A Coalitional Game Framework for Cooperative Secondary Spectrum Access

We consider a framework in which several service providers offer downlink wireless data access service in a certain area. Each provider serves its end-users through opportunistic secondary spectrum access of licensed spectrum, and needs to pay primary license holders of the spectrum usage based and membership based charges for such secondary spectrum access. In these circumstances, if providers pool their resources and allow end-users to be served by any of the cooperating providers, the total user satisfaction as well as the aggregate revenue earned by providers may increase. We use coalitional game theory to investigate such cooperation among providers, and show that the optimal cooperation schemes can be obtained as solutions of convex optimizations. We next show that under usage based charging scheme, if all providers cooperate, there always exists an operating point that maximizes the aggregate revenue of providers, while presenting each provider a share of the revenue such that no subset of providers has an incentive to leave the coalition. Furthermore, such an operating point can be computed in polynomial time. Finally, we show that when the charging scheme involves membership based charges, the above result holds in important special cases.

An analog MVUE for a wireless sensor network

An analog minimum-variance unbiased estimator(MVUE) over an asymmetric wireless sensor network is studied.Minimisation of variance is cast into a constrained non-convex optimisation problem. An explicit algorithm that solves the problem is provided. The solution is obtained by decomposing the original problem into a finite number of convex optimisation problems with explicit solutions. These solutions are then juxtaposed together by exploiting further structure in the objective function.

A Nash Bargaining Approach to Retention Enhancing Bid Optimization in Sponsored Search Auctions with Discrete Bids

Bid optimization is now becoming quite popular in sponsored search auctions on the Web. Given a keyword and the maximum willingness to pay of each advertiser interested in the keyword, the bid optimizer generates a profile of bids for the advertisers with the objective of maximizing customer retention without compromising the revenue of the search engine. In this paper, we present a bid optimization algorithm that is based on a Nash bargaining model where the first player is the search engine and the second player is a virtual agent representing all the bidders. We make the realistic assumption that each bidder specifies a maximum willingness to pay values and a discrete, finite set of bid values. We show that the Nash bargaining solution for this problem always lies on a certain edge of the convex hull such that one end point of the edge is the vector of maximum willingness to pay of all the bidders. We show that the other endpoint of this edge can be computed as a solution of a linear programming problem. We also show how the solution can be transformed to a bid profile of the advertisers.

Power control and transmission scheduling for network utility maximization in wireless networks

We consider a joint power control and transmission scheduling problem in wireless networks with average power constraints. While the capacity region of a wireless network is convex, a characterization of this region is a hard problem. We formulate a network utility optimization problem involving time-sharing across different "transmission modes," where each mode corresponds to the set of power levels used in the network. The structure of the optimal solution is a time-sharing across a small set of such modes. We use this structure to develop an efficient heuristic approach to finding a suboptimal solution through column generation iterations. This heuristic approach converges quite fast in simulations, and provides a tool for wireless network planning.

A Fast Linear Separability Test by Projection of Positive Points on Subspaces

A geometric and non parametric procedure for testing if two finite set of points are linearly separable is proposed. The Linear Separability Test is equivalent to a test that determines if a strictly positive point h > 0 exists in the range of a matrix A (related to the points in the two finite sets). The algorithm proposed in the paper iteratively checks if a strictly positive point exists in a subspace by projecting a strictly positive vector with equal co-ordinates (p), on the subspace. At the end of each iteration, the subspace is reduced to a lower dimensional subspace. The test is completed within r ≤ min(n, d + 1) steps, for both linearly separable and non separable problems (r is the rank of A, n is the number of points and d is the dimension of the space containing the points). The worst case time complexity of the algorithm is O(nr3) and space complexity of the algorithm is O(nd). A small review of some of the prominent algorithms and their time complexities is included. The worst case computational complexity of our algorithm is lower than the worst case computational complexity of Simplex, Perceptron, Support Vector Machine and Convex Hull Algorithms, if d

Pricing a Shared Access Link for Fair and Efficient Operation with Variable User Data Rates

