numerical method for general mixed quasi variational inequalities abstract in this paper we use the resolvent operator to suggest and analyze a new numerical method for solving general mixed quasi variational inequalities coupled with a new direction and a new step size k under certain conditions the global convergence of the proposed method is proved some preliminary computational results are given to illustrate the efficiency of the proposed method since the general mixed quasi variational inequalities include general variational inequalities quasi variational inequalities and nonlinear implicit complementarity problems as special cases results proved in this paper continue to hold for these problems results proved in this paper may be viewed as a refinement of the previous known results

the forward backward forward method from continuous and discrete perspective for pseudo monotone variational inequalities in hilbert spaces abstract tseng s forward backward forward algorithm is a valuable alternative for korpelevich s extragradient method when solving variational inequalities over a convex and closed set governed by monotone and lipschitz continuous operators as it requires in every step only one projection operation however it is well known that korpelevich s method converges and can therefore be used also for solving variational inequalities governed by pseudo monotone and lipschitz continuous operators in this paper we first associate to a pseudo monotone variational inequality a forward backward forward dynamical system and carry out an asymptotic analysis for the generated trajectories the explicit time discretization of this system results into tseng s forward backward forward algorithm with relaxation parameters which we prove to converge also when it is applied to pseudo monotone variational inequalities in addition we show that linear convergence is guaranteed under strong pseudo monotonicity numerical experiments are carried out for pseudo monotone variational inequalities over polyhedral sets and fractional programming problems

modified conditions for minimum of compliance we consider the problem of finding a design with minimum compliance using a simp material in particular we present a method to obtain competitive designs that do not include composite like material distributions the idea of this method is to modify a necessary condition for the optimal design of an object we prove that the modified condition defines a design and present numerical solutions

existence and regularity of solutions to nonlinear degenerate evolutionary variational inequalities with applications to dynamic network equilibrium problems abstract the aim of the paper is to consider a class of nonlinear degenerate evolutionary variational inequalities and to establish existence and regularity results for this class to obtain these results we make use of a regularization procedure and of the concept of mosco s convergence moreover taking into account the regularity result we present a method to solve a wide class of nonlinear degenerate evolutionary variational inequalities

solving linear variational inequality problems by a self adaptive projection method we propose a self adaptive projection method for solving linear variational inequality problems and show its global convergence under mild conditions some numerical results are also addressed which indicate that the method is quite robust and efficient

a projection method for convex constrained monotone nonlinear equations with applications in this paper we present a projection method to solve monotone nonlinear equations with convex constraints this method can be viewed as an extension of cg descent method which is one of the most effective conjugate gradient methods for solving unconstrained optimization problems because of derivative free and low storage the proposed method can be used to solve large scale nonsmooth monotone nonlinear equations its global convergence is established under some appropriate conditions preliminary numerical results show that the proposed method is effective and promising moreover we also successfully use the proposed method to solve the sparse signal reconstruction in compressive sensing

finding solutions of implicit complementarity problems by isotonicity of the metric projection abstract isac and nemeth g isac and a b nemeth projection method isotone projection cones and the complementarity problem j math anal app 153 258 275 1990 proved that solving a coincidence point equation fixed point problem in turn solves the corresponding implicit complementarity problem nonlinear complementarity problem and they exploited the isotonicity of the metric projection onto isotone projection cones to solve implicit complementarity problems nonlinear complementarity problems defined by these cones in this paper the notion of isotone projection cones is employed and an iterative algorithm is presented in connection with an implicit complementarity problem on isotone projection cones it is proved that if the sequence generated through the defined algorithm is convergent then its limit is a solution of the coincidence point equation and thus solves the implicit complementarity problem sufficient conditions are given for this sequence to be convergent for implicit complementarity problems defined by isotone projection cones the question of finding nonzero solutions of these problems is also studied

recursive hybrid algorithm for non linear system identification using radial basis function networks recursive identification of non linear systems is investigated using radial basis function networks a novel approach is adopted which employs a hybrid clustering and least squares algorithm the recursive clustering algorithm adjusts the centres of the radial basis function network while the recursive least squares algorithm estimates the connection weights of the network because these two recursive learning rules are both linear rapid convergence is guaranteed and this hybrid algorithm significantly enhances the real time or adaptive capability of radial basis function models the application to simulated real data are included to demonstrate the effectiveness of this hybrid approach

