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Showing papers on "Concave function published in 2002"


Book ChapterDOI
01 Jan 2002
TL;DR: In this article, it was shown that the Wigner lattice can exist in one dimension at least, and that the fundamental solution of the diffusion equation can be found in the same dimension.
Abstract: THE following is a preliminary report on some recent work, the full details of which will be published elsewhere. We have come across some inequalities about integrals and moments of log concave functions which hold in the multidimensional case and which are useful in obtaining estimates for multidimensional modified Gaussian measures. By making a small jump (we shall not go into the technical details) from the finite to the infinite dimensional case, upper and lower bounds to certain types of functional integrals can be obtained. As a non-trivial application of the latter we shall, for the first time, prove that the one-dimensional one-component quantummechanical plasma has long-range order when the interaction is strong enough. In other words, the Wigner lattice can exist, in one dimension at least. As another application we shall prove a log concavity theorem about the fundamental solution (Green’s function) of the diffusion equation.

69 citations


Journal ArticleDOI
TL;DR: This paper addresses itself to a portfolio optimization problem under nonconvex transaction costs and minimal transaction unit constraints and proposes a branch-and-bound algorithm for the resulting d.c. c. maximization problem subject to a constraint on the level of risk measured in terms of the absolute deviation of the rate of return.
Abstract: This paper addresses itself to a portfolio optimization problem under nonconvex transaction costs and minimal transaction unit constraints. Associated with portfolio construction is a fee for purchasing assets. Unit transaction fee is larger when the amount of transaction is smaller. Hence the transaction cost is usually a concave function up to certain point. When the amount of transaction increases, the unit price of assets increases due to illiquidity/market impact effects. Hence the transaction cost becomes convex beyond certain bound. Therefore, the net expected return becomes a general d.c. function (difference of two convex functions). We will propose a branch-and-bound algorithm for the resulting d.c. maximization problem subject to a constraint on the level of risk measured in terms of the absolute deviation of the rate of return of a portfolio. Also, we will show that the minimal transaction unit constraints can be incorporated without excessively increasing the amount of computation.

48 citations


Journal ArticleDOI
TL;DR: It is shown that the least exact penalty parameter of an equivalent parametric optimization problem can be diminished and the Lipschitz penalty function with a small penalty parameter is more suitable for solving some nonconvex constrained problems than the classical penalty function.
Abstract: In this article, we study the nonlinear penalization of a constrained optimization problem and show that the least exact penalty parameter of an equivalent parametric optimization problem can be diminished. We apply the theory of increasing positively homogeneous (IPH) functions so as to derive a simple formula for computing the least exact penalty parameter for the classical penalty function through perturbation function. We establish that various equivalent parametric reformulations of constrained optimization problems lead to reduction of exact penalty parameters. To construct a Lipschitz penalty function with a small exact penalty parameter for a Lipschitz programming problem, we make a transformation to the objective function by virtue of an increasing concave function. We present results of numerical experiments, which demonstrate that the Lipschitz penalty function with a small penalty parameter is more suitable for solving some nonconvex constrained problems than the classical penalty function.

30 citations


01 Jan 2002
TL;DR: In this paper, double integral inequalities for convex and concave functions are derived for numerical integration problems, and applications in numerical integration are also given, including the use of quadrature quadratures.
Abstract: Some double integral inequalities are established. These inequalities give upper and lower error bounds for the well-known mid-point and trapezoid quadrature rules. Some inequalities for convex and concave functions are derived. Applications in numerical integration are also given.

26 citations


Journal ArticleDOI
TL;DR: In this paper, a sigmoid utility function was proposed to model risk sensitivity in Andean pastoralists, Ache foragers, and Sulawesi Crested Black Macaques.

24 citations


Journal ArticleDOI
07 Aug 2002
TL;DR: A fluid-stochastic-event graph model is proposed that is a decision-free Petri net that determines continuous-state variables at epochs of failure/repair events and an optimization problem of maximizing a concave function of throughput rate and system parameters is addressed.
Abstract: This paper addresses the performance evaluation and optimization of failure-prone discrete-event systems. We propose a fluid-stochastic-event graph model that is a decision-free Petri net. Tokens are considered as continuous flows. A transition can be in operating state or in failure state. Jumps between failure and operating states do not depend on the firing conditions, and the sojourn time in each state is a random variable of general distribution. For performance evaluation, a set of evolution equations that determines continuous-state variables at epochs of failure/repair events is established. The cumulative firing quantity of each transition is proven to be concave in system parameters, including firing rates and initial marking. Gradient estimators are derived. Finally, an optimization problem of maximizing a concave function of throughput rate and system parameters is addressed.

