Showing papers on "Concave function published in 2009"
TL;DR: This work presents a subgradient algorithm for generating approximate saddle points and provides per-iteration convergence rate estimates on the constructed solutions, and focuses on Lagrangian duality, where it is shown this algorithm is particularly well-suited for problems where the subgradient of the dual function cannot be evaluated easily.
Abstract: We study subgradient methods for computing the saddle points of a convex-concave function. Our motivation comes from networking applications where dual and primal-dual subgradient methods have attracted much attention in the design of decentralized network protocols. We first present a subgradient algorithm for generating approximate saddle points and provide per-iteration convergence rate estimates on the constructed solutions. We then focus on Lagrangian duality, where we consider a convex primal optimization problem and its Lagrangian dual problem, and generate approximate primal-dual optimal solutions as approximate saddle points of the Lagrangian function. We present a variation of our subgradient method under the Slater constraint qualification and provide stronger estimates on the convergence rate of the generated primal sequences. In particular, we provide bounds on the amount of feasibility violation and on the primal objective function values at the approximate solutions. Our algorithm is particularly well-suited for problems where the subgradient of the dual function cannot be evaluated easily (equivalently, the minimum of the Lagrangian function at a dual solution cannot be computed efficiently), thus impeding the use of dual subgradient methods.
497 citations
TL;DR: In this article, the authors established the limiting distribution of the resulting estimator of the mode M(f0) and established 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.
Abstract: We find limiting distributions of the nonparametric maximum likelihood estimator (MLE) of a log-concave density, that is, a density of the form f0=exp ϕ0 where ϕ0 is a concave function on ℝ. The pointwise limiting distributions depend on the second and third derivatives at 0 of Hk, the “lower invelope” of an integrated Brownian motion process minus a drift term depending on the number of vanishing derivatives of ϕ0=log f0 at the point of interest. We also establish the limiting distribution of the resulting estimator of the mode M(f0) 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.
113 citations
TL;DR: This paper studies a joint pricing and product quality decision problem in a distribution channel, in which a manufacturer sells a product through a retailer, and finds that if the marginal revenue function is strictly concave, then the manufacturer chooses a lower product quality level than if selling the product directly to customers.
Abstract: This paper studies a joint pricing and product quality decision problem in a distribution channel, in which a manufacturer sells a product through a retailer. The manufacturer jointly determines the wholesale price and quality of the product, and the retailer determines the retail price. We find that if the marginal revenue function is strictly concave, then the manufacturer chooses a lower product quality level than if selling the product directly to customers. If the marginal revenue function is affine, then the manufacturer's optimal product quality decision is independent of the distribution channel structure. If the marginal revenue function is strictly convex, then the manufacturer chooses a higher product quality level than if selling the product directly to customers.
111 citations
TL;DR: This work provides recommended policies for convex, concave, and convex-concave lead-time cost functions and proves that these policies are asymptotically optimal and provide delay guarantees for all served customers.
Abstract: We consider a make-to-order system where customers are dynamically quoted lead times (and prices). Customers are homogenous but have general (nonlinear) disutility for delay. Because the firm is a monopolist, the pricing problem is trivial and the dynamic problem reduces to one of lead-time quotation and order sequencing. We also consider the (static) problem of up-front capacity installation. We use a large-capacity asymptotic regime to make the problem tractable. We provide recommended policies for convex, concave, and convex-concave lead-time cost functions and prove that these policies are asymptotically optimal. The policies are both highly intuitive and readily implementable. Moreover, they provide delay guarantees for all served customers. They are tested numerically; we find that significant benefits can accrue by using the prescribed dynamic policies instead of first-come-first-served type policies.
77 citations
TL;DR: This work provides a rare proof of convergence for an approximate dynamic programming algorithm using pure exploitation, where the states the authors visit depend on the decisions produced by solving the approximate problem.
Abstract: We consider a multistage asset acquisition problem where assets are purchased now, at a price that varies randomly over time, to be used to satisfy a random demand at a particular point in time in the future. We provide a rare proof of convergence for an approximate dynamic programming algorithm using pure exploitation, where the states we visit depend on the decisions produced by solving the approximate problem. The resulting algorithm does not require knowing the probability distribution of prices or demands, nor does it require any assumptions about its functional form. The algorithm and its proof rely on the fact that the true value function is a family of piecewise linear concave functions.