The literature on pricing implicitly assumes an "infinite data" model, in which sources can sustain any data rate indefinitely. We assume a more realistic "finite data" model, in which sources occasionally run out of data; this leads to variable user data rates. Further, we assume that users have contracts with the service provider, specifying the rates at which they can inject traffic into the network. Our objective is to study how prices can be set such that a single link can be shared efficiently and fairly among users in a dynamically changing scenario where a subset of users occasionally has little data to send. User preferences are modelled by concave increasing utility functions. Further, we introduce two additional elements: a convex increasing disutility function and a convex increasing multiplicative congestion-penally function. The disutility function takes the shortfall (contracted rate minus present rate) as its argument, and essentially encourages users to send traffic at their contracted rates, while the congestion-penalty function discourages heavy users from sending excess data when the link is congested. We obtain simple necessary and sufficient conditions on prices for fair and efficient link sharing; moreover, we show that a single price for all users achieves this. We illustrate the ideas using a simple experiment.

Relay Precoder Optimization in MIMO-Relay Networks With Imperfect CSI

In this paper, we consider robust joint designs of relay precoder and destination receive filters in a nonregenerative multiple-input multiple-output (MIMO) relay network. The network consists of multiple source-destination node pairs assisted by a MIMO-relay node. The channel state information (CSI) available at the relay node is assumed to be imperfect. We consider robust designs for two models of CSI error. The first model is a stochastic error (SE) model, where the probability distribution of the CSI error is Gaussian. This model is applicable when the imperfect CSI is mainly due to errors in channel estimation. For this model, we propose robust minimum sum mean square error (SMSE), MSE-balancing, and relay transmit power minimizing precoder designs. The next model for the CSI error is a norm-bounded error (NBE) model, where the CSI error can be specified by an uncertainty set. This model is applicable when the CSI error is dominated by quantization errors. In this case, we adopt a worst-case design approach. For this model, we propose a robust precoder design that minimizes total relay transmit power under constraints on MSEs at the destination nodes. We show that the proposed robust design problems can be reformulated as convex optimization problems that can be solved efficiently using interior-point methods. We demonstrate the robust performance of the proposed design through simulations.

Polynomial Approximation, Local Polynomial convexity, and Degenerate CR Singularities- II

We provide some conditions for the graph of a Holder-continuous function on (D) over bar, where (D) over bar is a closed disk in C, to be polynomially convex. Almost all sufficient conditions known to date - provided the function (say F) is smooth - arise from versions of the Weierstrass Approximation Theorem on (D) over bar. These conditions often fail to yield any conclusion if rank(R)DF is not maximal on a sufficiently large subset of (D) over bar. We bypass this difficulty by introducing a technique that relies on the interplay of certain plurisubharmonic functions. This technique also allows us to make some observations on the polynomial hull of a graph in C(2) at an isolated complex tangency.

Some Aspects of the Kobayashi and Carath,odory Metrics on Pseudoconvex Domains

The purpose of this article is to consider two themes, both of which emanate from and involve the Kobayashi and the Carath,odory metric. First, we study the biholomorphic invariant introduced by B. Fridman on strongly pseudoconvex domains, on weakly pseudoconvex domains of finite type in C (2), and on convex finite type domains in C (n) using the scaling method. Applications include an alternate proof of the Wong-Rosay theorem, a characterization of analytic polyhedra with noncompact automorphism group when the orbit accumulates at a singular boundary point, and a description of the Kobayashi balls on weakly pseudoconvex domains of finite type in C (2) and convex finite type domains in C (n) in terms of Euclidean parameters. Second, a version of Vitushkin's theorem about the uniform extendability of a compact subgroup of automorphisms of a real analytic strongly pseudoconvex domain is proved for C (1)-isometries of the Kobayashi and Carath,odory metrics on a smoothly bounded strongly pseudoconvex domain.

On concave univalent functions

We consider functions that map the open unit disc conformally onto the complement of an unbounded convex set with opening angle pa, a ? (1, 2], at infinity. In this paper, we show that every such function is close-to-convex of order (a - 1) and is included in the set of univalent functions of bounded boundary rotation. Many interesting consequences of this result are obtained. We also determine the extreme points of the set of concave functions with respect to the linear structure of the Hornich space.