22 citations


Book ChapterDOI
01 Jan 2002
TL;DR: The role of generalized support vector machines in separating massive and complex data using arbitrary nonlinear kernels is described and feature selection that improves generalization is implemented via an effective procedure that utilizes a polyhedral norm or a concave function minimization.
Abstract: We describe the role of generalized support vector machines in separating massive and complex data using arbitrary nonlinear kernels. Feature selection that improves generalization is implemented via an effective procedure that utilizes a polyhedral norm or a concave function minimization. Massive data is separated using a linear programming chunking algorithm as well as a successive overrelaxation algorithm, each of which is capable of processing data with millions of points.

19 citations


Proceedings ArticleDOI
TL;DR: This paper formulae the multiple capture time scheduling problem when the incident illumination probability density function (pdf) is completely known as a constrained optimization problem and aims to find the capture times that maximize the average signal SNR.
Abstract: Several papers have discussed the idea of extending image sensor dynamic range by capturing several images during a normal exposure time. Most of these papers assume that the images are captured according to a uniform or an exponentially increasing exposure time schedule. Even though such schedules can be justified by certain implementation considerations, there has not been any systematic study of how capture time schedules should be optimally determined. In this paper we formulae the multiple capture time scheduling problem when the incident illumination probability density function (pdf) is completely known as a constrained optimization problem. We aim to find the capture times that maximize the average signal SNR. The formulation leads to a general upper bound on achievable average SNR using multiple capture for any given illumination pdf. For a uniform pdf, the average SNR is a concave function in capture times and therefore well-known convex optimization techniques can be applied to find the global optimum. For a general piece-wise uniform pdf, the average SNR is not necessarily concave. The cost function, however, is a Difference of Convex (D.C.) function and well-established D.C. or global optimization techniques can be used.

18 citations


Journal ArticleDOI
TL;DR: In this article, the authors define criteria that indicate whether or not an experiment will lead to estimates with distributions well approximated by a normal distribution, based on the intrinsic and parameter effects curvatures.

17 citations


Book ChapterDOI
01 Jul 2002

12 citations


Journal ArticleDOI
TL;DR: In this article, the singular set of points at which a semiconcave function fails to be differentiable is studied, and the existence of arcs contained on this singular set is proved.


Journal ArticleDOI
TL;DR: In this article, the authors considered a forester who produces timber from land and labor, and whose objective is to maximize a discounted sum of one-period utilities given by a strictly concave function of timber output, land input, and labor input.

Journal ArticleDOI
TL;DR: In this article, the optimal portfolio management for the constant relative risk averse investor who maximizes an expected utility of his terminal wealth and who faces transaction costs during his trades is studied.
Abstract: In this paper we study the optimal portfolio management for the constant relative-risk averse investor who maximizes an expected utility of his terminal wealth and who faces transaction costs during his trades. In our model the investor's portfolio consists of one risky and one risk-free asset, and we assume that the transaction cost is a concave function of the traded volume of the risky asset. We find that under such transaction cost formulation the optimal trading strategies and boundaries of the no-transaction region are different than those when transaction costs are proportional, i.e. when they are linear in the traded volume. When transaction costs are concave, we show that the no-transaction region is narrower than when transaction costs are proportional, and it is not a positive cone. Under our transaction cost formulation, when the investor's wealth is relatively high, the optimal trading strategy consists in bringing the post-trade portfolio position inside the no-transaction region, whereas proportional transaction costs induce the investor trading to the boundary of the no-transaction region. We also examine the impact of the risky asset volatility and the risk aversion parameter on the shape of the no-transaction region. When comparing different transaction cost structures, we show that the financial securities' market tends to be more liquid with concave transaction costs than with alternative cost specifications.

Journal ArticleDOI
TL;DR: In this paper, the authors studied a one-sector model with externalities and nonlinear discounting and established the existence of a steady state under fairly general conditions and gave a necessary and sufficient condition for local indeterminacy.

Journal ArticleDOI
TL;DR: This article considers resource allocation with separable objective functions defined over subranges of the integers, with objective functions that are close to a concave function or some other smooth function, but with small irregularities in their shape.
Abstract: We consider resource allocation with separable objective functions defined over subranges of the integers. While it is well known that (the maximisation version of) this problem can be solved efficiently if the objective functions are concave, the general problem of resource allocation with functions that are not necessarily concave is difficult. In this article, we focus on a large class of problem instances, with objective functions that are close to a concave function or some other smooth function, but with small irregularities in their shape. It is described that these properties are important in many practical situations. The irregularities make it hard or impossible to use known, efficient resource allocation techniques. We show that, for this class of functions the optimal solution can be computed efficiently. We support our claims by experimental evidence. Our experiments show that our algorithm in hard and practically relevant cases runs up to 40–60 times faster than the standard method.

