Showing papers on "Concave function published in 2004"

Some inequalities for differentiable convex and concave mappings

Abstract: Several inequalities are obtained for some differentiable convex mappings that are connected with the celebrated Hermite-Hadamard integral inequality. Also a parallel development is made for concave functions.

Expected Profitability of Capital under Uncertainty – a Microeconomic Perspective

Abstract: Hartman (1972) and Abel (1983) showed that when firms are competitive and there is flexibility of labour relative to capital, marginal profitability of capital is a convex function of the stochastic variable (e.g., price); by Jensen’s inequality, this means that uncertainty increases the expected profitability of capital, which increases the incentive to invest. We argue that, besides factor substitutability, the relevant assumption for the convexity property to hold is the implicit assumption about the choice variable in the representative firm’s maximisation problem: the assumption of perfect competition implies that the choice variable is output and that price is exogenous. However, in the case of a firm facing a downward-sloping demand curve, both output and output price emerge as the possible choice variable. We show that, when price is the choice variable, marginal profitability of capital is a concave function of the stochastic variable; hence, by Jensen’s inequality, an increase in uncertainty decreases the expected profitability of capital. We also show that keeping the assumption of factor substitutability but changing the share of labour in the production function has an important impact on the degree of concavity/convexity of the capital profit function.

Consistency of Concave Regression with an Application to Current-Status Data

Abstract: We consider the problem of nonparametric estimation of a concave regression function F. We show that the supremum distance between the least square s estimatorand F on a compact interval is typically of order(log(n)/n)2/5. This entails rates of convergence for the estimator’s derivative. Moreover, we discuss the impact of additional constraints on F such as monotonicity and pointwise bounds. Then we apply these results to the analysis of current status data, where the distribution function of the event times is assumed to be concave.

Global convergence of Newton's method on an interval

Abstract: The solution of an equation f(x)=γ given by an increasing function f on an interval I and right-hand side γ, can be approximated by a sequence calculated according to Newton’s method. In this article, global convergence of the method is considered in the strong sense of convergence for any initial value in I and any feasible right-hand side. The class of functions for which the method converges globally is characterized. This class contains all increasing convex and increasing concave functions as well as sums of such functions on the given interval. The characterization is applied to Kepler’s equation and to calculation of the internal rate of return of an investment project.

Robust Control: A Note on the Response of the Control to Changes in the

Abstract: In this paper an analytical framework using robust control was developed for the one-state and one-control variable model to examine the response of the control to changes in the "free" parameter. However, in contrast to Gonzalez and Rodriguez (2003), the sign of the ``free" parameter in the criterion function of the min-max problem is negative. We find that this set up corresponds to the case where nature is benevolent while the problem posed by Gonzalez and Rodriguez (2003) corresponds to one of a malevolent nature. For the benevolent case, we show that: (i) more uncertainty produces a more cautious response of the control variable, and(ii) the control variable response is concave down for an increasing shape and concave up for a decreasing shape.

On Concavification and Convex Games

Abstract: We propose a new geometric approach for the analysis of cooperative games. A cooperative game is viewed as a real valued function $u$ defined on a finite set of points in the unit simplex. We define the \emph{concavification} of $u$ on the simplex as the minimal concave function on the simplex which is greater than or equal to $u$. The concavification of $u$ induces a game which is the \emph{totally balanced cover} of the game. The concavification of $u$ is used to characterize well-known classes of games, such as balanced, totally balanced, exact and convex games. As a consequence of the analysis it turns out that a game is convex if and only if each one of its sub-games is exact.

On algebraic connectivity as a function of an edge weight

Abstract: We consider the Laplacian matrix of a weighted graph, and how the algebraic connectivity, α, behaves when considered as a function of a single edge weight. Under suitable differentiability conditions, we bound the first derivative of α from above, show that α is necessarily concave down, and produce a lower bound on the second derivative of α. When α is simple, we discuss the effect of increasing an edge weight on the corresponding Fiedler vector. We also compute the limiting value of α as the edge weight increases to infinity.

Approximation Algorithms for Mixed Fractional Packing and Covering Problems.

