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Showing papers on "Concave function published in 2007"


Journal ArticleDOI
TL;DR: In this paper, a deterministic inventory model for deteriorating items with time-dependent backlogging rate is developed and the demand and deterioration rate are known, continuous, and differentiable function of price and time, respectively.
Abstract: In this paper, a deterministic inventory model for deteriorating items with time-dependent backlogging rate is developed. The demand and deterioration rate are known, continuous, and differentiable function of price and time, respectively. Under these general assumptions, we first prove that the optimal replenishment schedule not only exists but is unique, for any given selling price. Next, we show that the total profit is a concave function of price when the replenishment schedule is given. We then provide a simple algorithm to find the optimal selling price and replenishment schedule for the proposed model. Finally, we use a numerical example to illustrate the algorithm.

190 citations


Journal ArticleDOI
TL;DR: In this paper, an exact exchange theory and an exchange-hole sum rule are proposed to explain these failures and a way to correct them via a local hybrid functional is proposed to correct the failures.
Abstract: While the exact total energy of a separated open system varies linearly as a function of average electron number between adjacent integers, the energy predicted by semilocal density-functional approximations is concave up and the exact-exchange-only or Hartree-Fock energy is concave down. As a result, semilocal density functionals fail for separated open systems of fluctuating electron number, as in stretched molecular ions $\mathrm{A}_{2}{}^{+}$ and in solid transition-metal oxides. We develop an exact-exchange theory and an exchange-hole sum rule that explain these failures and we propose a way to correct them via a local hybrid functional.

138 citations


Journal ArticleDOI
TL;DR: A deterministic inventory model for deteriorating items with price-dependent demand is developed and it is proved that the optimal replenishment policy not only exists but also is unique.

107 citations


Proceedings ArticleDOI
TL;DR: In this article, the authors give a list of nonlinear contractive conditions which turn out to be mutually equivalent and derive them from general lemmas concerning subsets of the plane which may be applied both in the single-or set-valued case as well as for a family of mappings.
Abstract: We establish five theorems giving lists of nonlinear contractive conditions which turn out to be mutually equivalent. We derive them from some general lemmas concerning subsets of the plane which may be applied both in the singleor set-valued case as well as for a family of mappings. A separation theorem for concave functions is proved as an auxiliary result. Also, we discuss briefly the following problems for several classes of contractions: stability of procedure of successive approximations, existence of approximate fixed points, continuous dependence of fixed points on parameters, existence of invariant sets for iterated function systems. Moreover, James Dugundji’s contribution to the metric fixed point theory is presented. Using his notion of contractions, we also establish an extension of a domain invariance theorem for contractive fields.

105 citations


Journal ArticleDOI
TL;DR: It is shown that the limiting distributions of the nonparametric maximum likelihood estimator (MLE) of a log-concave density and its derivative are, under comparable smoothness assumptions, the same (up to sign) as in the convex density estimation problem.
Abstract: We find limiting distributions of the nonparametric maximum likelihood estimator (MLE) of a log-concave density, that is, a density of the form $f_0=\exp\varphi_0$ where $\varphi_0$ is a concave function on $\mathbb{R}$. The pointwise limiting distributions depend on the second and third derivatives at 0 of $H_k$, the "lower invelope" of an integrated Brownian motion process minus a drift term depending on the number of vanishing derivatives of $\varphi_0=\log f_0$ at the point of interest. We also establish the limiting distribution of the resulting estimator of the mode $M(f_0)$ and establish a new local asymptotic minimax lower bound which shows the optimality of our mode estimator in terms of both rate of convergence and dependence of constants on population values.

93 citations


Journal ArticleDOI
TL;DR: This article considers an infinite horizon, single product economic order quantity where demand and deterioration rate are continuous and differentiable function of price and time, respectively and provides a simple algorithm to find the optimal selling price and replenishment schedule.

76 citations


Journal ArticleDOI
TL;DR: It is shown that a correlation inequality of statistical mechanics can be applied to linear low-density parity-check codes and it is proved that part of the GEXIT curve can be computed exactly by iterative methods, at least on the interval where it is a concave function of the relative weight of code words.
Abstract: It is shown that a correlation inequality of statistical mechanics can be applied to linear low-density parity-check codes. Thanks to this tool we prove that, under a natural assumption, the exponential growth rate of regular low-density parity-check (LDPC) codes, can be computed exactly by iterative methods, at least on the interval where it is a concave function of the relative weight of code words. Then, considering communication over a binary input additive white Gaussian noise channel with a Poisson LDPC code we prove that, under a natural assumption, part of the GEXIT curve (associated to MAP decoding) can also be computed exactly by the belief propagation algorithm. The correlation inequality yields a sharp lower bound on the GEXIT curve. We also make an extension of the interpolation techniques that have recently led to rigorous results in spin glass theory and in the SAT problem

74 citations


Journal ArticleDOI
TL;DR: Maintenance of feature traces as defined by hint based approaches are implemented and represented in concave graph forms helping the recognition of interacting features with less computational effort.

