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Showing papers on "Concave function published in 1998"


Journal ArticleDOI
TL;DR: Three metaheuristics: tabu search, simulated annealing and genetic algorithm have been implemented and compared computationally with a random sampling technique to use heuristic search methods defined on the space of feasible sequences.

51 citations


Proceedings ArticleDOI
12 May 1998
TL;DR: A general affine scaling optimization algorithm is given that converges to a sparse solution for measures chosen from within this subclass of the Schur-concave functions.
Abstract: A general framework based on majorization, Schur-concavity, and concavity is given that facilitates the analysis of algorithm performance and clarifies the relationships between existing proposed diversity measures useful for best basis selection. Admissible sparsity measures are given by the Schur-concave functions, which are the class of functions consistent with the partial ordering on vectors known as majorization. Concave functions form an important subclass of the Schur-concave functions which attain their minima at sparse solutions to the basis selection problem. Based on a particular functional factorization of the gradient, we give a general affine scaling optimization algorithm that converges to a sparse solution for measures chosen from within this subclass.

37 citations


Journal ArticleDOI
TL;DR: The article describes the details of how the main idea of transformed density rejection can be used to construct algorithm TDRMV that generates random tuples from a multivariate log-concave distribution with a computable density.
Abstract: Different universal methods (also called automatic or black-box methods) have been suggested for sampling form univariate log-concave distributions. The descriptioon of a suitable universal generator for multivariate distributions in arbitrary dimensions has not been published up to now. The new algorithm is based on the method of transformed density rejection. To construct a hat function for the rejection algorithm the multivariate density is transformed by a proper transformation T into a concave function (in the case of log-concave density T(x) = log(x).) Then it is possible to construct a dominating function by taking the minimum of serveral tangent hyperplanes that are transformed back by T-1 into the original scale. The domains of different pieces of the hat function are polyhedra in the multivariate case. Although this method can be shown to work, it is too slow and complicated in higher dimensions. In this article we split the ℝn into simple cones. The hat function is constructed piecewise on each of the cones by tangent hyperplanes. The resulting function is no longer continuous and the rejection constant is bounded from below but the setup and the generation remains quite fast in higher dimensions; for example, n = 8. The article describes the details of how this main idea can be used to construct algorithm TDRMV that generates random tuples from a multivariate log-concave distribution with a computable density. Although the developed algorithm is not a real black box method it is adjustable for a large class of log-concave densities.

35 citations


Posted Content
Susan Athey1
TL;DR: In this article, the authors developed tools for analyzing properties of stochastic objective functions which take the form (formula) and analyzed the relationship between properties of the primitive functions, such as utility functions u and probability distributions F, and properties of (probability) functions.
Abstract: This paper develops tools for analyzing properties of stochastic objective functions which take the form (formula). The paper analyzes the relationship between properties of the primitive functions, such as utility functions u and probability distributions F, and properties of the stochastic objective. The methods are designed to address problems where the utility functions is restricted to lie in a set of functions which is a "closed convex cone" (examples of such sets include increasing functions, concave functions, or supermodular functions). It is shown that approaches previously applied to characterize monotonicity of V (that is, stochastic dominance theorems) can be used to establish other properties of V as well. The first part of the paper establishes necessary an sufficient conditions for V to satisfy "closed convex cone properties" such as monotonicity, supermodularity, and concavity, in the parameter (formula). Then, we consider necessary and sufficient conditions for monotone comparative statics predictions, building on the results of Milgrom and Shannon (1994). A new property of payoff functions is introduced, called l-supermodularity, which is shown to be necessary and sufficient for (formula) to be quasisupermodular in X (a property which is, in turn, necessary for comparative statics predictions). The results are illustrated with applications.

28 citations


Journal ArticleDOI
01 Dec 1998-Top
TL;DR: It is proved that there always exists a finite set that includes an optimal solution for the Huff and the Pareto-Huff competitive models on networks with the assumption of a concave function of the distance.
Abstract: In this paper we prove that there always exists a finite set that includes an optimal solution for the Huff and the Pareto-Huff competitive models on networks with the assumption of a concave function of the distance. In the Huff model, there is always a vertex of the network that belongs to the solution set. For the Pareto-Huff model, we prove that there is always an optimal solution at, or an e-optimal solution close to, a vertex or an isodistant point, a new concept introduced in this paper.

26 citations


Journal ArticleDOI
TL;DR: In this article, the integrals of powers of polar-conjugate concave functions are extended to integrals due to K. Mahler, together with its case of equality due to M. Meyer, and an application to estimation of the volume product of certain convex bodies is given.
Abstract: An inequality of K. Mahler, together with its case of equality, due to M. Meyer, are extended to integrals of powers of polar-conjugate concave functions. An application to estimation of the volume-product of certain convex bodies is given.