Abstract: We study general mixed fractional packing and covering problems (MPCe ) of the following form: Given a vector $f: B \rightarrow {\rm IR}^{M}_{+}$ of M nonnegative continuous convex functions and a vector $g: B \rightarrow {\rm IR}^{M}_{+}$ of M nonnegative continuous concave functions, two M – dimensional nonnegative vectors a,b, a nonempty convex compact set B and a relative tolerance e ∈ (0,1), find an approximately feasible vector x ∈ B such that f(x) ≤ (1 + e) a and g(x) ≥ (1 – e) b or find a proof that no vector is feasible (that satisfies x ∈ B, f(x) ≤ a and g(x) ≥ b).

Comparison theorems between several quasi-arithmetic means

Abstract: [1] S. ABRAMOVICH, J. E. PECARIC, Convex and Concave Functions and Generalized Holder Inequalities, Soochow Journal of Mathematics, Taiwan, 14 (1998), pp. 261–272. [2] YOUNG -HO KIM, Refinements and extension of an Inequality, Journal of Mathematical Analysis and Applications 245, 628–632 (2000). [3] D. S. MITRINOVIC, J. E. PECARIC AND A. M. FINK, Classical and New Inequalities in Analysis, Kluwer Academic, Dordrecht/Boston/London, 1993.

On the Two-Box Paradox

Abstract: On a game show, you are presented with two identical boxes. Both boxes contain positive monetary prizes, one twice the other. You are allowed to pick one box and observe the prize x > 0, after which you can choose to trade boxes. In terms of simple expected value, it is always better to trade since (2x) + () 5 > x. That is the paradox. Simple thought experiments suggest that a sufficiently large observed prize would cause a player not to trade, despite the mathematical computation of expected value. In individual cases, this creates some threshold, which depends on the observed prize, for ceasing to trade. A player may have in mind prior probabilities about what prizes the game show would offer, so that an observed prize of $10,000, for instance, would not yield equal judgmental odds of $20,000 or $5,000 in the unobserved box. The judgmental probability approach to the two-box problem seeks to develop optimal threshold strategies in terms of prior distributions on the set of possible prizes. Recent articles in this MAGAZINE have focused on the judgmental probability approach, although they have also discussed the second line of attack on this problem, expected utility [2, 3]. In expected utility theory, it is assumed than an individual has an underlying utility function for wealth. This utility function is increasing because it is presumed that an individual will always prefer more wealth to less wealth. In addition, the utility function is concave because it is presumed that an individual will have nonincreasing marginal utility for wealth. The utility function u is thus an increasing, concave function from the positive half line into the real line. The scaling on this function is unimportant because a positive linear transformation a + bu, with b > 0, is equivalent for individual REM 2. When n > 2 is even, integers a, b, and c satisfy a2 + b2 = cn if and 302 MATHEMATICS MAGAZINE

Linear programs with an additional separable concave constraint

Abstract: In this paper, we develop two algorithms for globally optimizing a special class of linear programs with an additional concave constraint. We assume that the concave constraint is defined by a separable concave function. Exploiting this special structure, we apply Falk-Soland's branch-and-bound algorithm for concave minimization in both direct and indirect manners. In the direct application, we solve the problem alternating local search and branch-and-bound. In the indirect application, we carry out the bounding operation using a parametric right-hand-side simplex algorithm.

The Max-Convolution Approach to Equilibrium Models with Indivisibilities

Abstract: This paper studies a competitive market model for trading indivisible commodities. Commodities can be desirable or undesirable. Agents' preferences depend on the bundle of commodities and the quantity of money they hold. We assume that agents have quasi-linear utilities in money. Using the max-convolution approach, we demonstrate that the market has a Walrasian equilibrium if and only if the potential market value function is concave with respect to the total initial endowment of commodities. We then identify sufficient conditions on each individual agent's behavior. In particular, we introduce a class of new utility functions, called the class of max-convolution concavity preservable utility functions. This class of utility functions covers both the class of functions which satisfy the gross substitutes condition of Kelso and Crawford (1982), or the single improvement condition, or the no complementarities condition of Gul and Stacchetti (1999), and the class of discrete concave functions of Murota and Shioura (1999).

