mixed quasi variational inequalities in this paper we use the resolvent operator and auxiliary principle techniques to suggest and analyze several iterative algorithms for solving mixed quasi variational inequalities and related problems we study the convergence criteria of these algorithms under mild conditions we also study the global stability and existence of a unique solution of these quasi variational inequalities by using the dynamical systems approach our results represent refinement and improvement of the previously known results for variational inequalities

on an iterative method for variational inequalities a number of numerical solutions are presented as examples of a new iterative method for variational inequalities the iterative method is based on the reduction of variational inequalities to the wiener hopf equations for obstacle problems the convergence is guaranteed inw 1 p spaces forp 2 the examples presented are one and two dimensional obstacle problems in cases when the greens function is known but the method is very general

perturbed proximal point algorithms for generalized quasivariational inclusions abstract in this paper we study a class of generalized quasivariational inclusions by using the properties of the resolvent operator associated with a maximal monotone mapping in hilbert space we have established an existence theorem of solutions for generalized quasivariational inclusions suggesting a new iterative algorithm and a perturbed proximal point algorithm for finding approximate solutions which strongly converge to the exact solution of the generalized quasivariational inclusions as special cases some known results in this field are also discussed

a modified forward backward splitting method for maximal monotone mappings we consider the forward backward splitting method for finding a zero of the sum of two maximal monotone mappings this method is known to converge when the inverse of the forward mapping is strongly monotone we propose a modification to this method in the spirit of the extragradient method for monotone variational inequalities under which the method converges assuming only the forward mapping is lipschitz continuous on some closed convex subset of its domain the modification entails an additional forward step and a projection step at each iteration applications of the modified method to decomposition in convex programming and monotone variational inequalities are discussed

on a generalization of a normal map and equation the class of normal maps was recently investigated by robinson and ralph in connection with the study of a variational inequality defined on a polyhedral set in this paper a generalization of such a map is considered and the associated generalized normal equation is studied the latter provides a unified formulation of several generalized variational inequality and complementarity problems using degree theory some sufficient conditions for the existence of a zero of a generalized normal map are established and the stability of a generalized normal equation at a solution is analyzed specializations of the results to various applications are discussed

cubic splines method for a system of third order boundary value problems we use uniform cubic polynomial splines to develop a numerical method for computing approximations to the solution of a system of third order boundary value problem associated with odd order obstacle problems such problems arise in physical oceanography and can be studied in the framework of variational inequality theory the convergence analysis of the new method is studied and an upper bound for the error is derived a numerical example is given to illustrate the efficiency of the new method

pattern formation in growing sandpiles with multiple sources or sinks adding sand grains at a single site in the abelian sandpile models produces beautiful but complex patterns we study the effect of sink sites on such patterns sinks change the scaling of the diameter of the pattern with the number n of sand grains added for example in two dimensions in the presence of a sink site the diameter of the pattern grows as sqrt n log n for large n whereas it grows as sqrt n if there are no sink sites in the presence of a line of sink sites this rate reduces to n 1 3 we determine the growth rates for various sink geometries along with the case when there are two lines of sink sites forming a wedge and generalizations to higher dimensions we characterize the asymptotic pattern in the large n limit for one such case the two dimensional f lattice with a single source adjacent to a line of sink sites the characterization is done in terms of the positions of different spatial features in the pattern for this lattice we also provide an exact characterization of the pattern with two sources when the line joining them is along one of the axes of the lattice

sensitivity analysis framework for general quasi variational inclusions abstract it is well known that the implicit resolvent equations are equivalent to the quasivariational inclusions we use this alternative equivalent formulation to study the sensitivity of the quasivariational inclusions without assuming the differentiability of the given data since the quasi variational inclusions include classical variational inequalities quasi mixed variational inequalities and complementarity problems as special cases results obtained in this paper continue to hold for these problems in fact our results can be considered as a significant extension of previously known results

co coercivity and its role in the convergence of iterative schemes for solving variational inequalities this paper stresses the role of co coercivity and related notions in the convergence of iterative schemes for solving monotone but not necessarily strongly or even strictly monotone variational inequalities the analysis will be conducted within the conceptual framework of the auxiliary problem principle