asymmetric variational inequality problems over product sets applications and iterative methods in this paper we i describe how several equilibrium problems can be uniformly modelled by a finite dimensional asymmetric variational inequality defined over a cartesian product of sets and ii investigate the local and global convergence of various iterative methods for solving such a variational inequality problem because of the special cartesian product structure these iterative methods decompose the original variational inequality problem into a sequence of simpler variational inequality subproblems in lower dimensions the resulting decomposition schemes often have a natural interpretation as some adjustment processes

projected reflected gradient methods for monotone variational inequalities this paper is concerned with some new projection methods for solving variational inequality problems with monotone and lipschitz continuous mapping in hilbert space first we propose the projected reflected gradient algorithm with a constant stepsize it is similar to the projected gradient method namely the method requires only one projection onto the feasible set and only one value of the mapping per iteration this distinguishes our method from most other projection type methods for variational inequalities with monotone mapping also we prove that it has r linear rate of convergence under the strong monotonicity assumption the usual drawback of algorithms with constant stepsize is the requirement to know the lipschitz constant of the mapping to avoid this we modify our first algorithm so that the algorithm needs at most two projections per iteration in fact our computational experience shows that such cases with two projections are very rare this scheme at least theoretically seems to be ve

numerical comparison between prediction correction methods for general variational inequalities in this paper we propose two prediction correction methods for solving general variational inequalities the first method makes use of a descent direction to produce the new iterate while the second method generates the corrector by the improved extragradient method under certain conditions the global convergence of both methods is proved it is proved theoretically that the lower bound of the progress obtained by the second method is greater than that by the first one numerical results are given to verify and compare the numerical efficiency of these two predictor corrector methods

solving nonlinear complementarity problems by isotonicity of the metric projection abstract the main motivation for introducing the notion of isotone projection cones was to solve nonlinear complementarity problems the notion of isotone projection cones is introduced by this paper in a similar fashion iterative methods for finding solutions of complementarity problems on isotone projection cones are presented the problem of finding nonzero solutions of these problems is also considered

a new inertial type hybrid projection proximal algorithm for monotone inclusions this paper investigates an enhanced proximal algorithm with interesting practical features and convergence properties for solving non smooth convex minimization problems or approximating zeroes of maximal monotone operators in hilbert spaces the considered algorithm involves a recent inertial type extrapolation technique the use of enlargement of operators and also a recently proposed hybrid strategy which combines inexact computation of the proximal iteration with a projection compared to other existing related methods the resulting algorithm inherits the good convergence properties of the inertial type extrapolation and the relaxed projection strategy it also inherits the relative error tolerance of the hybrid proximal projection method as a special result an update of inexact newton proximal method is derived and global convergence results are established

forcing strong convergence of korpelevich s method in banach spaces with its applications in game theory abstract we develop a variant of korpelevich s method for solving variational inequality problems with pseudomonotone operators in banach spaces we establish the strong convergence of the sequence generated by the method under reasonable assumptions on the problem data finally we justify the motivation of our theory by including compelling examples of infinite dimensional variational inequality problems for which our method is applicable

convergence property of gradient type methods with non monotone line search in the presence of perturbations in this paper two new kinds of methods which are called gradient type method and hybrid projection method with perturbations are proposed and non monotone line search technique is employed at the same time global convergence of these methods is proved only in the case where the gradient function is uniformly continuous on an open convex set containing the iteration sequence numerical examples are given at the end of this paper

applications of splitting algorithm to decomposition in convex programming and variational inequalities recently han and lou proposed a highly parallelizable decomposition algorithm for minimizing a strongly convex cost over the intersection of closed convex sets it is shown that their algorithm is in fact a special case of a splitting algorithm analyzed by gabay for finding a zero of the sum of two maximal monotone operators gabay s convergence analysis for the splitting algorithm is sharpened and new applications of this algorithm to variational inequalities convex programming and the solution of linear complementarily problems are proposed for convex programs with a certain separable structure a multiplier method that is closely related to the alternating direction method of multipliers of gabay mercier and of glowinski marrocco but which uses both ordinary and augmented lagrangians is obtained

option pricing in markov modulated exponential levy models with stochastic interest rates abstract in this paper we consider the problem of pricing european options namely vanilla options binary options and exchange options whose underlying assets prices dynamics follow markovian regime switching exponential levy models with stochastic interest rates where the stochastic interest rates are driven by markovian regime switching hull white process we obtain the integral representations of the option prices by fourier transform ft technique and some numerical results of 3 state case for merton jump diffusion model by the fast fourier transform fft approach the numerical results show that the pricing formulas are considerably accurate and easy to be implemented

