Concave function

About: Concave function is a research topic. Over the lifetime, 1415 publications have been published within this topic receiving 33278 citations.

Lightweight Monitoring of Distributed Streams

Abstract: As data becomes dynamic, large, and distributed, there is increasing demand for what have become known as distributed stream algorithms. Since continuously collecting the data to a central server and processing it there is infeasible, a common approach is to define local conditions at the distributed nodes, such that—as long as they are maintained—some desirable global condition holds.Previous methods derived local conditions focusing on communication efficiency. While proving very useful for reducing the communication volume, these local conditions often suffer from heavy computational burden at the nodes. The computational complexity of the local conditions affects both the runtime and the energy consumption. These are especially critical for resource-limited devices like smartphones and sensor nodes. Such devices are becoming more ubiquitous due to the recent trend toward smart cities and the Internet of Things. To accommodate for high data rates and limited resources of these devices, it is crucial that the local conditions be quickly and efficiently evaluated.Here we propose a novel approach, designated CB (for Convex/Concave Bounds). CB defines local conditions using suitably chosen convex and concave functions. Lightweight and simple, these local conditions can be rapidly checked on the fly. CB’s superiority over the state-of-the-art is demonstrated in its reduced runtime and power consumption, by up to six orders of magnitude in some cases. As an added bonus, CB also reduced communication overhead in all the tested application scenarios.

Nondifferentiable mathematical programming and convex-concave functions

Abstract: For a convex-concave functionL(x, y), we define the functionf(x) which is obtained by maximizingL with respect toy over a specified set. The minimization problem with objective functionf is considered. We derive necessary conditions of optimality for this problem. Based upon these necessary conditions, we define its dual problem. Furthermore, a duality theorem and a converse duality theorem are obtained. It is made clear that these results are extensions of those derived in studies on a class of nondifferentiable mathematical programming problems.

Quermassintegrals of quasi-concave functions and generalized Pr\'ekopa-Leindler inequalities

Abstract: We extend to a functional setting the concept of quermassintegrals, well-known within the Minkowski theory of convex bodies. We work in the class of quasi-concave functions defined on the Euclidean space, and with the hierarchy of their subclasses given by $\alpha$-concave functions. In this setting, we investigate the most relevant features of functional quermassintegrals, and we show they inherit the basic properties of their classical geometric counterpart. As a first main result, we prove a Steiner-type formula which holds true by choosing a suitable functional equivalent of the unit ball. Then, we establish concavity inequalities for quermassintegrals and for other general hyperbolic functionals, which generalize the celebrated Prekopa-Leindler and Brascamp-Lieb inequalities. Further issues that we transpose to this functional setting are: integral-geometric formulae of Cauchy-Kubota type, valuation property and isoperimetric/Uryshon-like inequalities.

A Distributed Probability Collectives Optimization Method for Multicast in CDMA Wireless Data Networks

Abstract: Recently, the problem of resource allocation for achieving optimal multicast throughput has become of high interest. In this paper, we consider scenarios in which the utility function is an arbitrary, rather than concave function of multicast rate and specifically apply the proposed algorithm to the multicast problem in CDMA wireless networks. By using a combination of probability collectives (PC) method and network coding, a new cross-layer optimization approach which converges to a sub-optimal solution is proposed. This solution has been achieved by decomposing the original problem into data routing sub-problems at the network layer and power allocation sub-problems at the physical layer. These sub-problems are then coupled through a set of Lagrangian multipliers and each sub- problem is solved in a distributed fashion. It will also be shown that if the utility function is concave or monotonic, the proposed approach converges to a near optimal solution.

A Robust Approach to Equilibrium Yield Curves

Abstract: The surplus yield models of fisheries management usually assume that a concave function of equilibrium yield versus fishing effort exists. However, this function is notoriously difficult to fit to real data for a number of reasons, including the fact that few fisheries are in equilibrium. A procedure for obtaining rough estimates of these equilibrium curves is introduced. Management strategies based on these estimates and the annual yield curves are also presented. The procedures are then applied to several fish stocks.