63 citations
TL;DR: For subclasses of functions 𝑓 including, for example, that of concave functions, the order of approximation 𝜔1(𝓓;1/𝑛), which for many functions � is essentially better than theOrder of approximation obtained by the linear Bernstein operators.
Abstract: Starting from the study of the Shepard nonlinear operator of max-prod type by Bede et al. (2006, 2008), in the book by Gal (2008), Open Problem 5.5.4, pages 324–326, the Bernstein max-prod-type operator is introduced and the question of the approximation order by this operator is raised. In recent paper, Bede and Gal by using a very complicated method to this open question an answer is given by obtaining an upper estimate of the approximation error of the form 𝐶𝜔1√(𝑓;1/𝑛) (with an unexplicit absolute constant 𝐶g0) and the question of improving the order of approximation 𝜔1√(𝑓;1/𝑛) is raised. The first aim of this note is to obtain this order of approximation but by a simpler method, which in addition presents, at least, two advantages: it produces an explicit constant in front of 𝜔1√(𝑓;1/𝑛) and it can easily be extended to other max-prod operators of Bernstein type. However, for subclasses of functions 𝑓 including, for example, that of concave functions, we find the order of approximation 𝜔1(𝑓;1/𝑛), which for many functions 𝑓 is essentially better than the order of approximation obtained by the linear Bernstein operators. Finally, some shape-preserving properties are obtained.
47 citations
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TL;DR: This paper shows that even when all the utilities are additively separable, piecewise-linear and concave, computing an approximate equilibrium in Fisher's model is PPAD-hard, which solves a long-term open question on the complexity of market equilibria.
Abstract: It is a common belief that computing a market equilibrium in Fisher's spending model is easier than computing a market equilibrium in Arrow-Debreu's exchange model. This belief is built on the fact that we have more algorithmic success in Fisher equilibria than Arrow-Debreu equilibria. For example, a Fisher equilibrium in a Leontief market can be found in polynomial time, while it is PPAD-hard to compute an approximate Arrow-Debreu equilibrium in a Leontief market.
In this paper, we show that even when all the utilities are additively separable, piecewise-linear, and concave functions, finding an approximate equilibrium in Fisher's model is complete in PPAD. Our result solves a long-term open question on the complexity of market equilibria. To the best of our knowledge, this is the first PPAD-completeness result for Fisher's model.
40 citations
TL;DR: Test results show that the e-dominance criterion can often significantly reduce the number of nondominated path vectors, compared to the standard dominance criterion, and there is a consequent reduction in both computation time and storage requirements.
39 citations
TL;DR: This paper proposes a finite element description of the Hessian, and proves convergence under very general conditions, even when the continuous solution is not smooth, working on any dimension, and requiring a linear number of constraints in the number of nodes.
Abstract: Many problems of theoretical and practical interest involve finding a convex or concave function. For instance, optimization problems such as finding the projection on the convex functions in $H^k(\Omega)$, or some problems in economics. In the continuous setting and assuming smoothness, the convexity constraints may be given locally by asking the Hessian matrix to be positive semidefinite, but in making discrete approximations two difficulties arise: the continuous solutions may be not smooth, and an adequate discrete version of the Hessian must be given. In this paper we propose a finite element description of the Hessian, and prove convergence under very general conditions, even when the continuous solution is not smooth, working on any dimension, and requiring a linear number of constraints in the number of nodes. Using semidefinite programming codes, we show concrete examples of approximations to optimization problems.
38 citations
TL;DR: It is proved it is a unimodal function and by taking advantage of this property, the optimal solution is determined and the model is developed mathematically and an analytical solution method is proposed.
31 citations
TL;DR: This work introduces new concave penalty functions and it is proved, using general equivalence results here derived, that the obtained continuous problems are equivalent to the original combinatorial problem.
Abstract: In this work, we study continuous reformulations of zero-one concave programming problems. We introduce new concave penalty functions and we prove, using general equivalence results here derived, that the obtained continuous problems are equivalent to the original combinatorial problem.
TL;DR: In this article, the authors considered the problem of maximizing a nondecreasing submodular set function under a matroid constraint, and proposed a framework based on the pipage rounding technique by Ageev and Sviridenko.