modified fixed point equations and related iterative methods for variational inequalities abstract in this paper we study the equivalence characterizations of several modified fixed point equations to variational inequalities vi based on these equations we give some applications in constructing iterative methods for the solution of the vi especially we show global convergence the sublinear convergence and the finite termination of a new iterative algorithm under certain conditions

fixed point optimization algorithm and its application to power control in cdma data networks we discuss the variational inequality problem for a continuous operator over the fixed point set of a nonexpansive mapping one application of this problem is a power control for a direct sequence code division multiple access data network for such a power control each user terminal has to be able to quickly transmit at an ideal power level such that it can get a sufficient signal to interference plus noise ratio and achieve the required quality of service iterative algorithms to solve this problem should not involve auxiliary optimization problems and complicated computations to ensure this we devise a fixed point optimization algorithm for the variational inequality problem and perform a convergence analysis on it we give numerical examples of the algorithm as a power control

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

generalized vector complementarity problem in fuzzy environment abstract this paper is devoted to introduce and study a new class of generalized vector complementarity problems gvcp for short and generalized vector variational inequalities gvvi for short in fuzzy environment under suitable conditions we prove the equivalence between gvcp and gvvi in banach spaces then without any monotonicity assumption we apply kkm technique to establish an existence theorem of solutions for gvvi finally we show that the solution set of gvcp is nonempty and closed

iterative methods for variational and complementarity problems in this paper we study both the local and global convergence of various iterative methods for solving the variational inequality and the nonlinear complementarity problems included among such methods are the newton and several successive overrelaxation algorithms for the most part the study is concerned with the family of linear approximation methods these are iterative methods in which a sequence of vectors is generated by solving certain linearized subproblems convergence to a solution of the given variational or complementarity problem is established by using three different yet related approaches the paper also studies a special class of variational inequality problems arising from such applications as computing traffic and economic spatial equilibria finally several convergence results are obtained for some nonlinear approximation methods

a new version of extragradient method for variational inequality problems abstract in this paper we propose a new version of extragradient method for the variational inequality problem the method uses a new searching direction which differs from any one in existing projection type methods and is of a better step size rule under a certain generalized monotonicity condition it is proved to be globally convergent

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

algorithms with strong convergence for a system of nonlinear variational inequalities in banach spaces abstract in this paper a general system of nonlinear variational inequality problem in banach spaces was considered which includes some existing problems as special cases for solving this nonlinear variational inequality problem we construct two methods which were inspired and motivated by korpelevich s extragradient method furthermore we prove that the suggested algorithms converge strongly to some solutions of the studied variational inequality

some new projection methods for variational inequalities in this paper we propose some new double projection methods for solving variational inequalities by using the wiener hopf equations technique it is shown that these methods converge linearly under mild conditions and include some existing projection methods as special cases some examples are given to illustrate the efficiency of the proposed methods

new projection type methods for monotone lcp with finite termination in this paper we establish two new projection type methods for the solution of monotone linear complementarity problem lcp the methods are a combination of the extragradient method and the newton method in which the active set strategy is used and only one linear system of equations with lower dimension is solved at each iteration it is shown that under the assumption of monotonicity these two methods are globally and linearly convergent furthermore under a nondegeneracy condition they have a finite termination property at last the methods are extended to solving monotone affine variational inequality problem

bond pricing under a markovian regime switching jump augmented vasicek model via stochastic flows in this article we shall explore the state of art of stochastic flows to derive an exponential affine form of the bond price when the short rate process is governed by a markovian regime switching jump diffusion version of the vasicek model we provide the flexibility that the market parameters including the mean reversion level the volatility rate and the intensity of the jump component switch over time according to a continuous time finite state markov chain the states of the chain may be interpreted as different states of an economy or different stages of a business cycle we shall provide a representation for the exponential affine form of the bond price in terms of fundamental matrix solutions of linear matrix differential equations

an algorithm for solving the obstacle problems in this paper we propose an algorithm for solving the obstacle problem we try tofind the approximated region of the contact in the obstacle problem by iteration numerical examples are given for the obstacle problem for a membrane and the elastic plastic torsion problem

a class of projection methods for general variational inequalities abstract in this paper we consider and analyze a new class of projection methods for solving pseudomonotone general variational inequalities using the wiener hopf equations technique the modified methods converge for pseudomonotone operators our proof of convergence is very simple as compared with other methods the proposed methods include several known methods as special cases

