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Showing papers on "Concave function published in 2000"


Journal ArticleDOI
TL;DR: This paper shows the equivalence between Murota's L-convex functions and Favati and Tardella’s submodular integrally convex functions, and gives alternative proofs of the separation theorems that provide a geometric insight by relating them to the ordinary separation theorem in convex analysis.
Abstract: The concepts of L-convex function and M-convex function have recently been introduced by Murota as generalizations of submodular function and base polyhedron, respectively, and discrete separation theorems are established for L-convex/concave functions and for M-convex/concave functions as generalizations of Frank’s discrete separation theorem for submodular/supermodular set functions and Edmonds’ matroid intersection theorem. This paper shows the equivalence between Murota’s L-convex functions and Favati and Tardella’s submodular integrally convex functions, and also gives alternative proofs of the separation theorems that provide a geometric insight by relating them to the ordinary separation theorem in convex analysis.

91 citations


Journal ArticleDOI
TL;DR: Designs that maximize the minimum efficiencies under the two criteria for optimality criteria are proposed along with a graphical method for finding these maximin designs.
Abstract: We consider the problem of designing an experiment when there are two competing optimality criteria. Designs that maximize the minimum efficiencies under the two criteria are proposed along with a graphical method for finding these maximin designs.

31 citations


Journal ArticleDOI
TL;DR: In this paper, the characterisation of geometrically convex and concave functions defined on (0,A] or (A, ∞) with multiplicative conditions was studied and unified proofs of some known and new inequalities were obtained.
Abstract: From the characterisation of geometrically convex and geometrically concave functions defined on (0,A] or \( [A,\infty) \) with \( A > 0 \), by means of their multiplicative conditions, we obtain unified proofs of some known and new inequalities. Functions of class C2 and strictly increasing on (a,b) fulfil some kind of supermultiplicativity and superadditivity. We have obtained a new constant determining the intervals of sub- and supermultiplicativity for the log function.

28 citations


Journal ArticleDOI
TL;DR: The proposed method first expresses a piecewise function as the summation of absolute terms, and searches for the interval where the optimal solution is allocated by finding the corresponding points with same value of membership functions.

25 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that for a nonconvex function defined on a real interval, there exists a point where this function behaves like a strictly concave function.
Abstract: In this note we show that, for a nonconvex function defined on a real interval, there exists a point where this function behaves like a strictly concave function. Due to this result, global convexity can be characterized as pointwise convexity everywhere. As an application, a necessary and sufficient condition for the separability of quasiarithmetic means with power means is obtained. Mathematics subject classification (1991): Primary 26A51, 26B25.

20 citations


Journal ArticleDOI
TL;DR: Given a nonlinear function h separating a convex and a concave function, this work provides various conditions under which there exists an affine separating function whose graph is somewhere almost parallel to the graph of h.
Abstract: Given a nonlinear function h separating a convex and a concave function, we provide various conditions under which there exists an affine separating function whose graph is somewhere almost parallel to the graph of h. Such results blend Fenchel duality with a variational principle and are closely related to the Clarke--Ledyaev mean value inequality.

20 citations


Journal ArticleDOI
A. M. Fink1
TL;DR: In this paper, the Hadamard inequalities for concave functions are extended to log-concave functions, where the Lebesgue measure is replaced by a signed measure.

15 citations


Journal ArticleDOI
TL;DR: In this paper, the authors give some inequalities of capacity in Gaussian channel with or without feedback, and show that the non-feedback capacity and the feedback capacity are concave functions of the same type.
Abstract: We give some inequalities of capacity in Gaussian channel with or without feedback. The nonfeedback capacity $C_{n,Z}(P)$ and the feedback capacity $C_{n,FB,Z}(P)$ are both concave functions of $P$. Though it is shown that $C_{n,Z}(P)$ is a convex function of $Z$ in some sense, $C_{n,FB,Z}(P)$ is a convex-like function of $Z$.

