Showing papers on "Concave function published in 2006"

Absolute value equation solution via concave minimization

Abstract: The NP-hard absolute value equation (AVE) Ax − |x| = b where \(A\in R^{n\times n}\) and \(b\in R^n\) is solved by a succession of linear programs The linear programs arise from a reformulation of the AVE as the minimization of a piecewise-linear concave function on a polyhedral set and solving the latter by successive linearization A simple MATLAB implementation of the successive linearization algorithm solved 100 consecutively generated 1,000-dimensional random instances of the AVE with only five violated equations out of a total of 100,000 equations

Oblivious network design

Abstract: Consider the following network design problem: given a network G = (V, E), source-sink pairs {si, ti} arrive and desire to send a unit of flow between themselves. The cost of the routing is this: if edge e carries a total of fe flow (from all the terminal pairs), the cost is given by Σ el(fe), where l is some concave cost function; the goal is to minimize the total cost incurred. However, we want the routing to be oblivious: when terminal pair {si, ti} makes its routing decisions, it does not know the current flow on the edges of the network, nor the identity of the other pairs in the system. Moreover, it does not even know the identity of the function l, merely knowing that l is a concave function of the total flow on the edge. How should it (obliviously) route its one unit of flow? Can we get competitive algorithms for this problem?In this paper, we develop a framework to model oblivious network design problems (of which the above problem is a special case), and give algorithms with poly-logarithmic competitive ratio for problems in this framework (and hence for this problem). Abstractly, given a problem like the one above, the solution is a multicommodity flow producing a "load" on each edge of Le = l(f1(e),f2(e), ..., fk(e)), and the total cost is given by an "aggregation function" agg (Le1,...,Lem) of the loads of all edges. Our goal is to develop oblivious algorithms that approximately minimize the total cost of the routing, knowing the aggregation function agg, but merely knowing that l lies in some class C, and having no other information about the current state of the network. Hence we want algorithms that are simultaneously "function-oblivious" as well as "traffic-oblivious".The aggregation functions we consider are the max and σ objective functions, which correspond to the well-known measures of congestion and total cost of a network; in this paper, we prove the following:• If the aggregation function is Σ, we give an oblivious algorithm with O(log2n) competitive ratio whenever the load function l is in the class of monotone sub-additive functions. (Recall that our algorithm is also "function-oblivious"; it works whenever each edge has a load function l in the class.)• For the case when the aggregation function is max, we give an oblivious algorithm with O(log2n log log n) competitive ratio, when the load function l is a norm; we also show that such a competitive ratio is not possible for general sub-additive functions.These are the first such general results about oblivious algorithms for network design problems, and we hope the ideas and techniques will lead to more and improved results in this area.

Greedy primal-dual algorithm for dynamic resource allocation in complex networks

Abstract: In Stolyar (Queueing Systems 50 (2005) 401---457) a dynamic control strategy, called greedy primal-dual (GPD) algorithm, was introduced for the problem of maximizing queueing network utility subject to stability of the queues, and was proved to be (asymptotically) optimal. (The network utility is a concave function of the average rates at which the network generates several "commodities.") Underlying the control problem of Stolyar (Queueing Systems 50 (2005) 401---457) is a convex optimization problem subject to a set of linear constraints. In this paper we introduce a generalized GPD algorithm, which applies to the network control problem with additional convex (possibly non-linear) constraints on the average commodity rates. The underlying optimization problem in this case is a convex problem subject to convex constraints. We prove asymptotic optimality of the generalized GPD algorithm. We illustrate key features and applications of the algorithm on simple examples.

Simultaneous Optimization via Approximate Majorization for Concave Profits or Convex Costs

