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Published in Philip A. Laplante, Comprehensive Dictionary of Electrical Engineering, 2018
probabilistic metric space a generalization of the notion of metric spaces onto the uncertain systems by replacing a metric on a given set S by a distance distribution function F, and a triangle inequality by a generalized inequality defined by triangle function . A distance distribution functions between two elements p, q S is defined as a real function F pq whose value F pq (x) for any real number x is interpreted as the probability, the membership function, or the grade of membership (depending on the type of the uncertainty model) that the distance between p and q is less than x. The simplest distance distribution function is given by the unit step (Heaviside) function 1 as follows: F pq (x) = 1(x - d( p, q)) where d is a standard metric. Then a probabilistic metric space reduces to the standard metric space. More precisely, a probabilistic metric space (PMS) is defined as a triple (S, F, ) endowed
A global optimality result in probabilistic spaces using control function
Published in Optimization, 2021
P. Saha, S. Guria, Samir Kumar Bhandari, Binayak S. Choudhury
Probabilistic metric spaces are mathematical structures in which the distance between any two points is given by a probability distribution function rather than by a non-negative number as in the case of metric spaces. The concept of this space was introduced by Karl Menger in 1942 [1]. The theory of this structure was developed mainly after 1960 through the works of different mathematicians. A comprehensive account of this development is given in the book of Schweizer and Sklar [2] published in 1983. Due to the inherent flexibility of a probabilistic extension, the probabilistic distance has been defined in various ways within two different types of definitions of the probabilistic metric. The essential feature of all these definitions is the inherent uncertainty build within the geometry of the space itself. The concept of probabilistic metric has been studied in the contexts of other mathematical structures like that in the recent work of Berckmoes and Lowen [3] where it has been considered in the categorical settings. It is also important in applications due to the inherently probabilistic nature of its geometry. A recent example of such an application in a nuclear fusion related problem is in [4]. There are also several studies of mathematics which have been extended to probabilistic metric spaces. An instance of such extensions is [5] where a probabilistic fixed point result has been used to establish a basic result on probabilistic differential equations in such spaces. In particular fixed point theory has developed very widely in the structure of probabilistic metric spaces. The book of Hazdzic and Pap [6] provide us with a comprehensive account of this theory up to 2001. Some more recent references are [7–10].
L 0-convex compactness and its applications to random convex optimization and random variational inequalities
Published in Optimization, 2021
Tiexin Guo, Erxin Zhang, Yachao Wang, Mingzhi Wu
Recently, random functional analysis and its applications to conditional risk measures naturally lead us to study the problem of random convex optimization and random variational inequalities. To let the reader have a good understanding on this problem, we first give a brief introduction of the closely related theoretical and financial backgrounds. Random functional analysis is functional analysis based on random metric spaces, random normed modules, random inner product modules and random locally convex modules, which are a random generalization of ordinary metric spaces, normed spaces, inner product spaces and locally convex spaces, respectively. The history of random functional analysis will unavoidably dates back to the theory of probabilistic metric spaces, which was initiated by K. Menger in 1942 and subsequently founded by B. Schweizer and A. Sklar, see [7] for details. The theory of probabilistic metric spaces is centred at the study of probabilistic metric spaces and probabilistic normed spaces, whose main idea is to use probability distribution functions to describe the probabilistic metric between two points or the probabilistic norm of a vector. Following the tradition from probability theory, random metric spaces and random normed spaces were presented in the course of the development of the theory of probabilistic metric spaces, where the random metric between two points or the random norm of a vector is described by a non-negative random variable, see [7, Chapters 9 and 15]. But random normed spaces had not obtained a substantial development up to 1989 since they are often endowed with the -topology, which are not locally convex in general and thus the traditional theory of conjugate spaces universally fails. The first substantial advance came in [8], where Guo introduced the notion of an almost surely bounded random linear functional on random normed spaces and proved the corresponding Hahn–Banach theorem, which leads to the study of random conjugate spaces. Further, Guo introduced the notions of random normed modules (a special class of random normed spaces) and random inner product modules in [9–11] (here, we also mention the work [12] of Haydon et al., who independently introduced the notion of random normed modules over the real number field in the name of randomly normed -modules, as a tool for the study of ultrapowers of Lebesgue–Bochner function spaces), which leads to a series of deep developments of random conjugate spaces [13–17] (here, we also mention the famous work [18] of Hansen and Richard, who independently proved the Riesz representation theorem of random conjugate spaces for a class of special complete random inner product modules–conditional Hilbert spaces, and gave its applications in representing the equilibrium price). As a random generalization of a locally convex space, random locally convex modules were introduced by Guo in [19] and deeply developed in [20–22]. It should be pointed out that random functional analysis was developed under the -topology before 2009.