Journal Article
TL;DR: In this article, the typical transaction cost model with no convex or no concave function is introduced, and a portfolio management model with typical transaction costs is put forward, and the influences of transaction cost models and risk levels on the effectiveness of portfolio are investigated through a numerical example.

Journal ArticleDOI
TL;DR: In this paper, an algebra approach for solving the linearly constrained continuous quasi-concave minimization problems is proposed, based on the fact that the optimal solutions can be achieved at an extreme point of the polyhedron.
Abstract: This paper proposes an algebra approach for solving the linearly constrained continuous quasi-concave minimization problems. The study involves a class of very generalized concave functions, continuous strictly quasi-concave functions. Based on the fact that the optimal solutions can be achieved at an extreme point of the polyhedron, we provide an algebra-based method for identifying the extreme points. The case on unbounded polyhedral constraints is also discussed and solved. Numerical examples are provided for illustration.

Journal ArticleDOI
TL;DR: The result is presented as a theorem, the Concavity Theorem, and a list of conditions that can easily be verified, and it is shown how the theorem can be extended to other applications, like in the area of road transportation.
Abstract: The Network Design Problem has been studied extensively and in many of these models the cost is assumed to be a concave function of the loads on the links. In this paper we investigate under which conditions this is indeed the case for the communication networks. The result is presented as a theorem, the Concavity Theorem, and a list of conditions that can easily be verified. It is also shown how the theorem can be extended to other applications, like in the area of road transportation.

Journal Article
TL;DR: In this article, the Popoviciu inequality for the concave function is established by using Schur's concavity of the elementary symmetric functions and a simple domination relation of the vector.
Abstract: An inequality for the concave function is established by using Schur's concavity of the elementary symmetric functions and a simple domination relation of the vector. As a consequence Popoviciu inequality is extended in many sides.

Posted Content
TL;DR: In this paper, the optimal portfolio management for the constant relative risk averse investor who maximizes an expected utility of his terminal wealth and who faces transaction costs during his trades is studied.
Abstract: In this paper we study the optimal portfolio management for the constant relative-risk averse investor who maximizes an expected utility of his terminal wealth and who faces transaction costs during his trades. In our model the investor's portfolio consists of one risky and one risk-free asset, and we assume that the transaction cost is a concave function of the traded volume of the risky asset. We find that under such transaction cost formulation the optimal trading strategies and boundaries of the no-transaction region are different than those when transaction costs are proportional, i.e. when they are linear in the traded volume. When transaction costs are concave, we show that the no-transaction region is narrower than when transaction costs are proportional, and it is not a positive cone. Under our transaction cost formulation, when the investor's wealth is relatively high, the optimal trading strategy consists in bringing the post-trade portfolio position inside the no-transaction region, whereas proportional transaction costs induce the investor trading to the boundary of the no-transaction region. We also examine the impact of the risky asset volatility and the risk aversion parameter on the shape of the no-transaction region. When comparing different transaction cost structures, we show that the financial securities' market tends to be more liquid with concave transaction costs than with alternative cost specifications.


Journal ArticleDOI
TL;DR: In this article, the authors used diffusion and heat conductivity equations for global optimization and showed that the superlinear rate of convergence can be achieved by using a programmer to instruct the computer as to which domains and distribution densities of random vectors are to be used.
Abstract: The use of Helmholtz (diffraction and diffusion) and heat conductivity equations for global optimization is described in Part I. Here numerical realization of these ideas is studied and the superlinear rate of convergence is demonstrated. Numerical methods are based on the idea that the solutions of diffraction (diffusion) equations p(x,ω) and heat conductivity equation U(x,t) are convex and concave functions, respectively, in the neighborhood of the point of global minimum in some modified aim function for which the point of global minimum does not change. An idea for the construction of iterative algorithms is developed, in which a programmer, using computation results, actively participates in computations, instructing the computer as to which domains and distribution densities of random vectors are to be used.

