Primal-dual fast gradient method with a model

Tyurin A.I.
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In this work we consider a possibility to use the conception of $(\delta, L)$-model of a function for optimization tasks, whereby solving a primal problem there is a necessity to recover a solution of a dual problem. The conception of $(\delta, L)$-model is based on the conception of $(\delta, L)$-oracle which was proposed by Devolder–Glineur–Nesterov, herewith the authors proposed approximate a function with an upper bound using a convex quadratic function with some additive noise $\delta$. They managed to get convex quadratic upper bounds with noise even for nonsmooth functions. The conception of $(\delta, L)$-model continues this idea by using instead of a convex quadratic function a more complex convex function in an upper bound. Possibility to recover the solution of a dual problem gives great benefits in different problems, for instance, in some cases, it is faster to find a solution in a primal problem than in a dual problem. Note that primal-dual methods are well studied, but usually each class of optimization problems has its own primal-dual method. Our goal is to develop a method which can find solutions in different classes of optimization problems. This is realized through the use of the conception of $(\delta, L)$-model and adaptive structure of our methods. Thereby, we developed primal-dual adaptive gradient method and fast gradient method with $(\delta, L)$-model and proved convergence rates of the methods, moreover, for some classes of optimization problems the rates are optimal. The main idea is the following: we find a dual solution to an approximation of a primal problem using the conception of $(\delta, L)$-model. It is much easier to find a solution to an approximated problem, however, we have to do it in each step of our method, thereby the principle of “divide and conquer” is realized.

Keywords: fast gradient method, model of the function, primal-dual method
Citation in English: Tyurin A.I. Primal-dual fast gradient method with a model // Computer Research and Modeling, 2020, vol. 12, no. 2, pp. 263-274
Citation in English: Tyurin A.I. Primal-dual fast gradient method with a model // Computer Research and Modeling, 2020, vol. 12, no. 2, pp. 263-274
DOI: 10.20537/2076-7633-2020-12-2-263-274
URL: http://crm-en.ics.org.ru/journal/article/2908/
Creative Commons License This work is licensed under a Creative Commons Attribution-NoDerivs 3.0 Unported License.

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