Abstract: For multicriteria problems and problems with a poorly characterized objective, it is often desirable to approximate simultaneously the optimum solution for a large class of objective functions. We consider two such classes: (1) Maximizing all symmetric concave functions. (2) Minimizing all symmetric convex functions. The first class corresponds to maximizing profit for a resource allocation problem (such as allocation of bandwidths in a computer network). The concavity requirement corresponds to the law of diminishing returns in economics. The second class corresponds to minimizing cost or congestion in a load balancing problem, where the congestion/cost is some convex function of the loads. Informally, a simultaneous α-approximation for either class is a feasible solution that is within a factor α of the optimum for all functions in that class. Clearly, the structure of the feasible set has a significant impact on the best possible α and the computational complexity of finding a solution that achieves (or nearly achieves) this α. We develop a framework and a set of techniques to perform simultaneous optimization for a wide variety of problems. We first relate simultaneous α-approximation for both classes to α-approximate majorization. Then we prove that α-approximately majorized solutions exist for logarithmic values of α for the concave profits case. For both classes, we present a polynomial-time algorithm to find the best α if the set of constraints is a polynomial-sized linear program and discuss several non-trivial applications. These applications include finding a (log n)-majorized solution for multicommodity flow, and finding approximately best α for various forms of load balancing problems. Our techniques can also be applied to produce approximately fair versions of the facility location and bi-criteria network design problems. In addition, we demonstrate interesting connections between distributional load balancing (where the sizes of jobs are drawn from known probability distributions but the actual size is not known at the time of placement) and approximate majorization.

Minimax Estimation of Location Parameters for Spherically Symmetric Distributions with Concave Loss

Abstract: For p >4 and one observation X on a p-dimensional spherically symmetric distribution, minimax estimators of Theta whose risks are smaller than the risk of X (the best invariant estimator) are found when the loss is a nondecreasing concave function of quadratic loss. For n observations X1, X2, ... Xn, we have classes of minimax estimators which are better than the usual procedures, such as the best invariant estimator, X-bar, or a maximum likelihood estimator.

Properties of game options

Abstract: A game option is an American option with the added feature that not only the option holder, but also the option writer, can exercise the option at any time. We characterize the value of a perpetual game option in terms of excessive functions, and we use the connection between excessive functions and concave functions to explicitly determine the value in some examples. Moreover, a condition on the two contract functions is provided under which the value is convex in the underlying diffusion value in the continuation region and increasing in the diffusion coefficient.

Functional Inequalities related to the Rogers-Shephard Inequality

Abstract: In this paper a notion of difference function Δ f is introduced for real-valued, non-negative and log-concave functions f defined in R n . The difference function represents a functional analogue of the difference body K + (− K ) of a convex body K. The main result is a sharp inequality which bounds the integral of Δ f from above in terms of the integral of f . Equality conditions are characterized. The investigation is extended to an analogous notion of difference function for α -concave functions, with α α -difference function of f in terms of the integral of f is proved. The bound is sharp in the case α = −∞ and in the one-dimensional case.

Nonexistence of solutions to KPP-type equations of dimension greater than or equal to one

Abstract: In this article, we consider a semilinear elliptic equations of the form u + f(u) = 0, where f is a concave function. We prove for arbitrary dimensions that there is no solution bounded in (0,1). The significance of this result in probability theory is also discussed.

Worst-case distribution analysis of stochastic programs

Abstract: We show that for even quasi-concave objective functions the worst-case distribution, with respect to a family of unimodal distributions, of a stochastic programming problem is a uniform distribution. This extends the so-called ``Uniformity Principle'' of Barmish and Lagoa (1997) where the objective function is the indicator function of a convex symmetric set.

A Geometric Study of the Split Decomposition

Abstract: This paper sheds new light on split decomposition theory and T-theory from the viewpoint of convex analysis and polyhedral geometry. By regarding finite metrics as discrete concave functions, Bandelt-Dress' split decomposition can be derived as a special case of more general decomposition of polyhedral/discrete concave functions introduced in this paper. It is shown that the combinatorics of splits discussed in connection with the split decomposition corresponds to the geometric properties of a hyperplane arrangement and a point configuration. Using our approach, the split decomposition of metrics can be naturally extended to distance functions, which may violate the triangle inequality, using partial split distances.

Bin-packing problem with concave costs of bin utilization

Abstract: We consider a generalized one-dimensional bin-packing model where the cost of a bin is a nondecreasing concave function of the utilization of the bin. Four popular heuristics from the literature of the classical bin-packing problem are studied: First Fit (FF), Best Fit (BF), First Fit Decreasing (FFD), and Best Fit Decreasing (BFD). We analyze their worst-case performances when they are applied to our model. The absolute worst-case performance ratio of FF and BF is shown to be exactly 2, and that of FFD and BFD is shown to be exactly 1.5. Computational experiments are also conducted to test the performance of these heuristics. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2006

Convex and Concave Parts of digital Curves

Abstract: Decomposition of a digital curve into convex and concave parts is of relevance in several scopes of image processing. In digital plane convexity cannot be observed locally. It becomes an interesting question, how far one can decide whether a part of a digital curve is convex or concave by a method which is "as local as possible". In a previous paper, it was proposed to define the meaningful parts of a digital curve as meaningful parts of the corresponding polygonal representation. This technique has an approximative character. In our considerations, we use geometry of arithmetical discrete line segments. We will introduce an exact method to define convex and concave parts of a digital curve.