15 citations


Journal ArticleDOI
TL;DR: In this article, the convergence of the simplicial branch-and-bound algorithm for minimizing a concave function over a polytope was proved. But the convergence was not proved for ω-subdivisions.
Abstract: The problem of minimizing a concave function over a polytope is considered. The simplicial branch-and-bound approach is presented and theoretical studies about the convergence of these algorithms are carried on. In particular, the convergence of the algorithm based on so-called ω-subdivisions is proved, which had been an open question for a long time.

11 citations


Journal ArticleDOI
TL;DR: In this paper, the Krein-Smulian theorem is used to verify the topological properties of sets, functions and correspondences needed in economic equilibrium analysis with infinite-dimensional commodity spaces.

9 citations


Journal ArticleDOI
TL;DR: In this article, a truncated uniform distribution is defined for the worst-case miss distance in a concave function of the engagement duration on a certain interval, and the worst case miss distance can be determined from a singe Monte Carlo trial.
Abstract: performance” satisfaction, is characterized as a truncated uniform distribution.This result was applied to the performance analysis of engagementdurationand of several error sources.Linear theory enables the formulation of analytical solutions to the problem.When numericalMonteCarlo simulationsare used to search for theworstcase distribution,a limited number of simulation runs typically are neededfor the search.Moreover,when themiss distanceis a concave function of the con ict duration on a certain interval, the worstcase miss distance can be determined from a singe Monte Carlo trial.

Journal ArticleDOI
TL;DR: In this article, the authors study the following stochastic investment model: opportunities occur randomly over time, following a renewal process with mean interarrival time d, and at each time the decision-maker can choose a distribution for an instantaneous net gain (or loss) from the set of all probability measures that have some prespecified expected value e and for which his maximum possible loss does not exceed his current capital.
Abstract: We study the following stochastic investment model: opportunities occur randomly over time, following a renewal process with mean interarrival time d, and at each of them the decision-maker can choose a distribution for an instantaneous net gain (or loss) from the set of all probability measures that have some prespecified expected value e and for which his maximum possible loss does not exceed his current capital. Between the investments he spends money at some constant rate. The objective is to avoid bankruptcy as long as possible. For the case e>d we characterize a strategy maximizing the probability that ruin never occurs. It is proved that the optimal value function is a concave function of the initial capital and uniquely determined as the solution of a fixed point equation for some intricate operator. In general, two-point distributions suffice; furthermore, we show that the cautious strategy of always taking the deterministic amount e is optimal if the interarrival times are hyperexponential, and, in the case of bounded interarrival times, is optimal ‘from some point on’, i.e. whenever the current capital exceeds a certain threshold. In the case e = 0 we consider a class of natural objective functions for which the optimal strategies are non-stationary and can be explicitly determined.

Book ChapterDOI
01 Jan 2000
TL;DR: In this article, it was shown that the global minimum of a piecewise concave function on a set X must be attained in a subset of X which is often finite and contains the extreme points of X.
Abstract: The problem of minimizing the maximum of a family of concave functions can be viewed as the problem of minimizing a piecewise concave function. In this paper, extending the theory of piecewise concavity introduced by Zangwill, we show that the global minimum of a piecewise concave function on a set X must be attained in a subset of X which is often finite and contains the extreme points of X. We then show that this result implies a recent result by Du and Hwang on minimax problems.

Journal ArticleDOI
TL;DR: The article describes the details how this main idea can be used to construct Algorithm ALC2D that can generate random pairs from all bivariate log-concave distributions with known domain, computable density, and computable partial derivatives.
Abstract: Different automatic (also called universal or black-box) methods have been suggested to sample from univariate log-concave distributions. Our new automatic algorithm for bivariate log-concave distributions is based on the method of transformed density rejection. In order to construct a hat function for a rejection algorithm the bivariate density is transformed by the logarithm into a concave function. Then it is possible to construct a dominating function by taking the minimum of several tangent planes, which are by exponentiation transformed back into the original scale. The choice of the points of contact is automated using adaptive rejection sampling. This means that points that are rejected by the rejection algorithm can be used as additional points of contact. The article describes the details how this main idea can be used to construct Algorithm ALC2D that can generate random pairs from all bivariate log-concave distributions with known domain, computable density, and computable partial derivatives.