Abstract: We consider the problem of maximizing a nondecreasing submodular set function under a matroid constraint. Recently, Calinescu et al. (2007) proposed an elegant framework for the approximation of this problem, which is based on the pipage rounding technique by Ageev and Sviridenko (2004), and showed that this framework indeed yields a (1 - 1/e)-approximation algorithm for the class of submodular functions which are represented as the sum of weighted rank functions of matroids. This paper sheds a new light on this result from the viewpoint of discrete convex analysis by extending it to the class of submodular functions which are the sum of M♮-concave functions. M♮-concave functions are a class of discrete concave functions introduced by Murota and Shioura (1999), and contain the class of the sum of weighted rank functions as a proper subclass. Our result provides a better understanding for why the pipage rounding algorithm works for the sum of weighted rank functions. Based on the new observation, we further extend the approximation algorithm to the maximization of a nondecreasing submodular function over an integral polymatroid. This extension has an application in multi-unit combinatorial auctions.
TL;DR: A study on the selection principle of mass on numerical optimisation problems and the performance of APO algorithm is compared with that of the modified electromagnetism-like (EM), differential evolution (DE), evolutionary algorithm (EA) and particle swarm optimisation (PSO) for multidimensional numeric benchmarks.
Abstract: Artificial physics optimisation (APO) algorithm is an optimisation algorithm based on physicomimetics framework. Driven by virtual force, a population of sample individuals searches a global optimum in the problem space. The mass of each individual corresponds to a user-defined function of the value of an objective function to be optimised. It is an important parameter to influence the performance of APO algorithm. Therefore, in this paper, the authors make a study on the selection principle of mass on numerical optimisation problems. According to the curvilinear style of the mass functions, they are classified into three different types of curvilinear functions: convex function, linear function and concave function. To make a deep insight, several versions of APO algorithm with different mass functions are used to solve two type benchmarks: unimodal and multimodal functions. Simulation results show the mass functions with concave curve may generally obtain the satisfied solution within the allowed iterations. In addition, the performance of APO algorithm is compared with that of the modified electromagnetism-like (EM), differential evolution (DE), evolutionary algorithm (EA) and particle swarm optimisation (PSO) for multidimensional numeric benchmarks. The simulation results show that APO algorithm is competitive.
TL;DR: The approach proposed outperforms other procedures for large-scaled instances (up to 2000 variables and constraints) and also includes pre-procedure techniques in order to reduce the problem size before starting the branch-and-bound process.
Abstract: In this paper we develop a branch-and-bound algorithm for solving a particular integer quadratic multi-knapsack problem. The problem we study is defined as the maximization of a concave separable quadratic objective function over a convex set of linear constraints and bounded integer variables. Our exact solution method is based on the computation of an upper bound and also includes pre-procedure techniques in order to reduce the problem size before starting the branch-and-bound process. We lead a numerical comparison between our method and three other existing algorithms. The approach we propose outperforms other procedures for large-scaled instances (up to 2000 variables and constraints).
01 Jan 2009
TL;DR: In this article, the authors investigate differentiability properties of monetary utility functions and give a counter-example to automatic continuity for concave functions, and show that continuity can be maintained by concave utility functions.
Abstract: We investigate differentiability properties of monetary utility functions At the same time we give a counter-example—important in finance—to automatic continuity for concave functions
TL;DR: In this article, it was shown that non-negative, increasing, convex (respectively, concave) functions are superadditive with respect to submajorisation on the positive cone of the space of all τ-measurable operators affiliated with a semidefinite von Neumann algebra.
Abstract: It is shown that non-negative, increasing, convex (respectively, concave) functions are superadditive (respectively, subadditive) with respect to submajorisation on the positive cone of the space of all τ-measurable operators affiliated with a semifinite von Neumann algebra. This extends recent results for n × n-matrices by Ando-Zhan, Kosem and Bourin-Uchiyama.
TL;DR: In this article, it was shown that the strong CE-property is equivalent to the openness of the barycentre map, while the strong convex envelopes (hulls) of continuous concave functions can be extended to non-compact convex sets, and that such an extension does exist.