implicit dynamical systems and quasi variational inequalities in this paper we propose and analyze some implicit projected dynamical systems associated with quasi variational inequalities by using the techniques of the projection and the wiener hopf equations we prove that the globally asymptotic stability of these dynamical systems requires only the pseudomonotonicity of the operator which is a weaker condition than monotonicity we also discuss some special cases which can be obtained from our main results the results obtained in this paper represent a significant improvement of the previously known results

implicit resolvent dynamical systems for quasi variational inclusions in this paper we suggest and analyze a class of implicit resolvent dynamical systems for quasi variational inclusions by using the resolvent operator technique we show that the trajectory of the solution of the implicit dynamical system converges globally exponentially to the unique solution of the quasi variational inclusions our results can be considered as a significant extension of the previously known results

tangent projection equations and general variational inequalities in this paper we establish the equivalence between the general variational inequalities and tangent projection equations this equivalence is used to discuss the local convergence analysis of a wide class of iterative methods for solving the general variational inequalities we show that some existing methods can identify the optimal face after finitely many iterations under the degenerate assumption

an iterative scheme for variational inequalities in this paper we introduce and study a general iterative scheme for the numerical solution of finite dimensional variational inequalities this iterative scheme not only contains as special cases the projection linear approximation and relaxation methods but also induces new algorithms then we show that under appropriate assumptions the proposed iterative scheme converges by establishing contraction estimates involving a sequence of norms in en induced by symmetric positive definite matrices gm thus in contrast to the above mentioned methods this technique allows the possibility of adjusting the norm at each step of the algorithm this flexibility will generally yield convergence under weaker assumptions

self adaptive projection algorithms for general variational inequalities in this paper we consider and analyze a new class of self adaptive projection algorithms for solving general variational inequalities by using the technique of updating the solution we prove that the convergence of these new methods only requires the pseudomonotonicity which is a weaker condition than monotonicity these new methods differ from the previously known splitting methods for solving variational inequalities and related complementarity problems proof of convergence is very simple as special cases we can obtain a number of four step forward backward splitting methods of noor for solving variational inequalities

normal maps inducted by linear transformations we study a certain piecewise linear manifold which we call the normal manifold associated with a polyhedral convex set and a family of continuous functions called normal maps that are induced on this manifold by continuous functions from rn to rn these normal maps occur frequently in optimization and equilibrium problems and the subclass of normal maps induced by linear transformations plays a key role our main result is that the normal map induced by a linear transformation is a lipschitzian homeomorphism if and only if the determinant of the map in each n cell of the normal manifold has the same nonzero sign

sensitivity analysis framework for variational inequalities in this paper a sensitivity analysis framework is developed for variational inequalities the perturbed solution to a parametric variational inequality problem is shown to be continuous and directionally differentiable under appropriate second order and regularity assumptions moreover this solution is once continuously differentiable if strict complementarity also holds

a new projection method for variational inequality problems we propose a new projection algorithm for solving the variational inequality problem where the underlying function is continuous and satisfies a certain generalized monotonicity assumption e g it can be pseudomonotone the method is simple and admits a nice geometric interpretation it consists of two steps first we construct an appropriate hyperplane which strictly separates the current iterate from the solutions of the problem this procedure requires a single projection onto the feasible set and employs an armijo type linesearch along a feasible direction then the next iterate is obtained as the projection of the current iterate onto the intersection of the feasible set with the halfspace containing the solution set thus in contrast with most other projection type methods only two projection operations per iteration are needed the method is shown to be globally convergent to a solution of the variational inequality problem under minimal assumptions preliminary computational experience is also reported

fixed point approach for complementarity problems abstract in this paper we use the fixed point technique to suggest a new unified and general algorithm for computing the approximate solution of a nonlinear complementarity problem of finding u such that u 0 tu a u 0 u tu a u 0 where t is a continuous mappping from r n into itself and a is a non linear transformation from r n into itself this algorithm includes many existing algorithms for complementarity problems as special cases convergence properties are also discussed and analyzed

a modified extragradient method for general monotone variational inequalities abstract in this paper we suggest and analyze a new iterative method for solving general monotone variational inequalities by using the updating technique the new method differs from the extragradient and other modified projection methods the new method is versatile and easy to implement our proof of convergence is very simple

