An Adaptive Proximal Method for Variational Inequalities

Computational Mathematics and Mathematical Physics - A novel analog of Nemirovski’s proximal mirror method with an adaptive choice of constants in the minimized prox-mappings at each...     

Generalized Mirror Prox Algorithm for Monotone Variational Inequalities: Universality and Inexact Oracle

Journal of Optimization Theory and Applications - We introduce an inexact oracle model for variational inequalities with monotone operators, propose a numerical method that solves such variational...     

Dual approaches to the minimization of strongly convex functionals with a simple structure under affine constraints

A strongly convex function of simple structure (for example, separable) is minimized under affine constraints. A dual problem is constructed and solved by applying a fast gradient method. The necessary properties of this method are established relying on which, under rather general conditions, the solution of the primal problem can be recovered with the same accuracy as the dual solution from the sequence generated by this method in the dual space of the problem. Although this approach seems natural, some previously unpublished rather subtle results necessary for its rigorous and complete theoretical substantiation in the required generality are presented.     

Oracle Complexity Separation in Convex Optimization

Journal of Optimization Theory and Applications - Many convex optimization problems have structured objective functions written as a sum of functions with different oracle types (e.g., full...     

Fast Gradient Descent for Convex Minimization Problems with an Oracle Producing a (δ, L)-Model of Function at the Requested Point

Computational Mathematics and Mathematical Physics - A new concept of $$(\delta ,L)$$ -model of a function that is a generalization of the Devolder–Glineur–Nesterov $$(\delta ,L)$$...     

Accelerated Gradient Sliding for Minimizing a Sum of Functions

Doklady Mathematics - We propose a new way of justifying the accelerated gradient sliding of G. Lan, which allows one to extend the sliding technique to a combination of an accelerated gradient...     

About the Power Law of the PageRank Vector Component Distribution. Part 2. The Buckley–Osthus Model,Verification of the Power Law for This Model,and Setup of Real Search Engines

This paper continues article [13]. Here we consider the Buckley–Osthus model for formation of an Internet network. We numerically calculate the PageRank vector for networks generated by this model.We show that the components of this vector are distributed according to a power law. We also discuss the computational aspects of this model with respect to the numerical methods for calculation of the PageRank vector, which were presented in the first part of the work. Finally, we describe a more general model for web-page ranking and some approaches to solve the optimization problem arising in this model learning.     

Universal Method for Stochastic Composite Optimization Problems

A fast gradient method requiring only one projection is proposed for smooth convex optimization problems. The method has a visual geometric interpretation, so it is called the method of similar triangles (MST). Composite, adaptive, and universal versions of MST are suggested. Based on MST, a universal method is proposed for the first time for strongly convex problems (this method is continuous with respect to the strong convexity parameter of the smooth part of the functional). It is shown how the universal version of MST can be applied to stochastic optimization problems.