Abstract: For a fixed convex domain in a linear metric space the problems of the continuity of convex envelopes (hulls) of continuous concave functions (the CE-property) and of convex envelopes (hulls) of arbitrary continuous functions (the strong CE-property) arise naturally. In the case of compact domains a comprehensive solution was elaborated in the 1970s by Vesterstrom and O'Brien. First Vesterstrom showed that for compact sets the strong CE-property is equivalent to the openness of the barycentre map, while the CE-property is equivalent to the openness of the restriction of this map to the set of maximal measures. Then O'Brien proved that in fact both properties are equivalent to a geometrically obvious 'stability property' of convex compact sets. This yields, in particular, the equivalence of the CE-property to the strong CE-property for convex compact sets. In this paper we give a solution to the following problem: can these results be extended to noncompact convex sets, and, if the answer is positive, to which sets? We show that such an extension does exist. This is an extension to the class of so-called μ-compact sets. Moreover, certain arguments confirm that this could be the maximal class to which such extensions are possible. Then properties of μ-compact sets are analysed in detail, several examples are considered, and applications of the results obtained to quantum information theory are discussed. Bibliography: 32 titles.
TL;DR: The authors show that pulsing can be superior if the product of the concave market potential function and the linear or concave advertising response function is convex in advertising, which is found to be superior to two rival models using a one-step-ahead forecasting procedure.
01 Jan 2009
TL;DR: In this article, the authors present a survey on some recent results about Newton's problem of minimal resistance in the calculus of variations, namely Newton's Problem of Minimal Resistance of Convex Variants.
Abstract: This chapter aims to present a survey on some recent results about one of the first problems in the calculus of variations, namely Newton’s problem of minimal resistance. Many variants of the problem can be studied, in relation to the various admissible classes of domains under consideration and to the different constraints that can be imposed. Here we limit ourselves essentially to the convex case. Other presentations in the workshop will deal with other kinds of domains.
TL;DR: Dominated convergence of the concave integral for capacities is characterized in terms of dual representation whenever sequences of functions converge pointwise outside a set of zero capacity.
TL;DR: In this paper, the existence of two positive solutions of the elliptic equation on R N with concave and convex nonlinearities was considered and shown to be true for both convex and concave nonlinearity.
TL;DR: From characterizations, it is obtained that a continuously differentiable function defined in an open interval is SOC-monotone (SOC-convex) of order n ≥ 3 if and only if it is 2-matrix monotones (matrix convex) and that Conjecture 4.2 proposed in Chen does not hold in general.
Abstract: We provide some characterizations for SOC-monotone and SOC-convex functions by using differential analysis. From these characterizations, we particularly obtain that a continuously differentiable function defined in an open interval is SOC-monotone (SOC-convex) of order n ? 3 if and only if it is 2-matrix monotone (matrix convex), and furthermore, such a function is also SOC-monotone (SOC-convex) of order n ? 2 if it is 2-matrix monotone (matrix convex). In addition, we also prove that Conjecture 4.2 proposed in Chen (Optimization 55:363---385, 2006) does not hold in general. Some examples are included to illustrate that these characterizations open convenient ways to verify the SOC-monotonicity and the SOC-convexity of a continuously differentiable function defined on an open interval, which are often involved in the solution methods of the convex second-order cone optimization.
19 Apr 2009
TL;DR: A new technique for minimizing a class of nonconvex functions for solving the problem of under-determined systems of linear equations is presented, based on locally replacing the nonconvergent objective function by a convex objective function.
Abstract: In this paper we present a new technique for minimizing a class of nonconvex functions for solving the problem of under-determined systems of linear equations. The proposed technique is based on locally replacing the nonconvex objective function by a convex objective function. The main property of the utilized convex function is that it is minimized at a point that reduces the original concave function. The resulting algorithm is iterative and outperforms some previous algorithms that have been applied to the same problem.
TL;DR: This paper forms the convexity and the continuity axioms for preference relations on @s-algebras with a metric topology and shows the existence of utility functions for convex continuous preference relations.
TL;DR: In this article, two nonparametric procedures for estimating a concave distribution function based on data corrupted with additive noise generated by a bounded decreasing density on (0, ∞) were considered.
