Showing papers on "Concave function published in 2022"
5 citations
4 citations
TL;DR: Under standard assumptions on the energy potential (Lipschitz continuity), it is demonstrated rigorously that the method converges optimally for symmetric schemes, and suboptimally for nonsymmetric schemes.
3 citations
TL;DR: In this paper , Colesanti and Fragalà studied the surface area measure of a log-concave function under minimal and optimal conditions, and extended the definition of functional surface area to the Lp-setting, generalizing a classic definition of Lutwak.
Abstract: This paper’s origins are in two papers: One by Colesanti and Fragalà studying the surface area measure of a log-concave function, and one by Cordero-Erausquin and Klartag regarding the moment measure of a convex function. These notions are the same, and in this paper we continue studying the same construction as well as its generalization. In the first half of the paper we prove a first variation formula for the integral of log-concave functions under minimal and optimal conditions. We also explain why this result is a common generalization of two known theorems from the above papers. In the second half we extend the definition of the functional surface area measure to the Lp-setting, generalizing a classic definition of Lutwak. In this generalized setting we prove a functional Minkowski existence theorem for even measures. This is a partial extension of a theorem of Cordero-Erausquin and Klartag that handled the case p = 1 for not necessarily even measures.
3 citations
TL;DR: In this paper , random concave functions are constructed on the unit simplex by taking a suitably scaled (mollified, or soft) minimum of random hyperplanes, and a transition from a deterministic almost sure limit to a nontrivial limiting distribution is characterized using convex duality and Poisson point processes.
Abstract: Spaces of convex and concave functions appear naturally in theory and applications. For example, convex regression and log-concave density estimation are important topics in nonparametric statistics. In stochastic portfolio theory, concave functions on the unit simplex measure the concentration of capital, and their gradient maps define novel investment strategies. The gradient maps may also be regarded as optimal transport maps on the simplex. In this paper we construct and study probability measures supported on spaces of concave functions. These measures may serve as prior distributions in Bayesian statistics and Cover’s universal portfolio, and induce distribution-valued random variables via optimal transport. The random concave functions are constructed on the unit simplex by taking a suitably scaled (mollified, or soft) minimum of random hyperplanes. Depending on the regime of the parameters, we show that as the number of hyperplanes tends to infinity there are several possible limiting behaviors. In particular, there is a transition from a deterministic almost sure limit to a nontrivial limiting distribution that can be characterized using convex duality and Poisson point processes.
2 citations
TL;DR: In this article , concentration inequalities in the class of ultra-log-concave distributions were established and it was shown that these distributions satisfy Poisson concentration bounds for the intrinsic ǫ > 0.
Abstract: We establish concentration inequalities in the class of ultra log-concave distributions. In particular, we show that ultra log-concave distributions satisfy Poisson concentration bounds. As an application, we derive concentration bounds for the intrinsic
2 citations
TL;DR: In this article , a new way of representing logarithmically concave functions on roads was introduced, which allows us to extend the notion of the largest volume ellipsoid contained in a convex body to the setting of concave function on roads.
2 citations
06 Apr 2022
TL;DR: This study proposes minimizing the worst-case objective function F(x) = maxy f (x, y) directly using the covariance matrix adaptation evolution strategy, in which the rankings of solution candidates are approximated by the proposed best-case ranking approximation (WRA) mechanism.
Abstract: In this study, we investigate the problem of min-max continuous optimization in a black-box setting minx maxy f (x,y). A popular approach updates x and y simultaneously or alternatingly. However, two major limitations have been reported in existing approaches. (I) As the influence of the interaction term between x and y (e.g., xTBy) on the Lipschitz smooth and strongly convex-concave function f increases, the approaches converge to an optimal solution at a slower rate. (II) The approaches fail to converge if f is not Lipschitz smooth and strongly convex-concave around the optimal solution. To address these difficulties, we propose minimizing the worst-case objective function F(x) = maxy f (x, y) directly using the covariance matrix adaptation evolution strategy, in which the rankings of solution candidates are approximated by our proposed worst-case ranking approximation (WRA) mechanism. Compared with existing approaches, numerical experiments show two important findings regarding our proposed method. (1) The proposed approach is eficient in terms of f-calls on a Lipschitz smooth and strongly convex-concave function with a large interaction term. (2) The proposed approach can converge on functions that are not Lipschitz smooth and strongly convex-concave around the optimal solution, whereas existing approaches fail.
