A Mann type iterative scheme for variational inequalities in noncompact subsets of Banach spaces

In this paper, we study the existence of solutions and approximation of the solutions by using a Mann type iterative scheme for variational inequalities in noncompact subsets of Banach spaces. The results presented in this paper generalize the corresponding results of J. Li [J. Li, On the existence of solutions of variational inequalities in Banach spaces, J. Math. Anal. Appl. 295 (2004) 115-126].     

Variational Inequalities for Leavable Bounded-Velocity Control

We study the variational inequality associated with a bounded-velocity control problem when discretionary stopping is allowed. We establish the existence of a strong solution by using the viscosity solution techniques. The optimal policy is shown to exist from the optimality conditions in the variational inequality.     

Robust Optimal Stopping-Time Control for Nonlinear Systems

Abstract. We formulate a robust optimal stopping-time problem for a state-space system and give the connection between various notions of lower value function for the associated games (and storage function for the associated dissipative system) with solutions of the appropriate variational inequality (VI) (the analogue of the Hamilton—Jacobi—Bellman—Isaacs equation for this setting). We show that the stopping-time rule can be obtained by solving the VI in the viscosity sense and a positive definite supersolution of the VI can be used for stability analysis.     

BECOOL: Ballooning eigensolver with COOL finite elements

An incompressible variational ideal ballooning mode equation is discretized with the COOL finite element discretization scheme using basis functions composed of variable order Legendre polynomials. This reduces the second order ordinary differential equation to a special block pentadiagonal matrix equation that is solved using an inverse vector iteration method. A benchmark test of BECOOL (Ballooning Eigensolver using COOL finite elements) with second order Legendre polynomials recovers precisely the eigenvalues computed by the VVBAL shooting code. Timing runs reveal the need to determine an optimal lower order case. Eigenvalue convergence runs show that cubic Legendre polynomials construct the optimal ballooning mode equation for intensive computations.     

Epidifferentiability of the map of infima in Hilbert spaces

This article deals with derivatives for set-valued maps that take values in ordered vector spaces, in particular it concerns about the relationship between the epiderivatives of a set-valued map and its associated map of infima. When the image space is a real separable Hilbert space ordered by an orthonormal basis, by using a variational technique based on a decoupling of the ordering cone into half-spaces, we show that both epiderivatives coincide under certain hypothesis of compactness and stability. Furthermore we obtain some computation formulas for these derivatives in terms of associated scalar set-valued maps.     

Convergence of solutions of kinetic variational inequalities in the rate-independent quasi-static limit

This paper discusses the convergence of kinetic variational inequalities to rate-independent quasi-static variational inequalities. Mathematical formulations as well as existence and uniqueness results for kinetic and rate-independent quasi-static problems are provided. Sharp a priori estimates for the kinetic problem are derived that imply that the kinetic solutions converge to the rate-independent ones, when the size of initial perturbations and the rate of application of the forces tend to 0. An application to three-dimensional elastic-plastic systems with hardening is given.     

Vector-valued perturbed equilibrium problems

The goal of this paper is to prove some general vector-valued perturbed equilibrium principles and some existence results of vector equilibrium points for bifunctions satisfying a new natural notion of lower semi-continuity. We obtain these results by going through a new concept of approximative equilibrium point.     

A note on finite termination of iterative algorithms in mathematical programming

Two well-known results on the unified treatment of finite termination of a class of algorithms for solving convex programming problems and for solving variational inequality problems are reconsidered. In particular, some of the underlying assumptions employed in the existing literature are shown to be redundant.     

Isomerization kinetics of small hydrocarbons in confinement

Chemical reactions are often carried out in nano-structured materials, which can enhance reactions due to their large specific surface area, their interactions with the reacting mixture and confinement effects. In this work, we present a systematic study of the effect that the geometrical restrictions imposed by the pore walls can have on reactions that involve a three dimensional rearrangement of the atoms in a molecule. In particular, we consider the isomerization of three 4-membered hydrocarbons—n-butane, 1-butene and 1,3-butadiene confined in carbon nanopores of slit geometry. Our results illustrate the fact that, in the molecular sieving limit, the reaction rates change as the double exponential of the pore size (Santiso et al., in J. Chem. Phys., 2007a, submitted), and therefore the transition rates in nanopores can be many orders of magnitude different from the corresponding bulk values. These results can be used as a guideline for the molecular-level design of improved catalytic materials.     

Gâteaux differentiability of the dual gap function of a variational inequality

In terms of the mapping involved in a variational inequality, we characterize the Gâteaux differentiability of the dual gap function G and present several sufficient conditions for its directional derivative expression, including one weaker than that of Danskin [J.M. Danskin, The theory of max–min, with applications, SIAM Journal on Applied Mathematics 14 (1966) 641–664]. When the solution set of a variational inequality problem is contained in that of its dual problem, the Gâteaux differentiability of G on the latter turns out to be equivalent to the conditions appearing in the authors’ recent results about the weakly sharp solutions of the variational inequality problem.