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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, such problems, both in convex and nonconvex cases, can be solved… (More)

- Yu. V. Nesterov
- 2005

- Yu. V. Nesterov
- 1998

We present a convex conic relaxation for a problem of maximizing 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 conic relaxation by another approach based on the second… (More)

- Yu. V. Nesterov
- 2004

In this paper we suggest a new efficient technique for solving integer knapsack problems. Our algorithms can be seen as application of Fast Fourier Transform to generating functions of integer polytopes. Using this approach, it is possible to count the number of boolean solutions of a single n-dimensional Diophantine equation 〈a, x〉 = b in O(‖a‖1 ln ‖a1‖ ln… (More)

In this paper we develop a new and efficient method for variational inequality with Lipschitz continuous strongly monotone operator. Our analysis is based on a new strongly convex merit function. We apply a variant of the developed scheme for solving quasivariational inequality. As a result, we significantly improve the standard sufficient condition for… (More)

Positive polynomial matrices play a fundamental role in systems and control theory: they represent e.g. spectral density functions of stochastic processes and show up in spectral factorizations, robust control and filter design problems. Positive polynomials obviously form a convex set and were recently studied in the area of convex optimization [1, 5]. It… (More)

- Yu. V. Nesterov
- 2005

In many applications it is possible to justify a reasonable bound for possible variation of subgradients of objective function rather than for their uniform magnitude. In this paper we develop a new class of efficient primal-dual subgradient schemes for such problem classes.

Positive polynomial matrices play a fundamental role in systems and control theory. We give here a simplified proof of the fact that the convex set of positive polynomial matrices can be parameterized using block Hankel and block Toeplitz matrices. We also show how to derive efficient computational algorithms for optimization problems over positive pseudo… (More)

- Yu. V. Nesterov
- 2006

In this paper we derive efficiency estimates of the regularized Newton’s method as applied to constrained convex minimization problems and to variational inequalities. We study a one-step Newton’s method and its multistep accelerated version, which converges on smooth convex problems as O( 1 k3 ), where k is the iteration counter. We derive also the… (More)

- Yu. V. Nesterov
- 2006

In this paper we develop a technique for constructing self-concordant barriers for convex cones. We start from a simple proof for a variant of standard result [1] on transformation of a ν-self-concordant barrier for a set into a self-concordant barrier for its conic hull with parameter (3.08 √ ν + 3.57)2. Further, we develop a convenient composition theorem… (More)