methods; Optimization problem; $$\epsilon $$ ϵ -Subdifferential; Inexact methods; Hilbert space; Accelerated methods; 65K05; 90C25; 90C30; 49J45 (search for similar items in EconPapers)
Date: 2019
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10898-019-00808-8

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