63 citations


Journal ArticleDOI
TL;DR: In this paper, the structure of approximate solutions of an autonomous discrete-time control system with a compact metric space of states X is studied and the turnpike properties of the approximate solutions which are independent of the length of the interval are investigated.
Abstract: In this work we study the structure of approximate solutions of an autonomous discrete-time control system with a compact metric space of states X . This control system is described by a bounded upper semicontinuous function v : X × X → R 1 which determines an optimality criterion and by a nonempty closed set Ω ⊂ X × X which determines a class of admissible trajectories (programs). We are interested in turnpike properties of the approximate solutions which are independent of the length of the interval, for all sufficiently large intervals. For when X is a compact convex subset of a finite-dimensional Euclidean space, the set Ω is convex and the function v is strictly concave we obtain a full description of the structure of approximate solutions.

54 citations


Journal ArticleDOI
TL;DR: In this paper, a matrix version of Choi's inequality for positive unital maps and operator convex functions remains valid for monotone convex function at the cost of unitary congruences.

43 citations


Journal ArticleDOI
TL;DR: In this paper, the authors consider a planner choosing treatments for observationally identical persons who vary in their response to treatment, and assume that the objective is to maximize a concave-monotone function f( ·) of the success rate and show that admissibility depends on the curvature of f ( ·).

Journal IssueDOI
01 Aug 2007
TL;DR: From the results reported, it can be shown that the hybrid methodology improves upon previous approaches in terms of efficiency and also on the pure genetic algorithm, i.e., without using the local search procedure.
Abstract: We address the single-source uncapacitated minimum cost network flow problem with general concave cost functions. Exact methods to solve this class of problems in their full generality are only able to address small to medium size instances, since this class of problems is known to be NP-Hard. Therefore, approximate methods are more suitable. In this work, we present a hybrid approach combining a genetic algorithm with a local search. Randomly generated test problems have been used to test the computational performance of the algorithm. The results obtained for these test problems are compared to optimal solutions obtained by a dynamic programming method for the smaller problem instances and to upper bounds obtained by a local search method for the larger problem instances. From the results reported it can be shown that the hybrid methodology improves upon previous approaches in terms of efficiency and also on the pure genetic algorithm, i.e., without using the local search procedure. © 2007 Wiley Periodicals, Inc. NETWORKS, Vol. 50(1), 67–76 2007

Journal ArticleDOI
TL;DR: In this article, a lower order lower bound for concave functions is introduced for the unit disk onto concave closed sets, which is the complements of convex closed sets.
Abstract: Conformal maps of the unit disk onto convex domains are a classical topic. Recently Avkhadiev and Wirths discovered that conformal maps onto concave domains (the complements of convex closed sets) have some novel properties, namely that there are non-trivial lower estimates, a rare thing for univalent functions (Avkhadiev, F.G. and Wirths, K.-J., 2002, Convex holes produce lower bounds for coefficients, Complex Variables, Theory and Application, 47, 556–563). We prove an inequality involving two variables and introduce a lower order for concave functions. §Dedicated to Professor Peter L. Duren on the occasion of his 70th birthday.

Journal ArticleDOI
TL;DR: In this paper, the authors show that there exists a regular diffusion process X and a differentiable gain function G such that the value function V of the optimal stopping problem fails to satisfy the smooth fit condition at the optimal point b.
Abstract: We show that there exists a regular diffusion process X and a differentiable gain function G such that the value function V of the optimal stopping problem fails to satisfy the smooth fit condition at the optimal stopping point b. On the other hand, if the scale function S of X is differentiable at b, then the smooth fit condition holds (whenever X is regular and G is differentiable at b). We give an example showing that the latter can happen even when at b.

Journal ArticleDOI
TL;DR: A concave function is constructed which is finitely defined on the whole space and can be considered as an extension of the existing function of the linear multicriteria programming problem.
Abstract: The efficient set of a linear multicriteria programming problem can be represented by a reverse convex constraint of the form g(z)≤0, where g is a concave function. Consequently, the problem of optimizing some real function over the efficient set belongs to an important problem class of global optimization called reverse convex programming. Since the concave function used in the literature is only defined on some set containing the feasible set of the underlying multicriteria programming problem, most global optimization techniques for handling this kind of reverse convex constraint cannot be applied. The main purpose of our article is to present a method for overcoming this disadvantage. We construct a concave function which is finitely defined on the whole space and can be considered as an extension of the existing function. Different forms of the linear multicriteria programming problem are discussed, including the minimum maximal flow problem as an example.

