Concave function

About: Concave function is a research topic. Over the lifetime, 1415 publications have been published within this topic receiving 33278 citations.

Modular inequalities in martingale Orlicz–Karamata spaces

Abstract: In this paper, some new martingale inequalities in the framework of Orlicz†Karamata spaces are provided. More precisely, we establish modular martingale inequalities associated with concave functions and slowly varying functions.

The selection strategy of mass functions in artificial physics optimisation algorithm

Abstract: Inspired by physicomimetics, artificial physics optimisation (APO) is a novel population-based stochastic algorithm. In APO framework, the mass of each individual corresponds to a user-defined function of the value of an objective to be optimised, which can supply some important information for searching global optima. There are many functions that can be used as mass function, and no doubt some will be better than others for specific optimisation problems or perhaps classes of problems. This paper proposes the basic requirement and design method of mass function, and classifies mass functions into four different types of curvilinear functions according to their curvilinear styles, such as linear function, convex function, and concave function, etc. Simulation results show the mass functions with concave curve may generally obtain the satisfied solution within the allowed iterations.

The Steiner symmetrization of log-concave functions and its applications

Abstract: In this paper, we give a new definition of functional Steiner symmetrizations on logconcave functions. Using the functional Steiner symmetrization, we give a new proof of the classical Prekopa-Leindler inequality on log-concave functions. Mathematics subject classification (2010): 46E30, 52A40.

The Dyck bound in the concave 1-dimensional random assignment model

Abstract: We consider models of assignment for random $N$ blue points and $N$ red points on an interval of length $2N$, in which the cost for connecting a blue point in $x$ to a red point in $y$ is the concave function $|x-y|^p$, for $0 1$, where the optimal matching is trivially determined, here the optimization is non-trivial. The purpose of this paper is to introduce a special configuration, that we call the \emph{Dyck matching}, and to study its statistical properties. We compute exactly the average cost, in the asymptotic limit of large $N$, together with the first subleading correction. The scaling is remarkable: it is of order $N$ for $p \frac{1}{2}$, and it is universal for a wide class of models. We conjecture that the average cost of the Dyck matching has the same scaling in $N$ as the cost of the optimal matching, and we produce numerical data in support of this conjecture. We hope to produce a proof of this claim in future work.

Опорные точки полунепрерывных снизу функций относительно множества липшицевых вогнутых функций

Abstract: For the functions defined on normed vector spaces, we introduce a new notion of the LC -convexity that generalizes the classical notion of convex functions. A function is called to be LC -convex if it can be represented as the upper envelope of some subset of Lipschitz concave functions. It is proved that the function is LC -convex if and only if it is lower semicontinuous and, in addition, it is bounded from below by a Lipschitz function. As a generalization of a global subdifferential of a classically convex function, we introduce the set of LC -minorants supported to a function at a given point and the set of LC -support points of a function that are then used to derive a criterion for global minimum points and a necessary condition for global maximum points of nonsmooth functions. An important result of the article is to prove that for a LC - convex function, the set of LC -support points is dense in its effective domain. This result extends the well-known Brondsted– Rockafellar theorem on the existence of the sub-differential for classically convex lower semicontinuous functions to a wider class of lower semicontinuous functions and goes back to the one of the most important results of the classical convex analysis – the Bishop–Phelps theorem on the density of support points in the boundary of a closed convex set.