On nonsmooth V-invexity and vector variational-like inequalities in terms of the Michel–Penot subdifferentials

In this paper, we establish some results which exhibit an application for Michel–Penot subdifferential in nonsmooth vector optimization problems and vector variational-like inequalities. We formulate vector variational-like inequalities of Stampacchia and Minty type in terms of the Michel–Penot subdifferentials and use these variational-like inequalities as a tool to solve the vector optimization problem involving nonsmooth V-invex function. We also consider the corresponding weak versions of the vector variational-like inequalities and establish various results for the weak efficient solutions.     

Generalized Nash equilibrium problems for lower semi-continuous strategy maps

We consider a class of generalized Ky Fan inequalities (quasi-variational inequalities) in which the involved multi-valued mapping is lower semi-continuous. We present a relaxed version of the generalized Nash equilibrium problem involving strategy maps, which are only lower semi-continuous. This relaxed version may have no exact Nash equilibrium, but we prove that it has an ε-Nash equilibrium for every ε > 0. Easy examples of such problems show no existence of exact solutions, but existence of ε-solutions for every ε > 0. We give positive answers to two questions (in the compact case) raised in a recent paper of Cubiotti and Yao.     

Sensitivity analysis based heuristic algorithms for mathematical programs with variational inequality constraints

In this paper we consider heuristic algorithms for a special case of the generalized bilevel mathematical programming problem in which one of the levels is represented as a variational inequality problem. Such problems arise in network design and economic planning. We obtain derivative information needed to implement these algorithms for such bilevel problems from the theory of sensitivity analysis for variational inequalities. We provide computational results for several numerical examples.     

Contrast structures in singularly perturbed quasilinear reaction-diffusion-advection equations

We consider stationary solutions with internal transition layers (contrast structures) for a singularly perturbed elliptic equation that is referred to in applications as the stationary reaction-diffusion-advection equation. We construct an asymptotic approximation of arbitrary-order accuracy to such solutions and prove the existence theorem. We suggest an efficient algorithm for constructing an asymptotic approximation to the localization curve of the transition layer. To justify the constructed asymptotics, we use and develop, to this class of problems, an asymptotic method of differential inequalities, which also permits one to prove the Lyapunov stability of such stationary solutions.     

Homogenization of boundary hemivariational inequalities in linear elasticity

In this paper we consider a mathematical model describing static elastic contact problems with the Hooke constitutive law and subdifferential boundary conditions. We treat boundary hemivariational inequalities which are weak formulations of contact problems. We establish existence and uniqueness of solutions to hemivariational inequalities. Using the notion of H-convergence of elasticity tensors we investigate the limit behavior of the sequence of solutions to hemivariational inequalities.     

Existence of Solutions of Systems of Generalized Implicit Vector Variational Inequalities

We consider five different types of systems of generalized vector variational inequalities and derive relationships among them. We introduce the concept of pseudomonotonicity for a family of multivalued maps and prove the existence of weak solutions of these problems under these pseudomonotonicity assumptions in the setting of Hausdorff topological vector spaces as well as real Banach spaces. We also establish the existence of a strong solution of our problems under lower semicontinuity for a family of multivalued maps involved in the formulation of the problems. By using a nonlinear scalar function, we introduce gap functions for our problems by which we can solve systems of generalized vector variational inequalities using optimization techniques. The first two authors were supported by SABIC and Fast Track Research Grants SAB-2006-05. They are grateful to the Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia for providing excellent research facilities.     

Stopping Problems of Certain Multiplicative Functionals and Optimal Investment with Transaction Costs

Optimal stopping and impulse control problems with certain multiplicative functionals are considered. The stopping problems are solved by showing the unique existence of the solutions of relevant variational inequalities. However, since functions defining the multiplicative costs change the signs, some difficulties arise in solving the variational inequalities. Through gauge transformation we rewrite the variational inequalities in different forms with the obstacles which grow exponentially fast but with positive killing rates. Through the analysis of such variational inequalities we construct optimal stopping times for the problems. Then optimal strategies for impulse control problems on the infinite time horizon with multiplicative cost functionals are constructed from the solutions of the risk-sensitive variational inequalities of "ergodic type" as well. Application to optimal investment with fixed ratio transaction costs is also considered.     

Continuity of solutions for parametric variational inequalities in Banach space

We consider here a type of pseudo-monotone parametric variational inequalities on a class of Banach spaces and show that such problems admit continuous (with respect to the parameter) solutions, as long as generic existence and uniqueness conditions for these solutions are satisfied. In particular, we show that such results are valid on a class of Banach spaces whenever we deal with strong pseudo-monotonicity, while others are valid in Hilbert spaces, whenever strict monotonicity is present. We also provide examples to illustrate the new results.     

Impulsive Boundary Value Problems for Two Classes of Fractional Differential Equation with Two Different Caputo Fractional Derivatives

We consider the impulsive boundary value problems for two classes of fractional differential equations with two different Caputo fractional derivatives and generalized boundary value conditions. Natural formulae of a solution for these problems are introduced, which can be regarded as a novelty item. Some sufficient conditions for existence and uniqueness of the solutions to this nonlinear equations are established by applying well-known Banach’s contraction mapping principle, Laplace transforms and some skills of inequalities. Finally, an example is given to illustrate the effectiveness of our results.     

