Concave function

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

Primal-dual and dual-fitting analysis of online scheduling algorithms for generalized flow-time problems

Abstract: We study online scheduling problems on a single processor that can be viewed as extensions of the well-studied problem of minimizing total weighted flow time. In particular, we provide a framework of analysis that is derived by duality properties, does not rely on potential functions and is applicable to a variety of scheduling problems. A key ingredient in our approach is bypassing the need for "black-box" rounding of fractional solutions, which yields improved competitive ratios. We begin with an interpretation of Highest-Density-First (HDF) as a primal-dual algorithm, and a corresponding proof that HDF is optimal for total fractional weighted flow time (and thus scalable for the integral objective). Building upon the salient ideas of the proof, we show how to apply and extend this analysis to the more general problem of minimizing $\sum_j w_j g(F_j)$, where $w_j$ is the job weight, $F_j$ is the flow time and $g$ is a non-decreasing cost function. Among other results, we present improved competitive ratios for the setting in which $g$ is a concave function, and the setting of same-density jobs but general cost functions. We further apply our framework of analysis to online weighted completion time with general cost functions as well as scheduling under polyhedral constraints.

A computability strategy for optimization of multiresolution broadcast systems: a layered energy distribution approach

Abstract: This paper investigates the possible system gain for a multiresolution broadcast system using multilayer transmission of multiresolution data by utilizing nonuniform layer transmission energies. It shows how to find the energy distribution that maximizes the system performance, measured in the form of a sum of a weighted layer values /spl times/ population product (representing possible revenue for service providers). Through the introduction of the relative population coverage function P(a/sub i/) it is shown for a N layer system that in many cases when P(a/sub i/) is a concave function (equivalent to -P(a/sub i/) being convex) it is possible to reduce what seems to be an N-dimensional problem to N line searches. The paper also shows how the relative population coverage function can be constructed in two ways. The first uses analytic models for signal strength and population coverage (Uniform and Rayleigh). The second uses numerical signal strength and population estimates in grid format. The paper also includes examples to illustrate how the method works and the performance gain it provides. One of the examples uses actual grid estimates for an example transmitter located in Lulea, Sweden.

Necessary and Sufficient Conditions for Dominance Using Generalized Lorenz Curves

Abstract: Almost all functions used in economics are concave, but there are important examples of convex functions, e. g., cost functions and poverty or concentration indices. In these cases one can choose between working with convex or concave functions (e.g., poverty indices or welfare functions).

A polynomial algorithm for resourse allocation problems with polymatroid constrains 1

Abstract: We investigate the following problem. It is necessary to allocate a discrete resource to a set of activities so as to maximize a separable concave function of a profit over a set of feasible resource allocations formed by integer points of a polymatroid. We present the Dichotomic Greedy Algorithm (DGA) for solving this problem and prove that the running time of DGA is polynomially bounded. The implementation of DGA is discussed for uniform and generalized symmetric polymatroids.