In the first part of this paper we prove that the global quadratic optimization problem over a simplex can be solved with a constant relative accuracy. In the second part we consider some natural extensions of the result.

We present a convex conic relaxation for a problem of maximising an indefinite quadratic form over a set of convex constraints on the squared variables. We show that for all these problems we get at least 12/37 relative accuracy of the approximation. In the second part of the paper we derive the...

In this paper we develop a new theory of static equilibrium in congested transportation networks. Our considerations are based on a physical meaning of the flows rather than on an artificially chosen model of travel time functionsl We introduce a concept of the stable equilibrium and prove the...

We propose an alternative approach to stochastic programming based on Monte-Carlo sampling and stochastic gradient optimization. The procedure is by essence probabilistic and the computed solution is a random variable.

In this paper we develop new primal-dual interior-point methods for linear programming, which are based on the concept of parabolic target space. We show that such schemes work in the infinity-neighborhood of the primal-dual central path. Nevertheless these methods possess the best known...

In this paper we present several "infeasible-start" path-following and potential-reduction primal-dual interior-point methods for non-linear conic problems. These methods try to find a recession direction of the feasible set of a self-dual homogeneous primal-dual problem.

In this paper we study the concepts of equilibrium and optimum in static transportation networks with elastic and non-elastic demands. The main mathematical tool of our paper is the theory of variational inequalities. We demonstrate that this theory is useful for proving the existence theorems....

In this paper we consider a new analytic center cutting plane method in a projective space. We prove the efficiency estimates for the general schemeand show that these results can be used in the analysis of a feasibility problem, the variational inequality problem and the problem of constrained...

We present a new class of transportation systems, the stable dynamics models, which provides a natural link between the static and dynamic traffic network models. They can be seen as steady states of dynamic networks (flows are constant in time). These models turn out to be very easy to study...