Absolute stability analysis through a connection to saturation nonlinearities

Abstract: A generalized sector was introduced recently for improved stability analysis of systems with nonlinearity and/or uncertainty. While providing more flexibility and admitting more accuracy in the description of the nonlinear/uncertain component, the generalized sector is almost as numerically tractable as the traditional conic sector - necessary and sufficient conditions for absolute quadratic stability were identified in the form of linear matrix inequalities (LMIs) for continuous-time systems with one nonlinear component. The objective of this paper is to develop a general framework for absolute stability analysis of systems with multiple nonlinear components under a generalized sector condition. Through a connection between saturation functions and piecewise linear convex/concave functions, the generalized sector is described in terms of a set of saturation functions. This transforms the problem of absolute stability analysis into one of stability analysis for systems with saturation nonlinearities, for which effective tools have recently been developed. Under the general framework, we develop explicit conditions for absolute quadratic stability of discrete-time systems with one nonlinear component.

Sequence Inequalities for the Logarithmic Convex (Concave) Function

Abstract: Let f be a positive strictly increasing logarithmic convex (or logarithmic concave) function on (0;1], then, for k being a nonnegative integer and n a natural number, the sequence 1 P n+k i=k+1 lnf( i

Integer concave cocirculations and Honeycombs

Abstract: A convex triangular grid is a planar digraph G embedded in the plane so that each bounded face is an equilateral triangle with three edges and their union R forms a convex polygon. A function h: E(G) → R is called a concave cocirculation if h(e) = g(v) - g(u) for each edge e = (u, v), where g is a concave function on R which is affinely linear within each bounded face of G. Knutson and Tao obtained an integrality result on so-called honeycombs implying that if an integer-valued function on the boundary edges is extendable to a concave cocirculation, then it is extendable to an integer one. We show a sharper property: for any concave cocirculation h, there exists an integer concave cocirculation h' satisfying h'(e) = h(e) for each edge e with h(e) E Z contained in the boundary or in a bounded face where h is integer on all edges. Also relevant polyhedral and algorithmic results are presented.

Integer Concave Cocirculations and Honeycombs

Abstract: A convex triangular grid is a planar digraph G embedded in the plane so that each bounded face is an equilateral triangle with three edges and their union \({\cal R}\) forms a convex polygon. A function \(h:E(G)\to{\mathbb R}\) is called a concave cocirculation if h(e)=g(v)–g(u) for each edge e=(u,v), where g is a concave function on \({\cal R}\) which is affinely linear within each bounded face of G. Knutson and Tao obtained an integrality result on so-called honeycombs implying that if an integer-valued function on the boundary edges is extendable to a concave cocirculation, then it is extendable to an integer one.

Convexification and concavification methods for some global optimization problems

Abstract: In this paper, firstly, we propose several convexification and concavification transformations to convert a strictly monotone function into a convex or concave function, then we propose several convexification and concavification transformations to convert a non-convex and non-concave objective function into a convex or concave function in the programming problems with convex or concave constraint functions, and propose several convexification and concavification transformations to convert a non-monotone objective function into a convex or concave function in some programming problems with strictly monotone constraint functions. Finally, we prove that the original programming problem can be converted into an equivalent concave minimization problem, or reverse convex programming problem or canonical D.C. programming problem. Then the global optimal solution of the original problem can be obtained by solving the converted concave minimization problem, or reverse convex programming problem or canonical D.C. programming problem using the existing algorithms about them.

The rate of convergence of Fourier series with respect to the eigenfunctions of a positive operator

Abstract: The paper develops some ideas of M. A. Krasnoselskii, expounded in the monograph [4], Sections 22.1-22.4. New results are obtained by the use of concave functions of a positive operator, which properties were studied in previous author's works.

Packet marking function of active queue management mechanism: should it be linear, concave, or convex?

Abstract: Recently, several gateway-based congestion control mechanisms have been proposed to support the end-to-end congestion control mechanism of TCP (Transmission Control Protocol). In this paper, we focus on RED (Random Early Detection), which is a promising gateway-based congestion control mechanism. RED randomly drops an arriving packet with a probability proportional to its average queue length (i.e., the number of packets in the buffer). However, it is still unclear whether the packet marking function of RED is optimal or not. In this paper, we investigate what type of packet marking function, which determines the packet marking probability from the average queue length, is suitable from the viewpoint of both steady state and transient state performances. Presenting several numerical examples, we investigate the advantages and disadvantages of three packet marking functions: linear, concave, and convex. We show that, although the average queue length in the steady state becomes larger, use of a concave function improves the transient behavior of RED and also realizes robustness against network status changes such as variation in the number of active TCP connections.

