Random Variational Inequalities Research Articles

On L0-convex compactness in random locally convex modules

For the study of some typical problems in finance and economics, Žitković introduced convex compactness and gave many remarkable applications. Recently, motivated by random convex optimization and random variational inequalities, Guo, et al. introduced L0-convex compactness, developed the related theory of L0-convex compactness in random normed modules and further applied it to backward stochastic equations. In this paper, we extensively study L0-convexly compact sets in random locally convex modules so that a series of fundamental results are obtained. First, we show that every L0-convexly compact set is complete (hence is also closed and has the countable concatenation property). Then, we prove that any L0-convexly compact set is linearly homeomorphic to a weakly compact subset of some locally convex space, and simultaneously establish the equivalence between L0-convex compactness and convex compactness for a closed L0-convex set. Finally, we establish Tychonoff type, James type and Banach-Alaoglu type theorems for L0-convex compactness, respectively.

Uniqueness and Hyers-Ulam Stability of Random Differential Variational Inequalities with Nonlocal Boundary Conditions

In this paper, we consider a new class of random differential variational inequalities (RDVIs) with nonlocal boundary conditions in Hilbert spaces. We apply the projection operator, Gronwall’s lemma and a result on the existence of a random differential inclusion to establish uniqueness and Hyers–Ulam stability results of the abstract inequality. As an illustrative application, the linear random differential complementarity systems and a price control model are investigated.

L 0-convex compactness and its applications to random convex optimization and random variational inequalities

ABSTRACT First, this paper introduces the notion of -convex compactness for a special class of closed convex subsets–closed -convex subsets of a Hausdorff topological module over the topological algebra , where is the algebra of equivalence classes of random variables from a probability space to the scalar field K of real numbers or complex numbers, endowed with the topology of convergence in probability. Then, this paper continues to develop the theory of -convex compactness by establishing various kinds of characterization theorems on -convex compactness for closed -convex subsets of a class of important topological modules – complete random normed modules, in particular, we make full use of the theory of random conjugate spaces to establish the characterization theorem of James type on -convex compactness for a closed -convex subset of a complete random normed module, which also surprisingly implies that our notion of -convex compactness coincides with Gordan Žitković's notion of convex compactness in the context of a closed -convex subset of a complete random normed module. As the first application of our results, we give a fundamental theorem on random convex optimization (or, -convex optimization), which includes Hansen and Richard's famous result as a special case. As the second application, we give an existence theorem of solutions of random variational inequalities, which generalizes H. Brezis' classical result from a reflexive Banach space to a random reflexive complete random normed module. It should be emphasized that a new method, namely the -convex compactness method, is presented for the second application since the usual weak compactness method is no longer applicable in the present case. Besides, our fundamental theorem on random convex optimization can be also applied in the study of optimization problems of conditional convex risk measures, which will be given in our future papers.

Stochastic approximation results for random variational inequality problem with monotone operator in Hilbert space and its application to optimal control

The main aim of this article is to show that the solution of a stochastic variational inequality problem obtained from a procedure based on extragradient-like scheme converges in quadratic mean to a unique random fixed point in Hilbert space. Furthermore, the result is applied to optimal control problem defined for a stochastic differential equation with a unit point delay in the state variable. The results generalize, extend, and unify many established results in the literature on deterministic variational inequality problems.

A migration equilibrium model with uncertain data and movement costs

In this paper, we consider a variational inequality model of migration in which populations move according to the utility functions corresponding to various locations and movement costs. In contrast to previously studied models based on variational inequalities, we take into account the possibility that some of the data in the model are not deterministic but, instead, are known through their probability distributions. Our theoretical framework is that of random variational inequalities in Lebesgue spaces.

FUZZY GENERAL NONLINEAR ORDERED RANDOM VARIATIONAL INEQUALITIES IN ORDERED BANACH SPACES

The main object of this work to introduced and studied a new class of fuzzy general nonlinear ordered random variational inequalities in ordered Banach spaces. By using the random B-restricted accretive mapping with measurable mappings <TEX>${\alpha},{\alpha}^{\prime}:{\Omega}{\rightarrow}(0,1)$</TEX>, an existence of random solutions for this class of fuzzy general nonlinear ordered random variational inequality (equation) with fuzzy mappings is established, a random approximation algorithm is suggested for fuzzy mappings, and the relation between the first value <TEX>$x_0(t)$</TEX> and the random solutions of fuzzy general nonlinear ordered random variational inequality is discussed.

Random Variational Inequalities and the Random Traffic Equilibrium Problem

In the paper we study, in a Hilbert space setting, a general random traffic equilibrium problem and characterize the random Wardrop equilibrium distribution by means of a random variational inequality. Some existence results are provided and the associated Lagrange function is studied. Some examples illustrate the random variational inequality and the counterintuitive behaviour of the traffic equilibrium problem is focused.