modified resolvent splitting algorithms for general mixed variational inequalities in this paper we suggest and analyze a number of four step resolvent splitting algorithms for solving general mixed variational inequalities by using the updating technique of the solution the convergence of these new methods requires either monotonicity or pseudomonotonicity of the operator proof of convergence is very simple our new methods differ from the existing splitting methods for solving variational inequalities and complementarity problems the new results are versatile and are easy to implement

some developments in general variational inequalities general variational inequalities provide us with a unified natural novel and simple framework to study a wide class of equilibrium problems arising in pure and applied sciences in this paper we present a number of new and known numerical techniques for solving general variational inequalities using various techniques including projection wiener hopf equations updating the solution auxiliary principle inertial proximal penalty function dynamical system and well posedness we also consider the local and global uniqueness of the solution and sensitivity analysis of the general variational inequalities as well as the finite convergence of the projection type algorithms our proofs of convergence are very simple as compared with other methods our results present a significant improvement of previously known methods for solving variational inequalities and related optimization problems since the general variational inequalities include quasi variational inequalities and quasi implicit complementarity problems as special cases results presented here continue to hold for these problems several open problems have been suggested for further research in these areas

some aspects of variational inequalities abstract in this paper we provide an account of some of the fundamental aspects of variational inequalities with major emphasis on the theory of existence uniqueness computational properties various generalizations sensitivity analysis and their applications we also propose some open problems with sufficient information and references so that someone may attempt solution s in his her area of special interest we also include some new results which we have recently obtained

on the identification of active constraints nondegeneracy conditions that guarantee that the optimal active constraints are identified in a finite number of iterations are studied results of this type have only been established for a few algorithms and then under restrictive hypothesis the main result is a characterization of those algorithms that identify the optimal constraints in a finite number of iterations this result is obtained with a nondegeneracy assumption which is equivalent in the standard nonlinear programming problem to the assumption that there is a set of strictly complementary lagrange multipliers as an important consequence of the authors results the way that this characterization applies to gradient projection and sequential quadratic programming algorithms is shown

new extragradient type methods for general variational inequalities in this paper we consider and analyze a new class of extragradient type methods for solving general variational inequalities the modified methods converge for pseudomonotone operators which is weaker condition than monotonicity our proof of convergence is very simple as compared with other methods the proposed methods include several new and known methods as special cases our results present a significant improvement of previously known methods for solving variational inequalities and related optimization problems

strongly nonlinear generalized equations we study a class of generalized equations which includes as a special case various classes of quasi variational inequalities introduced by noor isac siddiqi and ansari such problems occur in mathematical programming and in various other branches of pure and applied mathematics we also propose two iterative schemes for the numerical solution of the problems considered

day to day dynamic network disequilibria and idealized traveler information systems in this paper we present tatonnement models for calculating static wardropian user equilibria on congested networks with fully general demand and cost structures we present both a qualitative analysis of stability and numerical studies which show that such an approach provides a reliable means for determining static user equilibria we also describe circumstances for which these models depict day to day adjustments from one realizable disequilibrium state to another and how these adjustment processes differ depending on the quality of the information being provided by abstract traveler information systems specifically we demonstrate that such dynamic adjustment processes settle down to equilibria both when information is complete and when it is incomplete

general strongly nonlinear variational inequalities abstract in this paper we develop iterative algorithms for finding approximate solutions for new classes of variational and quasivariational inequalities which include as a special case some known results in this field it is shown that the solutions of the iterative schemes converge to the exact solutions

sensitivity analysis for quasi variational inclusions in this paper we develop the sensitivity analysis for quasi variational inclusions by using the implicit resolvent equations technique without assuming the differentiability of the given data

sensitivity analysis in nonlinear programs and variational inequalities via continuous selections sufficient conditions for solution lipschitz continuity piecewise differentiability and directional differentiability are presented for parametric nonlinear programs and variational inequalities using the idea of continuous selections the gaps between the sufficient conditions obtained here and the weakest possible conditions for the corresponding conclusions are discussed and measured by known regularity conditions