Estimating a Unimodal Distribution From Interval-Censored Data

Abstract: In this article we consider three nonparametric maximum likelihood estimators based on mixed-case interval-censored data. Apart from the unrestricted estimator, we consider estimators under the assumption that the underlying distribution function of event times is concave or unimodal. Characterizations of the estimates are derived, and algorithms are proposed for their computation. The estimators are shown to be asymptotically consistent, and the benefits of additional constraints are illustrated through simulations. Finally, the estimators are used as an ingredient in a nonparametric comparison of two samples.

Characterization of convexifiable functions

Abstract: A necessary and sufficient condition is given for a continuous function to be convexified, i.e., decomposed into the sum of a convex and a quadratic concave function. The class of smooth convexifiable functions is large. It includes all the continuously differentiable functions with derivatives satisfying the Lipschitz property, all twice continuously differentiable functions, and all analytic functions. The convexifiable functions are important in mathematical programming: Here we show that every program with such functions can be reduced to a partly linear-convex canonical form by the Liu–Floudas transformation. Hence, loosely speaking, almost all smooth programs of practical interest can be studied using only linear and convex programming, and the relationships between them.

Convex dominates concave: An exclusion principle in discrete-time Kolmogorov systems

Abstract: We establish an exclusion principle in discrete-time Kolmogorov systems by using average Liapunov functions. The exclusion principle shows that a weakly dominant species with a convex logarithmic growth rate function eliminates species with concave logarithmic growth rate functions. A general result is applied to specific population models. This application gives an improved exclusion principle for the specific population models.

Generalized quasilinearization and higher order of convergence for first order initial value problems

Abstract: In this paper the method of generalized quasilinearization for initial value problems has been extended when the forcing function is the sum of hyperconvex and hyperconcave functions of order m, m 0. The cases when m is even and m is odd has been discussed separately. AMS (MOS) Subject Classication. 34A12, 34A34, 34A45. The method of quazilinearization introduced by Bellman and Kalaba (1, 2) yields iterates which are lower bounds to the solutions of the nonlinear problem when the forcing function f is convex. Furthermore, this monotone sequence of approximate solutions converges uniformly, monotonically, and quadratically to the unique solution of the nonlinear problem on the interval of existence. However, if f is concave a dual result can be developed which yields upper bounds to the solution of the nonlinear problem. Recently, the method of quasilinearization combined with the method of upper and lower solutions has been extended, generalized, and rened so as to include the cases when the forcing function is the sum of convex and concave functions. See (4) for details. The method is extremely useful in scientic computations due to its accelerated rate of convergence as in (5, 6). In (3) Cabada and Nieto have obtained a higher order of convergence (an order more than 2). The idea used is on the same lines as monotone method, which requires the nonlinearity of the iterates to be the same as that of the order of convergence. However, in (7) they have extended the quasilinearization method of Bellman and Kalaba to obtain a higher order of convergence when the forcing function is either hyperconvex or hyperconcave. Furthermore, the nonlinearity of the iterates are one less than that of the iterates in (3). In this paper we consider the situation when the forcing function is the sum of a hyperconvex and a hyperconcave function of the same order. Further, we consider all

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.

An optimal adaptive algorithm for the approximation of concave functions

Abstract: Motivated by the study of parametric convex programs, we consider approximation of concave functions by piecewise affine functions. Using dynamic programming, we derive a procedure for selecting the knots at which an oracle provides the function value and one supergradient. The procedure is adaptive in that the choice of a knot is dependent on the choice of the previous knots. It is also optimal in that the approximation error, in the integral sense, is minimized in the worst case.

A contribution to duality theory, applied to the measurement of risk aversion

Abstract: This paper determines the precise connection between the curvature properties of an objective function and the ray-curvature properties of its dual. When the objective function is interpreted as a Bernoulli or cardinal utility function, our results characterize the relationship between an agent’s attitude towards income risks and her attitude towards risks in the underlying consumption space. We obtain these results by developing and applying a number of representation theorems for concave functions.