22 citations


Journal ArticleDOI
TL;DR: In this article, simple explicit estimates for the viscosity solution of the Cauchy problem for the Hamilton-Jacobi equation where either the Hamiltonian or the initial data are the sum of a convex and a concave function are presented.
Abstract: Simple explicit estimates are presented for the viscosity solution of the Cauchy problem for the Hamilton--Jacobi equation where either the Hamiltonian or the initial data are the sum of a convex and a concave function. The estimates become equalities whenever a "minmax" equals a "maxmin" and thus a representation formula for the solution is obtained, generalizing the classical Hopf formulas as well as some formulas of Kruzkov [Functional Anal. Appl., 2 (1969), pp. 128--136].

20 citations


Journal ArticleDOI
Jeff Kahn1, Yang Yu1
TL;DR: These results are mainly based on the Brunn–Minkowski Theorem and a theorem of Keith Ball, which allow us to reduce to a 2-dimensional version of the problem.
Abstract: elements of some (finite) poset , write for the probability that precedes in a random (uniform) linear extension of For define where the infimum is over all choices of and distinct Addressing an issue raised by Fishburn [6], we give the first nontrivial lower bounds on the function This is part of a more general geometric result, the exact determination of the function where the infimum is over chosen uniformly from some compact convex subset of a Euclidean space These results are mainly based on the Brunn–Minkowski Theorem and a theorem of Keith Ball [1], which allow us to reduce to a 2-dimensional version of the problem

17 citations


Journal ArticleDOI
TL;DR: A trust region affine scaling algorithm for solving the linearly constrained convex or concave programming problem is studied and it is shown that the limit point of the sequence of iterates satisfies the first and second order necessary conditions for optimality of the problem.
Abstract: We study a trust region affine scaling algorithm for solving the linearly constrained convex or concave programming problem. Under primal nondegeneracy assumption, we prove that every accumulation point of the sequence generated by the algorithm satisfies the first order necessary condition for optimality of the problem. For a special class of convex or concave functions satisfying a certain invariance condition on their Hessians, it is shown that the sequences of iterates and objective function values generated by the algorithm convergeR-linearly andQ-linearly, respectively. Moreover, under primal nondegeneracy and for this class of objective functions, it is shown that the limit point of the sequence of iterates satisfies the first and second order necessary conditions for optimality of the problem. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.

17 citations


Book
01 Jan 1998
TL;DR: The feature selection approach via concave minimization computes a separating-plane based classifier that improves upon the generalization ability of a separating plane computed without feature suppression, support the claim that mathematical programming is effective as the basis of data mining tools to extract patterns from a database which contain “knowledge” and thus achieve “ knowledge discovery in databases”.
Abstract: Machine learning problems of supervised classification, unsupervised clustering and parsimonious approximation are formulated as mathematical programs. The feature selection problem arising in the supervised classification task is effectively addressed by calculating a separating plane by minimizing separation error and the number of problem features utilized. The support vector machine approach is formulated using various norms to measure the margin of separation. The clustering problem of assigning m points in n-dimensional real space to k clusters is formulated as minimizing a piecewise-linear concave function over a polyhedral set. This problem is also formulated in a novel fashion by minimizing the sum of squared distances of data points to nearest cluster planes characterizing the k clusters. The problem of obtaining a parsimonious solution to a linear system where the right hand side vector may be corrupted by noise is formulated as minimizing the system residual plus either the number of nonzero elements in the solution vector or the norm of the solution vector. The feature selection problem, the clustering problem and the parsimonious approximation problem can all be stated as the minimization of a concave function over a polyhedral region and are solved by a theoretically justifiable, fast and finite successive linearization algorithm. Numerical tests indicate the utility and efficiency of these formulations on real-world databases. In particular, the feature selection approach via concave minimization computes a separating-plane based classifier that improves upon the generalization ability of a separating plane computed without feature suppression. This approach produces classifiers utilizing fewer original problem features than the support vector machine approaches, with comparable generalization ability. The clustering techniques are shown to be effective and efficient data mining tools in medical survival analysis applications. The parsimonious approximation methods yield improved results in a signal processing application, with high signal to noise ratio, over least squares and a lengthy combinatorial search. These results support the claim that mathematical programming is effective as the basis of data mining tools to extract patterns from a database which contain “knowledge” and thus achieve “knowledge discovery in databases”.