2 citations
TL;DR: In this article , the Riesz representation theorem was extended to the class of log-concave functions on R n , where "linear" is defined as linear with respect to the natural addition on a log-Concave function which is the sup-convolution.
1 citations
TL;DR: In this paper , the operator trace function Λr,s(A)[K,M]:=tr(K⁎ArMArK)s is introduced and its convexity and concavity properties are investigated.
1 citations
TL;DR: In this article , the theory of f-divergences for log-concave functions and their related inequalities was further developed for affine invariant entropy inequalities for convex bodies.
Abstract: In this paper we further develop the theory of f-divergences for log-concave functions and their related inequalities. We establish Pinsker inequalities and new affine invariant entropy inequalities. We obtain new inequalities on functional affine surface area and lower and upper bounds for the Kullback-Leibler divergence in terms of functional affine surface area. The functional inequalities lead to new inequalities for L_p-affine surface areas for convex bodies.
TL;DR: In this paper , the authors consider a family of functions called ray-concave functions and derive a closed-form expression for the convex envelope of these functions over arbitrary polytopes.
Abstract: Convexification based on convex envelopes is ubiquitous in the non-linear optimization literature. Thanks to considerable efforts of the optimization community for decades, we are able to compute the convex envelopes of a considerable number of functions that appear in practice, and thus obtain tight and tractable approximations to challenging problems. We contribute to this line of work by considering a family of functions that, to the best of our knowledge, has not been considered before in the literature. We call this family ray-concave functions. We show sufficient conditions that allow us to easily compute closed-form expressions for the convex envelope of ray-concave functions over arbitrary polytopes. With these tools, we are able to provide new perspectives to previously known convex envelopes and derive a previously unknown convex envelope for a function that arises in probability contexts.
22 Dec 2022
TL;DR: In this article , the authors consider the problem of finding the largest volume ellipsoid contained in a convex body under the constraint that it is pointwise below some given log-concave function.
Abstract: John's fundamental theorem characterizing the largest volume ellipsoid contained in a convex body $K$ in $\mathbb{R}^d$ has seen several generalizations and extensions. One direction, initiated by V. Milman is to replace ellipsoids by positions (affine images) of another body $L$. Another, more recent direction is to consider logarithmically concave functions on $\mathbb{R}^d$ instead of convex bodies: we designate some special, radially symmetric log-concave function $g$ as the analogue of the Euclidean ball, and want to find its largest integral position under the constraint that it is pointwise below some given log-concave function $f$. We follow both directions simultaneously: we consider the functional question, and allow essentially any meaningful function to play the role of $g$ above. Our general theorems jointly extend known results in both directions. The dual problem in the setting of convex bodies asks for the smallest volume ellipsoid, called \emph{L{\"o}wner's ellipsoid}, containing $K$. We consider the analogous problem for functions: we characterize the solutions of the optimization problem of finding a smallest integral position of some log-concave function $g$ under the constraint that it is pointwise above $f$. It turns out that in the functional setting, the relationship between the John and the L{\"o}wner problems is more intricate than it is in the setting of convex bodies.
TL;DR: In this paper , Chen and Li proposed a novel concept of S-convex functions defined on continuous spaces, which extends a key concept of M-natural-consistency in discrete convex analysis.
Abstract: A New Concept to Study Substitute Structures in Economics and Operations Models In “S-Convexity and Gross Substitutability,” Chen and Li propose a novel concept of S-convex functions defined on continuous spaces, which extends a key concept of M-natural-convex functions in discrete convex analysis. They develop a host of fundamental properties and characterizations of S-convex functions. In a parametric maximization model with a box constraint, they show that the set of optimal solutions is nonincreasing in the parameters if the objective function is S-concave and prove the necessity of S-concavity under some conditions. The monotonicity result finds notable inventory models. Interestingly, the authors show that S-concavity is the correct notion characterizing gross substitutability, an important concept in economics for markets with divisible goods.
