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Alexander Gasnikov

Publications and source records attributed to Alexander Gasnikov.

At least 289 records · Page 16Linked to original sources
arXiv2017

Stochastic Analysis in Problems, part 1 (in Russian)

This book contains a large number of exercises related to different stochastic disciplines. Difficulty of the problems varies from the basic level in the first chapter up to the analysis of articles in Probability, Statistics and Computer Science. Range of the given exercises covers most of the significant classical results of the area including techniques that arise in modern stochastic disciplines. The book is recommended for the course "Stochastic Analysis in problems" which is held by the Department of Mathematical Foundations of Control in Moscow Institute of Physics and Technology. Content of the book may be used for practical classes, exams and for self-education reasons in various stochastic disciplines.

math.PR↗
arXiv2017

Parallel algorithms and probability of large deviation for stochastic optimization problems

We consider convex stochastic optimization problems under different assumptions on the properties of available stochastic subgradient. It is known that, if the value of the objective function is available, one can obtain, in parallel, several independent approximate solutions in terms of the objective residual expectation. Then, choosing the solution with the minimum function value, one can control the probability of large deviation of the objective residual. On the contrary, in this short paper, we address the situation, when the value of the objective function is unavailable or is too expensive to calculate. Under "`light-tail"' assumption for stochastic subgradient and in general case with moderate large deviation probability, we show that parallelization combined with averaging gives bounds for probability of large deviation similar to a serial method. Thus, in these cases, one can benefit from parallel computations and reduce the computational time without loss in the solution quality.

math.OC↗
arXiv2017

Comparision of the definitions of generalized solution of the Cauchy problem for quasi-linear equation

In preprint we consider and compare different definitions of generalized solution of the Cauchy problem for 1d-scalar quasilinear equation (conservation law). We start from the classical approaches goes back to I.M. Gelfand, O.A. Oleinik, S.N. Kruzhkov and move to the modern finite-difference approximations approaches belongs to A.A. Shananin and G.M. Henkin. We discuss the conditions that provide definitions to be equivalent.

math-ph↗
arXiv2016

Searching equillibriums in Beckmann's and Nesterov--de Palma's models

In this paper we propose and develop classical Frank--Wolf algorithm for Beckmann's type models. This is not new, but we investigate details that allows us to speed up. We also consider stable dynamic like models. First model of this type was proposed 15 years ago by Yu. Nesterov and A. DePalma. We propose randomized dual averaging method with special (sum-type) randomization. For both of the problems we obtain the rates of convergences. It seems that this estimations to be unimprovable without additional assumption about problem formulation.

math.OC↗
arXiv2016

Efficient randomized mirror descents in stochastic online convex optimization

In the paper we consider an application of mirror descent (dual averaging) to the stochastic online convex optimization problems. We compare classical mirror descent (Nemirovski-Yudin, 1979) with dual averaging (Nesterov, 2005) and Grigoriadis-Khachiyan algorithm (1995). Grigoriadis-Khachiyan algorithm has just proved to be a randomized mirror descent (dual averaging) with randomization in KL-projection of (sub)gradient to a unit simplex. We found out that this randomization is an optimal way of solving sparse matrix games and some other problems arising in convex optimization and experts weighting.

math.OC↗
arXiv2016

Primal-Dual Method for Searching Equilibrium in Hierarchical Congestion Population Games

In this paper, we consider a large class of hierarchical congestion population games. One can show that the equilibrium in a game of such type can be described as a minimum point in a properly constructed multi-level convex optimization problem. We propose a fast primal-dual composite gradient method and apply it to the problem, which is dual to the problem describing the equilibrium in the considered class of games. We prove that this method allows to find an approximate solution of the initial problem without increasing the complexity.

