Yu. Nesterov

Optimization Problems over Positive Pseudopolynomial Matrices

The Nesterov characterizations of positive pseudopolynomials on the real line, the imaginary axis, and the unit circle are extended to the matrix case. With the help of these characterizations, a class of optimization problems over the space of positive pseudopolynomial matrices is considered. These problems can be solved in an efficient manner due to the inherent block Toeplitz or block Hankel structure induced by the characterization in question. The efficient implementation of the resulting algorithms is discussed in detail. In particular, the real line setting of the problem leads naturally to ill-conditioned numerical systems. However, adopting a Chebyshev basis instead of the natural basis for describing the polynomial matrix space yields a restatement of the problem and of its solution approach with much better numerical properties.

Homogeneous Analytic Center Cutting Plane Methods for Convex Problems and Variational Inequalities

In this paper we consider a new analytic center cutting plane method in an extended space. We prove the efficiency estimates for the general scheme and show that these results can be used in the analysis of a feasibility problem, the variational inequality problem, and the problem of constrained minimization. Our analysis is valid even for problems whose solution belongs to the boundary of the domain.

Efficient numerical methods for entropy-linear programming problems

Entropy-linear programming (ELP) problems arise in various applications. They are usually written as the maximization of entropy (minimization of minus entropy) under affine constraints. In this work, new numerical methods for solving ELP problems are proposed. Sharp estimates for the convergence rates of the proposed methods are established. The approach described applies to a broader class of minimization problems for strongly convex functionals with affine constraints.

On the efficiency of a randomized mirror descent algorithm in online optimization problems

A randomized online version of the mirror descent method is proposed. It differs from the existing versions by the randomization method. Randomization is performed at the stage of the projection of a subgradient of the function being optimized onto the unit simplex rather than at the stage of the computation of a subgradient, which is common practice. As a result, a componentwise subgradient descent with a randomly chosen component is obtained, which admits an online interpretation. This observation, for example, has made it possible to uniformly interpret results on weighting expert decisions and propose the most efficient method for searching for an equilibrium in a zero-sum two-person matrix game with sparse matrix.

Indirect reciprocity through gossiping can lead to cooperative clusters

Explaining how cooperation can emerge, and persist over time in various species is a prime challenge for both biologists and social scientists. Whereas cooperation in non-human species might be explained through mechanisms such as kinship selection or reciprocity, this is usually regarded as insufficient to explain the extent of cooperation observed in humans. It has been theorized that indirect reciprocity—I help you, and someone else later helps me—could explain the breadth of human cooperation. Reputation is central to this idea, since it conveys important information to third parties whether to cooperate or not with a focal player. In this paper we analyze a model for reputation dynamics through gossiping, and pay specific attention to the possible cooperation networks that may arise. We show that gossiping agents can organize into cooperative clusters, i.e. cooperate within clusters, and defect between them, which can be regarded as socially balanced. We also deduce conditions for these gossiping cooperators to be evolutionary stable.