The local polynomial hull near a degenerate CR singularity: Bishop discs revisited

Let be a smooth real surface in and let be a point at which the tangent plane is a complex line. How does one determine whether or not is locally polynomially convex at such a p-i.e. at a CR singularity? Even when the order of contact of with at p equals 2, no clean characterisation exists; difficulties are posed by parabolic points. Hence, we study non-parabolic CR singularities. We show that the presence or absence of Bishop discs around certain non-parabolic CR singularities is completely determined by a Maslov-type index. This result subsumes all known facts about Bishop discs around order-two, non-parabolic CR singularities. Sufficient conditions for Bishop discs have earlier been investigated at CR singularities having high order of contact with . These results relied upon a subharmonicity condition, which fails in many simple cases. Hence, we look beyond potential theory and refine certain ideas going back to Bishop.

Efficient Methods for Robust Classification Under Uncertainty in Kernel Matrices

In this paper we study the problem of designing SVM classifiers when the kernel matrix, K, is affected by uncertainty. Specifically K is modeled as a positive affine combination of given positive semi definite kernels, with the coefficients ranging in a norm-bounded uncertainty set. We treat the problem using the Robust Optimization methodology. This reduces the uncertain SVM problem into a deterministic conic quadratic problem which can be solved in principle by a polynomial time Interior Point (IP) algorithm. However, for large-scale classification problems, IP methods become intractable and one has to resort to first-order gradient type methods. The strategy we use here is to reformulate the robust counterpart of the uncertain SVM problem as a saddle point problem and employ a special gradient scheme which works directly on the convex-concave saddle function. The algorithm is a simplified version of a general scheme due to Juditski and Nemirovski (2011). It achieves an O(1/T-2) reduction of the initial error after T iterations. A comprehensive empirical study on both synthetic data and real-world protein structure data sets show that the proposed formulations achieve the desired robustness, and the saddle point based algorithm outperforms the IP method significantly.

Identification of Active Sources in Single-Channel Convolutive Mixtures Using Known Source Models

We address the problem of identifying the constituent sources in a single-sensor mixture signal consisting of contributions from multiple simultaneously active sources. We propose a generic framework for mixture signal analysis based on a latent variable approach. The basic idea of the approach is to detect known sources represented as stochastic models, in a single-channel mixture signal without performing signal separation. A given mixture signal is modeled as a convex combination of known source models and the weights of the models are estimated using the mixture signal. We show experimentally that these weights indicate the presence/absence of the respective sources. The performance of the proposed approach is illustrated through mixture speech data in a reverberant enclosure. For the task of identifying the constituent speakers using data from a single microphone, the proposed approach is able to identify the dominant source with up to 8 simultaneously active background sources in a room with RT60 = 250 ms, using models obtained from clean speech data for a Source to Interference Ratio (SIR) greater than 2 dB.

On the Erdos-Szekeres n-interior point problem

The n-interior point variant of the Erdos-Szekeres problem is to show the following: For any n, n-1, every point set in the plane with sufficient number of interior points contains a convex polygon containing exactly n-interior points. This has been proved only for n-3. In this paper, we prove it for pointsets having atmost logarithmic number of convex layers. We also show that any pointset containing atleast n interior points, there exists a 2-convex polygon that contains exactly n-interior points.

Loop transformations: convexity, pruning, and optimization

High-level loop transformations are a key instrument in mapping computational kernels to effectively exploit the resources in modern processor architectures. Nevertheless, selecting required compositions of loop transformations to achieve this remains a significantly challenging task; current compilers may be off by orders of magnitude in performance compared to hand-optimized programs. To address this fundamental challenge, we first present a convex characterization of all distinct, semantics-preserving, multidimensional affine transformations. We then bring together algebraic, algorithmic, and performance analysis results to design a tractable optimization algorithm over this highly expressive space. Our framework has been implemented and validated experimentally on a representative set of benchmarks running on state-of-the-art multi-core platforms.