TL;DR: In this article , a necessary and sufficient condition for a real-valued function defined on an open and convex subset of a Banach space to be quasi-concave is presented.
Abstract: This paper presents a necessary and sufficient condition for a real-valued function defined on an open and convex subset of a Banach space to be quasi-concave, and a sufficient condition for such a function to be strictly quasi-concave. These conditions are applicable to continuously differentiable functions that satisfy a mild additional assumption, and do not require the functions to be twice differentiable. Because this additional assumption is trivially satisfied for twice continuously differentiable functions, our results are pure extensions to classical results.
29 Jul 2022
TL;DR: In this paper , the authors considered De Finetti's control problem for continuous strategies with control rates bounded by a concave function and proved that a generalized mean-reverting strategy is optimal.
Abstract: We consider De Finetti's control problem for absolutely continuous strategies with control rates bounded by a concave function and prove that a generalized mean-reverting strategy is optimal. In order to solve this problem, we need to deal with a nonlinear Ornstein-Uhlenbeck process. Despite the level of generality of the bound imposed on the rate, an explicit expression for the value function is obtained up to the evaluation of two functions.This optimal control problem has those with control rates bounded by a constant and a linear function, respectively, as special cases.
08 Sep 2022
TL;DR: In this paper , a necessary and sufficient condition for a real-valued function defined on an open and convex subset of a Banach space to be quasi-concave is presented.
Abstract: This paper presents a necessary and sufficient condition for a real-valued function defined on an open and convex subset of a Banach space to be quasi-concave, and a sufficient condition for such a function to be strictly quasi-concave. These conditions are applicable to continuously differentiable functions that satisfy a mild additional assumption, and do not require the functions to be twice differentiable. Because this additional assumption is trivially satisfied for twice continuously differentiable functions, our results are pure extensions to classical results.
03 Oct 2022
TL;DR: In this article , nonparametric estimators for the conditional cumulative distribution functions Fx(y)=ℙ(Y≤y|X=x) of a response variable Y given a covariate X, solely under the assumption that the conditional distributions are increasing in x in the increasing concave or increasing convex order.
Abstract: A random variable Y1 is said to be smaller than Y2 in the increasing concave stochastic order if E[ϕ(Y1)]≤E[ϕ(Y2)] for all increasing concave functions ϕ for which the expected values exist, and smaller than Y2 in the increasing convex order if E[ψ(Y1)]≤E[ψ(Y2)] for all increasing convex ψ. This article develops nonparametric estimators for the conditional cumulative distribution functions Fx(y)=ℙ(Y≤y|X=x) of a response variable Y given a covariate X, solely under the assumption that the conditional distributions are increasing in x in the increasing concave or increasing convex order. Uniform consistency and rates of convergence are established both for the K-sample case X∈{1,…,K} and for continuously distributed X.
13 Dec 2022
TL;DR: In this paper , a family of entanglement monotones with the reduced functions are explored and it is shown that they are not strictly concave and therefore cannot be considered to be a monotone.
Abstract: Quantum entanglement is known to be monogamous, i.e., it obeys strong constraints on how the entanglement can be distributed among multipartite systems. Almost all the entanglement monotones so far are shown to be monogamous. We explore here a family of entanglement monotones with the reduced functions are concave but not strictly concave and show that they are not monogamous. They are defined by four kinds of the ``partial-norm'' of the reduced state, which we call them \textit{partial-norm of entanglement}, minimal partial-norm of entanglement, reinforced minimal partial-norm of entanglement, and \textit{partial negativity}, respectively. This indicates that, the previous axiomatic definition of the entanglement monotone needs supplemental agreement that the reduced function should be strictly concave since such a strict concavity can make sure that the corresponding convex-roof extended entanglement monotone is monogamous. Here, the reduced function of an entanglement monotone refers to the corresponding function on the reduced state for the measure on bipartite pure states.