15 citations


Journal ArticleDOI
TL;DR: The concept of total unimodularity is extended to infinite systems of linear equalities in nonnegative variables where it is shown when extreme points inherit integrality from approximating finite systems.
Abstract: The property that an optimal solution to the problem of minimizing a continuous concave function over a compact convex set in Rn is attained at an extreme point is generalized by the Bauer Minimum Principle to the infinite dimensional context. The problem of approximating and characterizing infinite dimensional extreme points thus becomes an important problem. Consider now an infinite dimensional compact convex set in the nonnegative orthant of the product space Rinfinity. We show that the sets of extreme points EN of its corresponding finite dimensional projections onto RN converge in the product topology to the closure of the set of extreme points E of the infinite dimensional set. As an application, we extend the concept of total unimodularity to infinite systems of linear equalities in nonnegative variables where we show when extreme points inherit integrality from approximating finite systems. An application to infinite horizon production planning is considered.

Journal ArticleDOI
TL;DR: In this paper, a concave relationship between the equality of personal income and the per capita growth rate is derived in conjunction with a social welfare function which is bounded by the concave function.
Abstract: The core of this paper (in a microeconomic foundation on three levels—income generation, income spending, and social climate of the society) consists of a concave relationship between the (in) equality of personal income and the per capita growth rate. The results mentioned are derived in conjunction with a social welfare function which is bounded by the concave function. Furthermore, ample empirical evidence is shown for the suggestions made in this paper.

Book ChapterDOI
01 Jan 1998
TL;DR: In this paper, the authors compared vector valued generalized concave functions in the bicriteria case, that is when the images of the functions are contained in ℜ2.
Abstract: In this paper some classes of vector valued generalized concave functions will be compared in the bicriteria case, that is when the images of the functions are contained in ℜ2. We will prove that, in the bicriteria case, continuous (C,C)-quasiconcave functions coincide with C-quasiconcave functions introduced by Luc; we will also prove that (C,C)-quasiconcave functions have a first order characterization and that they can be characterized by means of their increasness and decreasness.

Journal ArticleDOI
TL;DR: In this paper, a non-trivial set φ of extended-real valued functions on R n, containing all affine functions, such that an extended real valued function f is convex if and only if it is φ-convex in the sense of Dolecki and Kurcyusz, i.e., the (pointwise) supremum of some subset of φ.

Proceedings ArticleDOI
01 Nov 1998
TL;DR: Investigations into the use of concave/Schur-concave functions as regularizing sparsity measures and their application to the problem of obtaining sparse representations, x, of environmentally generated signals y, and theproblem of learning environmentally adapted overcomplete dictionaries are discussed.
Abstract: Given a very overcomplete m/spl times/n dictionary of representation vectors a/sub i/, A=[a/sub 1/,...,a/sub n/], n/spl Gt/m, an environmentally generated signal, y, can be succinctly represented within the dictionary by obtaining a sparse solution, x, to the linear inverse problem Ax/spl ap/y using various previously proposed methodologies. In particular, sparse solutions can be found by an appropriately regularized minimization of the error e=y-Ax. In this paper we briefly discuss our investigations into the use of concave/Schur-concave functions as regularizing sparsity measures, and their application to the problem of obtaining sparse representations, x, of environmentally generated signals y, and the problem of learning environmentally adapted overcomplete dictionaries.

Journal ArticleDOI
TL;DR: In this article, the existence of a decomposition of a given germ of 1-form on an affine space as a linear combination with positive coefficients of the differentials of concave functions was studied.
Abstract: We study the existence of a decomposition of a given germ of 1-form on an affine space as a linear combination with positive coefficients of the differentials of concave functions. The number of entries should be minimal. Necessary and sufficient conditions upon the form germ with regular degeneracy to have such a disaggregation are found. This problem originates from recent papers of P.A. Chiappori and I. Ekeland on the mathematical economy.


Journal ArticleDOI
TL;DR: The dual of a generalized fractional programming problem in which the ratio is the quotient of a quadratic form and a positive concave function is constructed and a numerical example is given.
Abstract: In generalized fractional programming, one seeks to minimize the maximum of a finite number of ratios. Such programs are, in general, nonconvex and consequently are difficult to solve. Here, we consider a particular case in which the ratio is the quotient of a quadratic form and a positive concave function. The dual of such a problem is constructed and a numerical example is given.

Journal ArticleDOI
TL;DR: In this article, the Legendre-Fenchel conjugate (max(g, −h))* in terms of g* and h* was derived in reverse convex optimization.

Journal ArticleDOI
Hiro-O Kitax1
TL;DR: In this paper, the authors considered the Orlicz space Lπ and gave a generalization of Soria's result [S1], where π (t) is a concave function with some nice properties.
Abstract: Let S* (f be the majorant function of the partial sums of the trigonometric Fourier series of f. In this paper we consider the Orlicz space Lπ and give a generalization of Soria's result [S1]. Let π (t) be a concave function with some nice properties and . If there exists a positive constant a0 < 1 such that then we have .