Solving nonlinear multi-objective optimization problems with fuzzy relation inequality constraints regarding Archimedean triangular norm compositions

We propose an approach to solve a nonlinear multi-objective problem subject to fuzzy relation inequalities with max-Archimedean-t-norm composition by a genetic algorithm. The additive generator of Archimedean t-norms is utilized to reform the existent genetic algorithm to solve the constrained nonlinear multi-objective optimization problems. We consider thoroughly the feasible set of the fuzzy relation inequality systems in three possible cases, namely “≤”, “≥” and the combination of them. In general, their feasible sets are nonconvex which are completely determined by one vector as their maximum solution and a finite number of minimal solutions. The maximum and minimal solutions are formulated by using the additive generator. Additionally, we present some conditions for each case under which the problem can be reduced. Finally, each reduced problem is solved by the genetic algorithm and the efficiency of the proposed method is shown by some numerical examples.     

EXISTENCE AND DUALITY OF IMPLICIT VECTOR VARIATIONAL PROBLEMS*

In this paper, we consider implicit vector variational problems which contain vector equilibrium problems and vector variational inequalities as special cases. The existence of solutions of implicit vector variational problems and vector equilibrium problems have been established. As a special case, we derive some existence results for a solution of vector variational inequalities. We also study the duality of implicit vector variational problems and discuss the relationship between solutions of dual and primal problems. Our results on duality contains known results in the literature as special cases.

     

Nonexistence theorem except the out-of-phase and in-phase solutions in the coupled van der Pol equation system

We consider a coupled van der Pol equation system. Our coupled system consists of two van der Pol equations that are connected with each other by linear terms. We assume that two distinctive solutions (out-of-phase and in-phase solutions) exist in the dynamical system of coupled equations and give answers to some problems.     

Karush-Kuhn-Tucker systems: regularity conditions,error bounds and a class of Newton-type methods

 We consider optimality systems of Karush-Kuhn-Tucker (KKT) type, which arise, for example, as primal-dual conditions characterizing solutions of optimization problems or variational inequalities. In particular, we discuss error bounds and Newton-type methods for such systems. An exhaustive comparison of various regularity conditions which arise in this context is given. We obtain a new error bound under an assumption which we show to be strictly weaker than assumptions previously used for KKT systems, such as quasi-regularity or semistability (equivalently, the R ₀-property). Error bounds are useful, among other things, for identifying active constraints and developing efficient local algorithms. We propose a family of local Newton-type algorithms. This family contains some known active-set Newton methods, as well as some new methods. Regularity conditions required for local superlinear convergence compare favorably with convergence conditions of nonsmooth Newton methods and sequential quadratic programming methods. Received: December 10, 2001 / Accepted: July 28, 2002 Published online: February 14, 2003 Key words. KKT system – regularity – error bound – active constraints – Newton method Mathematics Subject Classification (1991): 90C30, 65K05     

Studies on anti‐periodic boundary value problems for two classes of special second order impulsive differential equations

We consider anti‐periodic boundary value problems for two classes of special second order impulsive differential equations. On the basis of several important impulsive differential inequalities, by using the monotone iterative technique coupled with lower and upper solutions, we obtain sufficient conditions to guarantee the existence and uniqueness of solutions for such problems. Further, we give two examples to illustrate our conclusions. Copyright © 2013 John Wiley & Sons, Ltd.     

On Approximately Star-Shaped Functions and Approximate Vector Variational Inequalities

In this paper, we consider a vector optimization problem involving approximately star-shaped functions. We formulate approximate vector variational inequalities in terms of Fréchet subdifferentials and solve the vector optimization problem. Under the assumptions of approximately straight functions, we establish necessary and sufficient conditions for a solution of approximate vector variational inequality to be an approximate efficient solution of the vector optimization problem. We also consider the corresponding weak versions of the approximate vector variational inequalities and establish various results for approximate weak efficient solutions.     

On formulations of the stochastic uncapacitated lot-sizing problem

We consider two formulations of a stochastic uncapacitated lot-sizing problem. We show that by adding (?,S) inequalities to the one with the smaller number of variables, both formulations give the same LP bound. Then we show that for two-period problems, adding another class of inequalities gives the convex hull of integral solutions.     

Finite element discretization of parabolic-elliptic interface problems

We consider mixed boundary value problems for partial differential equations which are parabolic in one part of the underlying domain and elliptic in another part. Such problems arise in eddy current simulations in electromagnetics. A fully discrete scheme for the numerical solution of such problems is presented. Particularly, adaptivity in space and time is discussed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Teachers’ metacognitive and heuristic approaches to word problem solving: analysis and impact on students’ beliefs and performance

We conducted a 7-month video-based study in two sixth-grade classrooms focusing on teachers’ metacognitive and heuristic approaches to problem solving. All problem-solving lessons were analysed regarding the extent to which teachers implemented a metacognitive model and addressed a set of eight heuristics. We observed clear differences between both teachers’ instructional approaches. Besides, we examined teachers’ and students’ beliefs about the degree to which metacognitive and heuristic skills were addressed in their classrooms and observed that participants’ beliefs were overall in line with our observations of teachers’ instructional approaches. In addition, we investigated how students’ problem-solving skills developed as a result of teachers’ instructional approaches. A positive relationship between students’ spontaneous application of heuristics to solve non-routine word problems and teachers’ references to these skills in their problem-solving lessons was found. However, this increase in the application of heuristics did not result in students’ better performance on these non-routine word problems.     

A symmetry problem from probability

We examine solutions of two related boundary value problems for smooth domains in Euclidean space which arise from variational problems in probability. We show that the existence of solutions to each problem implies that the domain is a sphere.