21 Jul 2022
TL;DR: In this article , it was shown that all level sequences in codimension two are log-concave, and that every compressed level Hilbert function in any codin-tion is also linear in length.
Abstract: We show here that codimension three Artinian Gorenstein sequences are log-concave, and that there are codimension four Artinian Gorenstein sequences that are not log-concave. We also show that all level sequences in codimension two, and every compressed level Hilbert function in any codimension is log-concave.
19 Aug 2022
TL;DR: In this article , a piecewise inner-approximation of the concave function is achieved using an auxiliary linear program that leads to a bilevel program, which provides a lower bound to the original problem.
Abstract: In this article, we discuss an exact algorithm for solving mixed integer concave minimization problems. A piecewise inner-approximation of the concave function is achieved using an auxiliary linear program that leads to a bilevel program, which provides a lower bound to the original problem. The bilevel program is reduced to a single level formulation with the help of Karush-Kuhn-Tucker (KKT) conditions. Incorporating the KKT conditions lead to complementary slackness conditions that are linearized using BigM, for which we identify a tight value for general problems. Multiple bilevel programs, when solved over iterations, guarantee convergence to the exact optimum of the original problem. Though the algorithm is general and can be applied to any optimization problem with concave function(s), in this paper, we solve two common classes of operations and supply chain problems; namely, the concave knapsack problem, and the concave production-transportation problem. The computational experiments indicate that our proposed approach outperforms the customized methods that have been used in the literature to solve the two classes of problems by an order of magnitude in most of the test cases.
TL;DR: In this article , two computer programming languages, Wolfram Mathematica and Anaconda Python, were used to solve a transportation problem involving a concave cost function and achieved an optimal value of N253,000 with an optimal solution as z12 = 13, z22 = 5, z23 = 8, z31 = 11 and z33 = 4.
Abstract: The work is on computer programming language solution to a transportation problem involving a concave cost function. Two computer programming languages; Wolfram Mathematica and Anaconda Python programming tools were employed in this study to effectively solve four real life examples from published works. The results from the programming languages yielded an optimal value of N253,000 with an optimal solution as z12 = 13, z22 = 5, z23 = 8, z31 = 11 and z33 = 4 in the first example and the remaining three examples were successfully solved with optimal values of N377,000, GH¢ 236,000 and N509,000 respectively, and the results agreed with the results of existing Karush-Kuhn-Tucker (KKT) procedure of Modified Distribution Method.
TL;DR: The result is a MATLAB code that can be used to compute the maximum length of the crossbar as a function of the width of the channel (its two parts) and the angle between them.
Abstract: In this short paper, we study the problem of traversing a crossbar through a bent channel, which has been formulated as a nonlinear convex optimization problem. The result is a MATLAB code that we can use to compute the maximum length of the crossbar as a function of the width of the channel (its two parts) and the angle between them. In case they are perpendicular to each other, the result is expressed analytically and is closely related to the astroid curve (a hypocycloid with four cusps).
TL;DR: In this paper , the authors give some singular value inequalities for sector matrices involving operator concave function, which are generalizations of some existing results and present some unitarily invariant norm inequalities for sectors matrices.
Abstract: In this paper, we give some singular value inequalities for sector matrices involving operator concave function, which are generalizations of some existing results. Moreover, we present some unitarily invariant norm inequalities for sector matrices.
11 Jul 2022
TL;DR: In this paper , the minimax identity for a non-decreasing upper-semicontinuous utility function satisfying mild growth assumption was studied and inequalities between the robust utility functionals of an initial utility function and its concavification were obtained.
Abstract: We study the minimax identity for a non-decreasing upper-semicontinuous utility function satisfying mild growth assumption. In contrast to the classical setting, we do not impose the assumption that the utility function is concave. By considering the concave envelope of the utility function we obtain equalities and inequalities between the robust utility functionals of an initial utility function and its concavification. Furthermore, we prove similar equalities and inequalities in the case of implementing an upper bound on the final endowment of the initial model.