Proceedings ArticleDOI
04 Dec 2000
TL;DR: This paper defines the notion of a PCFB for a class of nonuniform orthonormal Fbs, and shows how it generalizes the uniform PCFBs by being optimal for a certain family of concave objectives.
Abstract: The optimality of principal component filter banks (PCFBs) for data compression has been observed in many works to varying extents. Recent work by the authors has made explicit the precise connection between the optimality of uniform orthonormal filter banks (FBs) and the principal component property: The PCFB is optimal whenever the minimization objective is a concave function of the subband variances of the FB. This gives a unified explanation of PCFB optimality for compression, denoising and progressive transmission. However not much is known for the case when the optimization is over a class of nonuniform Fbs. In this paper we first define the notion of a PCFB for a class of nonuniform orthonormal Fbs. We then show how it generalizes the uniform PCFBs by being optimal for a certain family of concave objectives. Lastly, we show that existence of nonuniform PCFBs could imply severe restrictions on the input power spectrum. For example, for the class of unconstrained orthonormal nonuniform Fbs with any given set of decimators that are not all equal, there is no PCFB if the input spectrum is strictly monotone.

Journal ArticleDOI
TL;DR: The extended quasilinearization method of Lakshmikantham et al. as mentioned in this paper for the first order initial value problems is applied to the nonlinear systems and it is shown that there exist monotone sequences which converge uniformly to the unique solution of the system and the convergence is quadratic.
Abstract: The extended quasilinearization method of Lakshmikantham et al. (J. Optim. Theory Appl. 87 (1995), 379-401) for the first order initial value problems is applied to the nonlinear systems. It is shown that there exist monotone sequences which converge uniformly to the unique solution of the system and the convergence is quadratic. Futhermore, a variety of results are obtained by splitting the functions involved into the difference of two convex or two concave functions, each of which is interesting by itself, with the same conclusion. Moreover, new results are extracted, as a byproduct, from the present results which offer simultaneous bounds for the cases where there is no splitting involved

Proceedings ArticleDOI
12 Dec 2000
TL;DR: It is proved that the cumulative firing of transitions are Lipschitz continuous, non-decreasing and concave functions of system parameters including maximal firing rates and the initial marking.
Abstract: Performance evaluation and optimization of failure-prone discrete event systems are addressed. Our analysis is based on a fluid stochastic event graph model that is a decision-free Petri net. In fluid Petri nets, each place holds a continuous flow instead of discrete tokens of conventional Petri nets. A transition can be in operating state or in failure state. A transition in operating state can fire at its maximal speed and a transition in failure state cannot fire. Jumps between failure and operating states are independent of 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 discrete events is established. Based on the evolution equations, we prove that the cumulative firing of transitions are Lipschitz continuous, non-decreasing and concave functions of system parameters including maximal firing rates and the initial marking. Gradient estimators are derived and their properties established. Finally, an optimization problem that maximizes a concave function of throughput rate and the system parameters is addressed.

01 Jan 2000
TL;DR: In this article, the multifractal spectrum of local entropies is analyzed for arbitrary-invariant measures and possible singularities in the spectrum are discussed in terms of their relation to phase transitions.
Abstract: Weaddresstheproblemofthemultifractalanalysisoflocalentropiesfor arbitraryinvariantmeasures. Weobtainanupperestimateonthemultifractal spec- trumoflocalentropies,whichissimilartotheestimateforlocaldimensions. Weshow thatinthecaseofGibbsmeasurestheaboveestimatebecomesanexactequality. In this case the multifractal spectrumof local entropies is a smooth concave function. Wediscuss possible singularities in themultifractal spectrum andtheir relation to phase transitions.

01 Jan 2000
TL;DR: In this article, the authors studied the concavity of some functions that can be written as powers of linear combinations of powers of some concave functions and showed that the concaveness of these functions can be expressed as a linear combination of the powers of the functions.
Abstract: In this paper, one studies the concavity of some functions that can be written as powers of linear combinations of powers of some concave functions.