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Author: Dvurechensky, Pavel × Author: Stonyakin, Fedor × End date: 2024 × Start date: 2020 × Type: report × Version: publishedVersion ×

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Now showing 1 - 4 of 4
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    Gradient methods for problems with inexact model of the objective
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Stonyakin, Fedor; Dvinskikh, Darina; Dvurechensky, Pavel; Kroshnin, Alexey; Kuznetsova, Olesya; Agafonov, Artem; Gasnikov, Alexander; Tyurin, Alexander; Uribe, Cesar A.; Pasechnyuk, Dmitry; Artamonov, Sergei
    We consider optimization methods for convex minimization problems under inexact information on the objective function. We introduce inexact model of the objective, which as a particular cases includes inexact oracle [19] and relative smoothness condition [43]. We analyze gradient method which uses this inexact model and obtain convergence rates for convex and strongly convex problems. To show potential applications of our general framework we consider three particular problems. The first one is clustering by electorial model introduced in [49]. The second one is approximating optimal transport distance, for which we propose a Proximal Sinkhorn algorithm. The third one is devoted to approximating optimal transport barycenter and we propose a Proximal Iterative Bregman Projections algorithm. We also illustrate the practical performance of our algorithms by numerical experiments.
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    Inexact model: A framework for optimization and variational inequalities
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Stonyakin, Fedor; Gasnikov, Alexander; Tyurin, Alexander; Pasechnyuk, Dmitry; Agafonov, Artem; Dvurechensky, Pavel; Dvinskikh, Darina; Piskunova, Victorya
    In this paper we propose a general algorithmic framework for first-order methods in optimization in a broad sense, including minimization problems, saddle-point problems and variational inequalities. This framework allows to obtain many known methods as a special case, the list including accelerated gradient method, composite optimization methods, level-set methods, proximal methods. The idea of the framework is based on constructing an inexact model of the main problem component, i.e. objective function in optimization or operator in variational inequalities. Besides reproducing known results, our framework allows to construct new methods, which we illustrate by constructing a universal method for variational inequalities with composite structure. This method works for smooth and non-smooth problems with optimal complexity without a priori knowledge of the problem smoothness. We also generalize our framework for strongly convex objectives and strongly monotone variational inequalities.
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    Inexact relative smoothness and strong convexity for optimization and variational inequalities by inexact model
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Stonyakin, Fedor; Gasnikov, Alexander; Tyurin, Alexander; Pasechnyuk, Dmitry; Agafonov, Artem; Dvurechensky, Pavel; Dvinskikh, Darina; Artamonov, Sergei; Piskunova, Victorya
    In this paper we propose a general algorithmic framework for first-order methods in optimization in a broad sense, including minimization problems, saddle-point problems and variational inequalities. This framework allows to obtain many known methods as a special case, the list including accelerated gradient method, composite optimization methods, level-set methods, Bregman proximal methods. The idea of the framework is based on constructing an inexact model of the main problem component, i.e. objective function in optimization or operator in variational inequalities. Besides reproducing known results, our framework allows to construct new methods, which we illustrate by constructing a universal conditional gradient method and universal method for variational inequalities with composite structure. These method works for smooth and non-smooth problems with optimal complexity without a priori knowledge of the problem smoothness. As a particular case of our general framework, we introduce relative smoothness for operators and propose an algorithm for VIs with such operator. We also generalize our framework for relatively strongly convex objectives and strongly monotone variational inequalities.
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    Advances in low-memory subgradient optimization
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Dvurechensky, Pavel; Gasnikov, Alexander; Nurminski, Evgeni; Stonyakin, Fedor
    One of the main goals in the development of non-smooth optimization is to cope with high dimensional problems by decomposition, duality or Lagrangian relaxation which greatly reduces the number of variables at the cost of worsening differentiability of objective or constraints. Small or medium dimensionality of resulting non-smooth problems allows to use bundle-type algorithms to achieve higher rates of convergence and obtain higher accuracy, which of course came at the cost of additional memory requirements, typically of the order of n2, where n is the number of variables of non-smooth problem. However with the rapid development of more and more sophisticated models in industry, economy, finance, et all such memory requirements are becoming too hard to satisfy. It raised the interest in subgradient-based low-memory algorithms and later developments in this area significantly improved over their early variants still preserving O(n) memory requirements. To review these developments this chapter is devoted to the black-box subgradient algorithms with the minimal requirements for the storage of auxiliary results, which are necessary to execute these algorithms. To provide historical perspective this survey starts with the original result of N.Z. Shor which opened this field with the application to the classical transportation problem. The theoretical complexity bounds for smooth and non-smooth convex and quasi-convex optimization problems are briefly exposed in what follows to introduce to the relevant fundamentals of non-smooth optimization. Special attention in this section is given to the adaptive step-size policy which aims to attain lowest complexity bounds. Unfortunately the non-differentiability of objective function in convex optimization essentially slows down the theoretical low bounds for the rate of convergence in subgradient optimization compared to the smooth case but there are different modern techniques that allow to solve non-smooth convex optimization problems faster then dictate lower complexity bounds. In this work the particular attention is given to Nesterov smoothing technique, Nesterov Universal approach, and Legendre (saddle point) representation approach. The new results on Universal Mirror Prox algorithms represent the original parts of the survey. To demonstrate application of non-smooth convex optimization algorithms for solution of huge-scale extremal problems we consider convex optimization problems with non-smooth functional constraints and propose two adaptive Mirror Descent methods. The first method is of primal-dual variety and proved to be optimal in terms of lower oracle bounds for the class of Lipschitz-continuous convex objective and constraints. The advantages of application of this method to sparse Truss Topology Design problem are discussed in certain details. The second method can be applied for solution of convex and quasi-convex optimization problems and is optimal in a sense of complexity bounds. The conclusion part of the survey contains the important references that characterize recent developments of non-smooth convex optimization.