math.OC↗
arXiv2016

On the three-stage version of stable dynamic model

An attempt to merge into a single model, which reduces to the solution of non-smooth convex optimization problem: calculation model of OD-matrix (entropy model), the mode split model and the model of the equilibrium distribution of flows (Stable dynamic model, Nesterov - de Palma, 2003). To best of our knowledge, this is the first attempt to combine this three models. Previously such attempts were done for other types of equlibrium models, mainly with the BMW-model (1955), the calibration of which is significantly more difficult. We also remark, that our model much better then traditional from computational point of view.

math.OC↗
arXiv2016

Efficient randomized algorithms for PageRank problem

In the paper we compare well known numerical methods of finding PageRank vector. We propose Markov Chain Monte Carlo method and obtain a new estimation for this method. We also propose a new method for PageRank problem based on the reduction of this problem to the matrix game. We solve this (sparse) matrix game with randomized mirror descent. It should be mentioned that we used non-standard randomization (in KL-projection) goes back to Grigoriadis-Khachiayn (1995).

math.OC↗
arXiv2016

Fast Primal-Dual Gradient Method for Strongly Convex Minimization Problems with Linear Constraints

In this paper we consider a class of optimization problems with a strongly convex objective function and the feasible set given by an intersection of a simple convex set with a set given by a number of linear equality and inequality constraints. A number of optimization problems in applications can be stated in this form, examples being the entropy-linear programming, the ridge regression, the elastic net, the regularized optimal transport, etc. We extend the Fast Gradient Method applied to the dual problem in order to make it primal-dual so that it allows not only to solve the dual problem, but also to construct nearly optimal and nearly feasible solution of the primal problem. We also prove a theorem about the convergence rate for the proposed algorithm in terms of the objective function and the linear constraints infeasibility.

math.OC↗
arXiv2016

Efficient calculation of stochastic equilibriums in the Beckmann's and stable dynamic models

We propose composite approache to the special sum-type convex optimization problem with affine restriction and special entropy type regularization. Since the fuctional has a penalty type form, we reformulate initial conditional optimization problem in a special unconstrained form that allows us to put the penalty type functional into the composite term. We also describe the characteristic functions on graphs technique (Yu. Nesterov, 2007) in application to the dual problem.

math.OC↗
arXiv2016

About reduction of searching competetive equillibrium to the minimax problem in application to different network problems

In the paper we show that an important class of multistage traffic equillibrium models (including correspondence matrix calculation, traffic assignment problem etc) and their economic generalizations can be considered as proper population games with the minimax structure of equillibriums. This is not trivial because at the first sight we have to consider competetive (Valras) equilibrium and Nash-Vardroop equillibrium.

math.OC↗
arXiv2016

Gradient-free prox-methods with inexact oracle for stochastic convex optimization problems on a simplex

In the paper we show that euclidian randomization in some situations (i.e. for gradient-free method on a simplex) can be as good as the randomization on the unit sphere in 1-norm. That is on the simplex example we show that for gradient-free methods the choise of the prox-structure and the choise of a way of randomization have to be connected to each other. We demonstrate how it can be done in an optimal way. It is important that we consider inexact oracle.

math.OC↗
arXiv2016

Universal method with inexact oracle and its applications for searching equillibriums in multistage transport problems

In this paper we propose a new efficient approach for numerical calculation of equillibriums in multistage transport problems. In the very core of our approach lies the proper combination of Universal Gradient Method proposed by Yu. Nesterov (2013) and conception of inexact oracle (Devolder--Glineur--Nesterov, 2011). In particular our technique allows us to calculate Wasserstein's Barycenter in a fast manner (this results generalized M. Cuturi et al. (2014)).

math.OC↗
arXiv2016

Efficient numerical algorithms for regularized regression problem with applications to traffic matrix estimations

In this work we collect and compare to each other many different numerical methods for regularized regression problem and for the problem of projection on a hyperplane. Such problems arise, for example, as a subproblem of demand matrix estimation in IP- networks. In this special case matrix of affine constraints has special structure: all elements are 0 or 1 and this matrix is sparse enough. We have to deal with huge-scale convex optimization problem of special type. Using the properties of the problem we try "to look inside the black-box" and to see how the best modern methods work being applied to this problem.

math.OC↗