Tensor Complementarity Problem and Semi-positive Tensors

The tensor complementarity problem (\q, \mathcal{A}) is to \mbox{ find } \x \in \mathbb{R}^n\mbox{ such that }\x \geq \0, \q + \mathcal{A}\x^{m-1} \geq \0, \mbox{ and }\x^\top (\q + \mathcal{A}\x^{m-1}) = 0. We prove that a real tensor $\mathcal{A}$ is a (strictly) semi-positive tensor if and only if the tensor complementarity problem (\q, \mathcal{A}) has a unique solution for \q>\0 (\q\geq\0), and a symmetric real tensor is a (strictly) semi-positive tensor if and only if it is (strictly) copositive. That is, for a strictly copositive symmetric tensor $\mathcal{A}$, the tensor complementarity problem (\q, \mathcal{A}) has a solution for all \q \in \mathbb{R}^n

Real and Complex Monotone Communication Games

Noncooperative game-theoretic tools have been increasingly used to study many important resource allocation problems in communications, networking, smart grids, and portfolio optimization. In this paper, we consider a general class of convex Nash Equilibrium Problems (NEPs), where each player aims to solve an arbitrary smooth convex optimization problem. Differently from most of current works, we do not assume any specific structure for the players' problems, and we allow the optimization variables of the players to be matrices in the complex domain. Our main contribution is the design of a novel class of distributed (asynchronous) best-response- algorithms suitable for solving the proposed NEPs, even in the presence of multiple solutions. The new methods, whose convergence analysis is based on Variational Inequality (VI) techniques, can select, among all the equilibria of a game, those that optimize a given performance criterion, at the cost of limited signaling among the players. This is a major departure from existing best-response algorithms, whose convergence conditions imply the uniqueness of the NE. Some of our results hinge on the use of VI problems directly in the complex domain; the study of these new kind of VIs also represents a noteworthy innovative contribution. We then apply the developed methods to solve some new generalizations of SISO and MIMO games in cognitive radios and femtocell systems, showing a considerable performance improvement over classical pure noncooperative schemes.Comment: to appear on IEEE Transactions in Information Theor

Hamming-like distances for ill-defined strings in linguistic classification

Ill-defined strings often occur in soft sciences, e.g. in linguistics or in biology. In this paper we consider l-length strings which have in each position one of the three symbols 0 or false, 1 or true, b or irrelevant. We tackle some generalisations of the usual Hamming distance between binary crisp strings which were recently used in computational linguistics. We comment on their metric properties, since these should guide the selection of the clustering algorithm to be used for language classification. The concluding section is devoted to future work, and the string approach, as currently pursued, is compared to alternative approaches

Hamming-like Distances for Ill-defined Strings in Linguistic Classification

Ill-defined strings often occur in soft sciences, e.g. in linguistics or in biology. In this paper we consider \ell-length strings which have in each position one of the three symbols 0 or false, 1 or true, \flat or irrelevant. We tackle some generalisations of the usual Hamming distance between binary crisp strings which were recently used in computational linguistics. We comment on their metric properties, since these should guide the selection of the clustering algorithm to be used for language classification. The concluding section is devoted to future work, and the string approach, as currently pursued, is compared to alternative approaches

An Inferentially Many-Valued Two-Dimensional Notion of Entailment

Starting from the notions of q-entailment and p-entailment, a two-dimensional notion of entailment is developed with respect to certain generalized q-matrices referred to as B-matrices. After showing that every purely monotonic singleconclusion consequence relation is characterized by a class of B-matrices with respect to q-entailment as well as with respect to p-entailment, it is observed that, as a result, every such consequence relation has an inferentially four-valued characterization. Next, the canonical form of B-entailment, a two-dimensional multiple-conclusion notion of entailment based on B-matrices, is introduced, providing a uniform framework for studying several different notions of entailment based on designation, antidesignation, and their complements. Moreover, the two-dimensional concept of a B-consequence relation is defined, and an abstract characterization of such relations by classes of B-matrices is obtained. Finally, a contribution to the study of inferential many-valuedness is made by generalizing Suszko’s Thesis and the corresponding reduction to show that any B-consequence relation is, in general, inferentially four-valued