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This article is cited in 3 scientific papers (total in 3 papers)
Adaptive mirror descent algorithms for convex and strongly convex optimization problems with functional constraints
F. S. Stonyakinab, M. Alkousab, A. N. Stepanova, A. A. Titovb a Vernadsky Crimean Federal University,
4 Vernadsky Avenue, 295007 Simferopol, Russia
b Moscow Institute of Physics and Technologies,
9 Institutskii Bystreet, 141701 Dolgoprudnyi, Russia
Abstract:
Under consideration are some adaptive mirror descent algorithms
for the problems of minimization of a convex objective functional
with several convex Lipschitz (generally, nonsmooth)
functional constraints.
It is demonstrated that
the methods are applicable to the objective functionals
of various levels of smoothness:
The Lipschitz condition holds either
for the objective functional itself
or for its gradient or Hessian
(while the functional itself can fail to satisfy the Lipschitz condition).
The main idea is the adaptive adjustment of the method
with respect to the Lipschitz constant of the objective functional
(its gradient or Hessian),
as well as the Lipschitz constant of the constraint.
The two types of methods are considered:
adaptive (not requiring the knowledge of the Lipschitz constants
neither for the objective functional nor for constraints)
and partially adaptive
(requiring the knowledge of the Lipschitz constant for constraints).
Using the restart technique,
some methods are proposed for strongly convex minimization problems.
Some estimates of the rate of convergence are obtained
for all algorithms under consideration
in dependence on the level of smoothness of the objective functional.
Numerical experiments are presented to illustrate
the advantages of the proposed methods for some examples. Tab. 3, bibliogr. 22.
Keywords:
adaptive Mirror Descent, Lipschitz condition, Lipschitz gradient, Lipschitz Hessian, strongly convex function, the technique of restarts.
Received: 17.10.2018 Revised: 24.02.2019 Accepted: 27.02.2019
Citation:
F. S. Stonyakin, M. Alkousa, A. N. Stepanov, A. A. Titov, “Adaptive mirror descent algorithms for convex and strongly convex optimization problems with functional constraints”, Diskretn. Anal. Issled. Oper., 26:3 (2019), 88–114; J. Appl. Industr. Math., 13:3 (2019), 557–574
Linking options:
https://www.mathnet.ru/eng/da932 https://www.mathnet.ru/eng/da/v26/i3/p88
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