Some equilibrium problems under uncertainty and random variational inequalities

In this paper we describe some nonlinear equilibrium problems under uncertainty arising from economics and operations research. In particular we treat Wardrop equilibria in traffic networks. We show how the theory of monotone random variational inequalities, where random variables occur both in the operator and the constraint set, can be applied to model these problems. Therefore in this contribution we introduce the topic of random variational inequalities and present some of our recent results in this field. In particular, we treat the more structured case where a finite Karhunen-Loeve expansion leads to a separation of the random and the deterministic variables. Here we describe a norm convergent approximation procedure based on averaging and truncation. We illustrate this procedure by means of some small sized numerical examples.

A variational inequality model of the spatial price network problem with uncertain data

We consider the multicriteria spatial price network equilibrium model in which consumers are allowed to weight both the transportation cost and the transportation time associated with the shipment of a given commodity. In particular, we assume that weights are not deterministic but subject to random fluctuations. Therefore, we formulate the problem as a random variational inequality and prove that under natural assumptions the associated random operator satisfies a uniform monotonicity property. The subsequent analysis is carried out in the framework of the theory of random variational inequalities.

VISCOSITY APPROXIMATION METHODS OF RANDOM FIXED POINT SOLUTIONS AND RANDOM VARIATIONAL INEQUALITIES IN HILBERT SPACES

In this paper, we construct random iterative processes for nonexpansive random operators and study necessary conditions for these processes. It is shown that these random iterative processes converge to random fixed points of nonexpansive random operators and solve some random variational inequalities. We also proved that an implicit random iterative process converges to the random fixed point and solves these random variational inequalities. Our results can be viewed as a refinement and improvement of the previously known results for variational inequality theory and also give generalization stochastic version of some results of Xu [23].

On a Class of Random Variational Inequalities on Random Sets

We study a class of random variational inequalities on random sets and give measurability, existence, and uniqueness results in a Hilbert space setting. In the special case where the random and the deterministic variables are separated, we present a discretization technique based on averaging and truncation, prove a Mosco convergence result for the feasible random set, and establish norm convergence of the approximation procedure.

Random Nonlinear Variational Inequalities for Mappings of Monotone Type in Banach Spaces

Some new concepts of β-semimonotone, and weakly semiaccretive random mappings are introduced and then some nonlinear variational inequalities for such random mappings are solved.

Random equilibrium problems on networks

We apply the theory of random variational inequalities to study a class of random equilibrium problems on networks. By means of two classical test problems we treat the case of random demand and random cost and compute mean values and variances for two special probability distributions.

Random Variational Inequality and Complementarity Problem

In this paper we have defined Random complementarity problem and obtained some existence results for Random variational inequality and complementarity problem in reflexive Banach space and Hilbert space.

Some Deterministic and Random Vector Equilibrium Problems

In this paper, some generalized concepts about semicontinuity and convexity for vector-valued bifunctions are introduced. A new existence theorem for vector equilibrium problems is proved. Further, a random version of this result is considered. Applications to cone saddle points, random vector optimization, random vector variational inequalities, and random vector best approximation problems are finally given.

A class of random variational inequalities and simple random unilateral boundary value problems - existence, discretization, finite element approximation

The paper deals with a class of random variational inequalities and simple random elliptic boundary value problems with unilateral conditions. Here randomness enters in the coefficient of the elliptic operator and in the right hand side of the p.d.e. In addition to existence and uniqueness results a theory of combined probabilistic deterministic discretization is developped that includes nonconforming approxima¬tion of unilateral constraints. Without any regularity assumptions on the solution, norm convergence of the full approximation process is established. The theory is applied to a Helmholtz like elliptic equation with Signorini boundary conditions as a simple model problem, where Galerkin discretization is realized by finite element approximation

Nonlinear Random Variational Inequality Problem of Mathematical Programming in Banach Spaces

A random version of Browder’s variational inequality is presented by introducing a random operator. The existence of a random solution is established by considering the random operator to be completely continuous. 1991 Mathematics subject classification (AMS)

General algorithm of random solutions for random nonlinear variational inequalities in banach Spaces*

In this paper, we study a class of random nonlinear variational inequalities in Banach spaces. By applying a random minimax inequahty obtained by Tarafdar and Yuan, some existence uniqueness theorems of random solutions for the random nonhnear variational inequalities are proved. Next, by applying the random auxiliary problem technique, we suggest an innovative iterative algorithm to compute the random approximate solutions of the random nonlinear variational inequahty. Finally, the convergence criteria is also discussed

Minimization of stochastic functionals and random variational inequalities

We examine the problem of minimizing a stochastic functional on a closed convex subset of a Hilbert space and provide sufficient conditions for the existence of a solution. As a departure from the literature, none of these conditions is imposed on all sample paths. To argue for the need to avoid such conditions, we prove that the Ito integral on L 2(0, 1) possesses neither l.s.c. nor convex sample paths. The theory is utilized to verify the existence of a minimum for a natural extension of a functional determined by randomly perturbing the energy in the 1 dimensional obstacle problem. In the second part of the paper we establish a link between random variational inequalities and minimization of stochastic functional, the link is employed to establish the existence of solutions for some cases of the former

On Random Variational Inequalities

In this paper, we consider a random variational inequality. An existence theorem for a unique solution of a random variational inequality is proved, which includes the results of Noor, Lions, and Stampacchia. Several special cases, which can be obtained from our results, are also discussed.