In this paper we develop a new primal-dual subgradient method for nonsmooth convex optimization problems. This scheme is based on a self-concordant barrier for the basic feasible set. It is suitable for finding approximate solutions with certain relative accuracy. We discuss some applications of...

In this paper, we develop new subgradient methods for solving nonsmooth convex optimization problems. These methods are the first ones, for which the whole sequence of test points is endowed with the worst-case performance guarantees. The new methods are derived from a relaxed estimating...

We consider a new class of huge-scale problems, the problems with sparse subgradients. The most important functions of this type are piece-wise linear. For optimization problems with uniform sparsity of corresponding linear operators, we suggest a very efficient implementation of subgradient...

In this paper we suggest a new framework for constructing mathematical models of market activity. Contrary to the majority of the classical economical models (e.g. Arrow- Debreu, Walras, etc.), we get a characterization of general equilibrium of the market as a saddle point in a convex-concave...

In this paper, we establish a local quadratic convergence of polynomial-time interior-point methods for general conic optimization problems. The main structural property used in our analysis is the logarithmic homogeneity of self-concordant barrier functions. We propose new path-following...

In this paper we analyze several new methods for solving optimization problems with the objective function formed as a sum of two convex terms: one is smooth and given by a black-box oracle, and another is general but simple and its structure is known. Despite to the bad properties of the sum,...

In this paper we propose a new approach for constructing efficient schemes for nonsmooth convex optimization. It is based on a special smoothing technique, which can be applied to the functions with explicit max-structure. Our approach can be considered as an alternative to black-box...

In this paper we propose two new nonsymmetric primal-dual potential-reduction methods for conic problems. The methods are based on the primal-dual lifting [5]. This procedure allows to construct a strictly feasible primal-dual pair related by an exact scaling relation even if the cones are not...

In this paper we propose an accelerated version of the cubic regularization of Newton's method [6]. The original version, used for minimizing a convex function with Lipschitz-continuous Hessian, guarantees a global rate of convergence of order O(1/k exp.2), where k is the iteration counter. Our...

In this paper we analyze computational performance of dual trigonometric generating functions on some integer programming problems. We show that if the number of equality constraints is fixed, then this technique allows to solve the problems in time, which is polynomial in the dimension of the...