Abstract: We consider two nonparametric procedures for estimating a concave distribution function based on data corrupted with additive noise generated by a bounded decreasing density on (0, ∞). For the maximum likelihood (ML) estimator and least squares (LS) estimator, we state qualitative properties, prove consistency and propose a computational algorithm. For the LS estimator and its derivative, we also derive the pointwise asymptotic distribution. Moreover, the rate n−2/5 achieved by the LS estimator is shown to be minimax for estimating the distribution function at a fixed point.
TL;DR: In this paper, an instantaneous replenishment policy for deteriorating items with price-dependent demand was developed, where a variable proportion of the items will deteriorate per time, where shortages are permissible and completely backordered.
Abstract: In this paper, we have developed an instantaneous replenishment policy for deteriorating items with price-dependent demand. The demand and deterioration rates are continuous and differentiable function of price and time respectively. A variable proportion of the items will deteriorate per time, where shortages are permissible and completely backordered. We have developed a policy with price-dependent demand under profit maximization. The net profit per unit time is a concave function. Further, it is illustrated with the help of a numerical example. Keywords: Price-dependent; Weibull distribution; Varying rate of deterioration. © 2010 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved. DOI: 10.3329/jsr.v2i1.2764 J. Sci. Res. 2 (1), 24-36 (2010)
21 Aug 2009
TL;DR: This work gives an efficient sampling algorithm based on a random walk for *** 1/(n *** 1)-concave probability densities satisfying a smoothness criterion, which includes heavy-tailed densities such as the Cauchy density.
Abstract: Efficient sampling, integration and optimization algorithms for logconcave functions [BV04, KV06, LV06a] rely on the good isoperimetry of these functions. We extend this to show that *** 1/(n *** 1)-concave functions have good isoperimetry, and moreover, using a characterization of functions based on their values along every line, we prove that this is the largest class of functions with good isoperimetry in the spectrum from concave to quasi-concave. We give an efficient sampling algorithm based on a random walk for *** 1/(n *** 1)-concave probability densities satisfying a smoothness criterion, which includes heavy-tailed densities such as the Cauchy density. In addition, the mixing time of this random walk for Cauchy density matches the corresponding best known bounds for logconcave densities.
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TL;DR: In this paper, the authors studied the maximal operator for functions taking values in a Banach space and showed that the L^p-boundedness of this operator depends on the range space; certain requirements on type and cotype are present for instance.
Abstract: This paper studies a new maximal operator introduced by Hyt\"onen, McIntosh and Portal in 2008 for functions taking values in a Banach space. The L^p-boundedness of this operator depends on the range space; certain requirements on type and cotype are present for instance. The original Euclidean definition of the maximal function is generalized to sigma-finite measure spaces with filtrations and the L^p-boundedness is shown not to depend on the underlying measure space or the filtration. Martingale techniques are applied to prove that a weak type inequality is sufficient for L^p-boundedness and also to provide a characterization by concave functions.
03 Jul 2009
TL;DR: An altruistic designer who knows the utility functions of the users and who can dictate the allocation x = (x 1,...,x n ) can easily choose the allocation that maximizes the welfare, since this is a simple convex optimization problem.
Abstract: Consider n users vying for shares of a divisible good. Every user i wants as much of the good as possible but has diminishing returns, meaning that its utility U i (x i ) for x i *** 0 units of the good is a nonnegative, nondecreasing, continuously differentiable concave function of x i . The good can be produced in any amount, but producing $X = \sum_{i=1}^n x_i$ units of it incurs a cost C (X ) for a given nondecreasing and convex function C that satisfies C (0) = 0. Cost might represent monetary cost, but other interesting interpretations are also possible. For example, x i could represent the amount of traffic (measured in packets, say) that user i injects into a queue in a given time window, and C (X ) could denote aggregate delay (X ·c (X ), where c (X ) is the average per-unit delay). An altruistic designer who knows the utility functions of the users and who can dictate the allocation x = (x 1 ,...,x n ) can easily choose the allocation that maximizes the welfare $W(x) = \sum_{i=1}^n U_i(x_i) - C(X)$, where $X = \sum_{i=1}^n x_i$, since this is a simple convex optimization problem.
TL;DR: In this article, a global minimizer of a concave function over a sphere was found by constructing a canonical dual function, and two theorems concerning the global optimality of an extrema of the optimization problem were presented.