Journal ArticleDOI
TL;DR: An effective solution procedure, particularly useful for an approximation scheme, is proposed for the economic production quantity model with shortages under a general inventory cost rate function and piecewise linear concave production costs.

Journal ArticleDOI
TL;DR: The conjecture that any continuous even function can be uniformly approximated by homogeneous polynomials of even degree on K is proven in the following cases: (a) if d = 2, (b) if K is twice continuously differentiable and has positive curvature in every point; or (c) if k is the boundary of a convex polytope.
Abstract: Let \(K\subset\mathbb{R}^d\) be the boundary of a convex domain symmetric to the origin. The conjecture that any continuous even function can be uniformly approximated by homogeneous polynomials of even degree on K is proven in the following cases: (a) if d = 2; (b) if K is twice continuously differentiable and has positive curvature in every point; or (c) if K is the boundary of a convex polytope.

Proceedings ArticleDOI
01 May 2007
TL;DR: A new stochastic tool, called convex ordering, is presented that provides an ordering of any convex function of transmission rates of two protocols and valuable insights into high order behaviors of protocols.
Abstract: Window growth function for congestion control is a strong determinant of protocol behaviors, especially its second and higher-order behaviors associated with the distribution of transmission rates, its variances, and protocol stability. This paper presents a new stochastic tool, called convex ordering, that provides an ordering of any convex function of transmission rates of two protocols and valuable insights into high order behaviors of protocols. As the ordering determined by this tool is consistent with any convex function of rates, it can be applied to any unknown metric for protocol performance that consists of some high-order moments of transmission rates, as well as those already known such as rate variance. Using the tool, it is analyzed that a protocol with a growth function that starts off with a concave function and then switches to a convex function (e.g., an odd order function such as x3 and x5) around the maximum window size in the previous loss epoch, gives the smallest rate variation under a variety of network conditions. Among existing protocols, BIC and CUBIC have this window growth function. Experimental and simulation results confirm the analytical findings.

Journal ArticleDOI
TL;DR: A loop analysis of transportation cost is performed, and an alternative mathematical proof of the optimality of tree-formed networks is given, which leads to an efficient global algorithm for the searching of optimal structures for a given transportation system with concave cost functions.
Abstract: Transportation networks play a vital role in modern societies. Structural optimization of a transportation system under a given set of constraints is an issue of great practical importance. For a general transportation system whose total cost C is determined by C=Sigma C-i < j(ij)(I-ij), with C-ij (I-ij) being the cost of the flow I-ij between node i and node j, Banavar and co-workers [Phys. Rev. Lett. 84, 4745 (2000)] proved that the optimal network topology is a tree if C-ij proportional to parallel to I-ij parallel to(gamma) with 0 < 1. The same conclusion also holds in the more general case where all the flow costs are strictly concave functions of the flow I-ij. To further understand the qualitative difference between systems with concave and convex cost functions, a loop analysis of transportation cost is performed in the present paper, and an alternative mathematical proof of the optimality of tree-formed networks is given. The simple intuitive picture of this proof then leads to an efficient global algorithm for the searching of optimal structures for a given transportation system with concave cost functions.

Proceedings ArticleDOI
04 Dec 2007
TL;DR: Scenarios in which the utility function is an arbitrary, rather than concave function of multicast rate are considered and a new cross-layer optimization approach which converges to a sub-optimal solution is proposed by using a combination of probability collectives (PC) method and network coding.
Abstract: Recently, the problem of resource allocation for achieving optimal multicast throughput has become of high interest. In this paper, we consider scenarios in which the utility function is an arbitrary, rather than concave function of multicast rate and specifically apply the proposed algorithm to the multicast problem in CDMA wireless networks. By using a combination of probability collectives (PC) method and network coding, a new cross-layer optimization approach which converges to a sub-optimal solution is proposed. This solution has been achieved by decomposing the original problem into data routing sub-problems at the network layer and power allocation sub-problems at the physical layer. These sub-problems are then coupled through a set of Lagrangian multipliers and each sub- problem is solved in a distributed fashion. It will also be shown that if the utility function is concave or monotonic, the proposed approach converges to a near optimal solution.

Proceedings ArticleDOI
24 Jun 2007
TL;DR: Convexity/concavity properties of symbol error rates (SER) of the maximum likelihood detector operating in the AWGN channel (non-fading and fading) are studied and universal bounds for the SER 1st and 2nd derivatives are obtained.
Abstract: Convexity/concavity properties of symbol error rates (SER) of the maximum likelihood detector operating in the AWGN channel (non-fading and fading) are studied. Generic conditions are identified under which the SER is a convex/concave function of the SNR. Universal bounds for the SER 1st and 2nd derivatives are obtained, which hold for arbitrary constellations and are tight for some of them. Applications of the results are discussed, which include optimum power allocation in spatial multiplexing systems, optimum power/time sharing to decrease or increase (jamming problem) error rate, and implication for fading channels.