Approximation algorithms for mixed fractional packing and covering problems

Abstract: We study general mixed fractional packing and covering problems (MPCe ) of the following form: Given a vector $f: B \rightarrow {\rm IR}^{M}_{+}$ of M nonnegative continuous convex functions and a vector $g: B \rightarrow {\rm IR}^{M}_{+}$ of M nonnegative continuous concave functions, two M – dimensional nonnegative vectors a,b, a nonempty convex compact set B and a relative tolerance e ∈ (0,1), find an approximately feasible vector x ∈ B such that f(x) ≤ (1 + e) a and g(x) ≥ (1 – e) b or find a proof that no vector is feasible (that satisfies x ∈ B, f(x) ≤ a and g(x) ≥ b).

Discrete strip-concave functions, Gelfand-Tsetlin patterns, and related polyhedra

Abstract: Discrete strip-concave functions considered in this paper are, in fact, equivalent to an extension of Gelfand-Tsetlin patterns to the case when the pattern has a not necessarily triangular but convex configuration. They arise by releasing one of the three types of rhombus inequalities for discrete concave functions (or ``hives'') on a ``convex part'' of a triangular grid. The paper is devoted to a combinatorial study of certain polyhedra related to such functions or patterns, and results on faces, integer points and volumes of these polyhedra are presented. Also some relationships and applications are discussed. In particular, we characterize, in terms of valid inequalities, the polyhedral cone formed by the boundary values of discrete strip-concave functions on a grid having trapezoidal configuration. As a consequence of this result, necessary and sufficient conditions on a pair of vectors to be the shape and content of a semi-standard skew Young tableau are obtained.

Integer concave cocirculations and honeycombs

Abstract: A convex triangular grid is represented by a planar digraph $G$ embedded in the plane so that (a) each bounded face is surrounded by three edges and forms an equilateral triangle, and (b) the union $\Rscr$ of bounded faces is a convex polygon. A real-valued function $h$ on the edges of $G$ is called a concave cocirculation if $h(e)=g(v)-g(u)$ for each edge $e=(u,v)$, where $g$ is a concave function on $\Rscr$ which is affinely linear within each bounded face of $G$. Knutson and Tao [J. Amer. Math. Soc. 12 (4) (1999) 1055--1090] proved an integrality theorem for so-called honeycombs, which is equivalent to the assertion that an integer-valued function on the boundary edges of $G$ is extendable to an integer concave cocirculation if it is extendable to a concave cocirculation at all. In this paper we show a sharper property: for any concave cocirculation $h$ in $G$, there exists an integer concave cocirculation $h'$ satisfying $h'(e)=h(e)$ for each boundary edge $e$ with $h(e)$ integer and for each edge $e$ contained in a bounded face where $h$ takes integer values on all edges. On the other hand, we explain that for a 3-side grid $G$ of size $n$, the polytope of concave cocirculations with fixed integer values on two sides of $G$ can have a vertex $h$ whose entries are integers on the third side but $h(e)$ has denominator $\Omega(n)$ for some interior edge $e$. Also some algorithmic aspects and related results on honeycombs are discussed.

Network flow problems with step cost functions

Abstract: Network flow problems are widely studied, especially for those having convex cost functions, fixed-charge cost functions, and concave functions. However, network flow problems with general nonlinear cost functions receive little attention. The problems with step cost functions are important due to the many practical applications. In this paper, these problems are discussed and formulated as equivalent mathematical mixed 0-1 linear programming problems. Computational results on randomly generated test beds for these exact approached solution procedure are reported in the paper.

Reducing queue-fill variability for emergency traffic in a differentiated services network

Abstract: Supporting emergency services applications using the Internet has gained considerable interest in the recent past. A simple multiple average multiple threshold (MAMT) random early detection (RED) scheme is used to provide the necessary quality of service for emergency traffic. We have studied the RED dropping function as one of the contributing factors towards the variation observed in the queue fill. This may cause interactive emergency applications to experience large delay and delay variation thus deteriorating the service provided to them. Hence, we compare the performance of the linear dropping function in terms of average delay and variation in queue fill with that of the three other dropping functions namely: concave, convex and step. With the help of an exhaustive simulation study conducted using the network simulator-2, we conclude that the concave function provides the best performance