01 Jan 2002
TL;DR: Under smoothness conditions on the function and the feasible set, the algorithm computes a near-optimal point in a number of operations which is bounded by a polynomial function of all relevant input parameters and the reciprocal of the desired precision, with high probability.
Abstract: We consider maximising a concave function over a convex set by a simple randomised algorithm. The strength of the algorithm is that it requires only approximate function evaluations for the concave function and a weak membership oracle for the convex set. Under smoothness conditions on the function and the feasible set, we show that our algorithm computes a near-optimal point in a number of operations which is bounded by a polynomial function of all relevant input parameters and the reciprocal of the desired precision, with high probability. As an application to which the features of our algorithm are particularly useful we study two-stage stochastic programming problems. These problems have the property that evaluation of the objective function is #P-hard under appropriate assumptions on the models. Therefore, as a tool within our randomised algorithm, we devise a fully polynomial randomised approximation scheme for these function evaluations, under appropriate assumptions on the models. Moreover, we deal with smoothing the feasible set, which in two-stage stochastic programming is a polyhedron.

Posted Content
TL;DR: In this article, a disaggregated nonparametric analysis in which subjects were classified according to which transformation function is most consistent with their revealed choice behavior was conducted. And they found essentially no evidence of an S-shaped transformation function for choice under risk.
Abstract: We conduct an experiment in an attempt to (i) measure the structure of preferences over lotteries and (ii) test for stability of the probability transformation functions over different choice sets. The design is based on manipulations of the “probability triangle” A disaggregated nonparametric analysis in which we classify subjects according to which transformation function is most consistent with their revealed choice behavior shows that a linear and a strictly concave transformation function are the most common for risky choice. We find essentially no evidence of an S-shaped transformation function for choice under risk. Formal econometric estimation clearly rejects the S-shaped function in favor a strictly concave function. A formal econometric analysis exploiting the ordered discrete nature of the data leads us to infer a strictly concave probability transformation function, consistent with the nonparametric analysis. The difference between our results and those of previous studies can be attributed to the choice of functional forms used in estimating the transformation function, to the limited space of lotteries upon which estimates have been based.

Journal ArticleDOI
TL;DR: In this paper, it was shown that the free energy and the cost associated to a bipartite matching problem can be explicitly estimated in terms of the solution of a suitable system of equations (cavity equations in the following).
Abstract: In this paper I show that the free energy F and the cost C associated to a bipartite matching problem can be explicitly estimated in term of the solution of a suitable system of equations (cavity equations in the following). The proof of these results relies on a well known result in combinatorics: the Van der Waerden conjecture (Egorychev–Falikman Theorem). Cavity equations, derived by a mean field argument by Mezard and Parisi, can be considered as a smoothed form of the dual formulation for the bipartite matching problem. Moreover cavity equation are the Euler–Lagrange equations of a convex functional G parameterized by the temperature T. In term of their unique solution it is possible to define a free-energy-like function of the temperature g(T). g is a strictly decreasing concave function of T and C=g(0). The convexity of G allows to define an explicit algorithm to find the solution of the cavity equations at a given temperature T. Moreover, once the solution of the cavity equations at a given temperature T is known, the properties of g allow to find exact estimates from below and from above of the cost C.

Journal ArticleDOI
Tao Mei1, Peide Liu1
TL;DR: In this paper, the authors prove the equivalence of the following two conditions: (1) the condition that Φ 1 (M f ) ≤ c E Φ 2 (Z 0 + A ∞ ) for every nonnegative submartingale f = ( f n ) n ≥ 0 with it's Doob's Decomposition: f = Z + A, where Z is a martingale in L 1 and A is a nonnegative incrasing and predictable process.


01 Jan 2002
TL;DR: In this article, the authors give a new characterization of excessive functions with respect to arbitrary one-dimensional regular diffusion processes, using the notion of concavity, and show that excessive functions are essentially concave functions, in some generalized sense, and vice-versa.
Abstract: Contributions to the Theory of Optimal Stopping for One–Dimensional Diffusions Savas Dayanik Advisor: Ioannis Karatzas We give a new characterization of excessive functions with respect to arbitrary one–dimensional regular diffusion processes, using the notion of concavity. We show that excessive functions are essentially concave functions, in some generalized sense, and vice–versa. This, in turn, allows us to characterize the value function of the optimal stopping problem, for the same class of processes, as “the smallest nonnegative concave majorant of the reward function”. In this sense, we generalize results of Dynkin– Yushkevich for the standard Brownian motion. Moreover, we show that there is essentially one class of optimal stopping problems, namely, the class of undiscounted optimal stopping problems for the standard Brownian motion. Hence, optimal stopping problems for arbitrary diffusion processes are not inherently more difficult than those for Brownian motion. The concavity of the value functions also allows us to draw sharper conclusions about their smoothness, thanks to the nice properties of concave functions. We can therefore offer a new perspective and new facts about the smooth–fit principle and the method of variational inequalities in the context of optimal stopping. The results are illustrated in detail on a number of non–trivial, concrete optimal stopping problems, both old and new.