Jensen's Inequality for g-Expectation on Convex (Concave) Function

Abstract: Under the most elementary conditions with respect to g-expectation introduced by Peng S.,based on [8],this paper prove that Jensen's inequality for g-expectation on convex(resp.concave)function holds in general if and only if the g is a super-homogeneous (resp.sub-homogeneous)generator in(y,z)and does not depend on y.

The cutoff transaction size of a quadratic concave holding and penalty cost functions to the information value applying to the newsboy model

Abstract: This investigation adopts the perspective of the retailer and incorporates information flow between a retailer and customers. Two new models are considered that differ in terms of information completeness. The first model involves the retailer having incomplete information regarding the state of customers demand, namely extending the model of Dekker et al. (IIE Transactions, 32, 2000, 461) by considering the quadratic concave holding and penalty cost functions based on the law of diminishing marginal cost to fit in with the some practical situations. Meanwhile, the second model involves the retailer having full information on the state of customers demand. Precise expressions are derived for the expected total profit of these two newsboy models with a cutoff transaction size and compound Poisson demand distribution. It is worthwhile to measure the value of information and identify the effect of factors for enterprise decision making regarding whether or not to pay for information that can help increase profits. Moreover, we adopt and modify the golden section search technique (Haftka et al., 1990) to determine an optimal order-up-to level S and a cutoff transaction size q systematically. Finally, numerical examples are given to illustrate the result derived.

Integral inequalities for concave functions

Abstract: The aim of this paper is to extend to the context of several variables a number of results related to the Favard-Berwald inequalities.

Convex and concave functions

Abstract: This chapter introduces convex and concave functions defined on convex sets in Rn and gives some of their basic properties and provides some fundamental theorems involving these functions. These theorems are very important in deriving optimality conditions for nonlinear programming problems and developing suitable computational schemes. A function of f defined on a convex set S in Rn, is said to be a convex function on S, if it satisfies the equation f [(X1 + (1 – (,)X2] ≤ (f(X1) + (1 - () f(X2). The function f is said to be strictly convex on S if the above inequality is strict for X1≠ X 2, and 0 < ( < 1. A function f is said to be concave (strictly concave) if –f is convex (strictly convex). It is clear that a linear function is convex as well as concave but neither strictly convex nor strictly concave. Alternatively, a function f defined on a convex set S in Rn is convex (concave) if linear interpolation between the values of the function never underestimates the actual value at the interpolated point.

A Price Discrimination Modeling Using Geometric Programming

Abstract: This paper presents a price discrimination model which determines the product's selling price and marketing expenditure in two markets. We assume production as a function of price and marketing cost in two states. The cost of production is also assumed to be a function of production in both markets. We propose a Multi Objective Decision Making (MODM) method called Lexicograph Approach (LA) in order to find an efficient solution between two objectives for each market. Geometric Programming is used to solve the resulted model. In our GP implementation, we use a transformed dual problem in order to reduce the model to an optimization of a nonlinear concave function subject to some linear constraints and solved the resulted model using a simple grid line search.

The convex-concave characteristics of Gaussian channel capacity functions

Abstract: In this correspondence, we give several inherent properties of the capacity function of a Gaussian channel with and without feedback by using some operator inequalities and matrix analysis. We give a new proof method which is different from the method appearing in: K. Yanagi and H. W. Chen, "Operator inequality and its application to information theory," Taiwanese J. Math., vol. 4, no. 3, pp. 407-416, Sep. 2000. We obtain the following results: C/sub n,Z/(P) and C/sub n,FB,Z/(P) are both concave functions of P, C/sub n,Z/(P) is a convex function of the noise covariance matrix and C/sub n,FB,Z/(P) is a convex-like function of the noise covariance matrix. This new proof method is very elementary and the results shall help study the capacity of Gaussian channel. Finally, we state a conjecture concerning the convexity of C/sub n,FB,/spl middot//(P).

Location of CODP based on postponement manufacture

Abstract: Through formulating a quantitative model based on cost,the total cost of the initial working procedures of production process is analyzed.The delivery lead time in customer order is taken as a restriction condition into account.The optimal position of CODP under 4 kinds of situations is acquired according to its range.The optimal value of CODP is expressed with a formula on the basis of analyzing 4 kinds of situations.Finally,the optimal value of CODP is acquired through the analysis of the total cost function respectively for constant function,strictly increasing function,strictly decreasing function,convex function,concave function and wave shape function,and a practical case is studied.