Journal ArticleDOI
TL;DR: A new class of capacitated economic lot-sizing problems is studied and it is shown that the problem is NP-hard in general and a fully polynomial-time approximation algorithm under mild conditions on the cost functions is derived.

Book ChapterDOI
TL;DR: In this paper, the authors presented a confidence band for a regression function f using suitable multiscale sign tests and showed that good approximations can be obtained in O(n2) steps.
Abstract: Suppose that one observes pairs (x1,Y1), (x2,Y2), ..., (xn,Yn), where x1 < x2 < ... < xn are fixed numbers while Y1, Y2, ..., Yn are independent random variables with unknown distributions. The only assumption is that Median(Yi) = f(xi) for some unknown convex or concave function f. We present a confidence band for this regression function f using suitable multiscale sign tests. While the exact computation of this band seems to require O(n4) steps, good approximations can be obtained in O(n2) steps. In addition the confidence band is shown to have desirable asymptotic properties as the sample size n tends to infinity.

Patent
21 Mar 2007
TL;DR: In this paper, a transductive support vector machine is trained based on labeled training data and unlabeled test data and a non-convex objective function which optimizes a hyperplane classifier for classifying the unlabelled test data is decomposed into a convex function and a concave function.
Abstract: Disclosed is a method for training a transductive support vector machine The support vector machine is trained based on labeled training data and unlabeled test data A non-convex objective function which optimizes a hyperplane classifier for classifying the unlabeled test data is decomposed into a convex function and a concave function A local approximation of the concave function at a hyperplane is calculated, and the approximation of the concave function is combined with the convex function such that the result is a convex problem The convex problem is then solved to determine an updated hyperplane This method is performed iteratively until the solution converges

Posted Content
TL;DR: In this paper, a non-negative concave function on the positive half-line was shown to be operator concave when f is concave and the symmetric norms are symmetric.
Abstract: Let f be a non-negative concave function on the positive half-line. Let A and B be two positive matrices. Then, for all symmetric norms, || f(A+B) || is less than || f(A)+f(B) ||. When f is operator concave, this was proved by Ando and Zhan. Our method is simpler. Several related results are presented.


Posted Content
TL;DR: The optimal switching problem for one-dimensional diffusions is solved by directly using the dynamic programming principle and the excessive characterization of the value function using the properties of concave functions.
Abstract: We explicitly solve the optimal switching problem for one-dimensional diffusions by directly employing the dynamic programming principle and the excessive characterization of the value function. The shape of the value function and the smooth fit principle then can be proved using the properties of concave functions.

Journal ArticleDOI
TL;DR: This paper presents necessary and sufficient conditions for boundedness of a feasible region defined by reverse convex constraints and establishes sufficient and necessary conditions for existence of an upper bound for a convex objective function defined over the system of concave inequality constraints.
Abstract: In this paper, we are concerned with the problem of boundedness in the constrained global maximization of a convex function. In particular, we present necessary and sufficient conditions for boundedness of a feasible region defined by reverse convex constraints and we establish sufficient and necessary conditions for existence of an upper bound for a convex objective function defined over the system of concave inequality constraints. We also address the problem of boundedness in the global maximization problem when a feasible region is convex and unbounded.

Journal Article
TL;DR: In this article, it was shown that g expectation is equal to the classical expectation when g is a linear generator and Jensen's inequality of general bivariate concave function for g expectation holds.
Abstract: Using Girsanov transformation,the paper presents an attempt to prove that g expectation is equal to the classical expectation when g is a linear generator,thus Jensen's inequality of general bivariate concave function for g expectation holdsThen the paper,by exploring the representation theorem for generators,leads to conclusion that g is linear when Jensen's inequality of general bivariate concave function holds;it is proved that Jensen's inequality of bivariate increasing concave function holds if and only if g is a sublinear generator

Posted Content
TL;DR: This article showed that the number of isolated roots of a system of polynomials in a torus is bounded by the mixed volume of the Newton polytopes of the given polynomial, and this upper bound is generically exact.
Abstract: A theorem of Kushnirenko and Bernstein shows that the number of isolated roots of a system of polynomials in a torus is bounded above by the mixed volume of the Newton polytopes of the given polynomials, and this upper bound is generically exact. We improve on this result by introducing refined combinatorial invariants of polynomials and a generalization of the mixed volume of convex bodies: the mixed integral of concave functions. The proof is based on new techniques and results from relative toric geometry.