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

numerical approach to monotone variational inequalities by a one step projected reflected gradient method with line search procedure abstract in this paper we revisit a recent approach to classical monotone variational inequalities by means of a projected reflected gradient type method in r n a line search procedure is incorporated for possible varying step sizes and no lipschitz continuity condition on the operator is required a main feature of the proposed method is that it requires only one projection step onto the feasible set at each iteration

convergence rate analysis of iteractive algorithms for solving variational inequality problems we present a unified convergence rate analysis of iterative methods for solving the variational inequality problem our results are based on certain error bounds they subsume and extend the linear and sublinear rates of convergence established in several previous studies we also derive a new error bound for gamma strictly monotone variational inequalities the class of algorithms covered by our analysis in fairly broad it includes some classical methods for variational inequalities e g the extragradient matrix splitting and proximal point methods for these methods our analysis gives estimates not only for linear convergence which had been studied extensively but also sublinear depending on the properties of the solution in addition our framework includes a number of algorithms to which previous studies are not applicable such as the infeasible projection methods a separation projection method inexact hybrid proximal point methods and some splitting techniques finally our analysis covers certain feasible descent methods of optimization for which similar convergence rate estimates have been recently obtained by luo 14

monotonicity of fixed point and normal mappings associated with variational inequality and its application we prove sufficient conditions for the monotonicity and the strong monotonicity of fixed point and normal maps associated with variational inequality problems over a general closed convex set sufficient conditions for the strong monotonicity of their perturbed versions are also shown these results include some well known in the literature as particular instances inspired by these results we propose a modified solodov and svaiter iterative algorithm for the variational inequality problem whose fixed point map or normal map is monotone

a truly globally convergent newton type method for the monotone nonlinear complementarity problem the josephy newton method for solving a nonlinear complementarity problem consists of solving possibly inexactly a sequence of linear complementarity problems under appropriate regularity assumptions this method is known to be locally superlinearly convergent to enlarge the domain of convergence of the newton method some globalization strategy based on a chosen merit function is typically used however to ensure global convergence to a solution some additional restrictive assumptions are needed these assumptions imply boundedness of level sets of the merit function and often even global uniqueness of the solution we present a new globalization strategy for monotone problems which is not based on any merit function our linesearch procedure utilizes the regularized newton direction and the monotonicity structure of the problem to force global convergence by means of a computationally explicit projection step which reduces the distance to the solution set of the problem the resulting algorithm is truly globally convergent in the sense that the subproblems are always solvable and the whole sequence of iterates converges to a solution of the problem without any regularity assumptions in fact the solution set can even be unbounded each iteration of the new method has the same order of computational cost as an iteration of the damped newton method under natural assumptions the local superlinear rate of convergence is also achieved

a new double projection algorithm for variational inequalities we present a modification of a double projection algorithm proposed by solodov and svaiter for solving variational inequalities the main modification is to use a different armijo type linesearch to obtain a hyperplane strictly separating current iterate from the solutions of the variational inequalities our method is proven to be globally convergent under very mild assumptions if in addition a certain error bound holds we analyze the convergence rate of the iterative sequence we use numerical experiments to compare our method with that proposed by solodov and svaiter

multivariate european option pricing in a markov modulated lvy framework this paper studies the pricing of some multivariate european options namely exchange options and quanto options when the risky assets involved are modelled by markov modulated lvy processes mmlps pricing formulae are based upon the characteristic exponents by using the well known fft methodology we study these pricing issues both under a risk neutral martingale measure and the historical measure the dependence between the assets components is incorporated in the joint characteristic function of the mmlps as an example we concentrate upon a regime switching version of the model of ballotta etal 2015 in which the dependence structure is introduced in a flexible way several numerical examples are provided to illustrate our results

a sequential linear programming algorithm for solving monotone variational inequalities applied to strongly monotone variational inequalities newton s algorithm achieves local quadratic convergence in this paper it is shown how the basic newton method can be modified to yield an algorithm whose global convergence can be guaranteed by monitoring the monotone decrease of the gap function associated with the variational inequality each iteration consists in the solution of a linear program in the space of primal dual variables and of a linesearch convergence does not depend on strong monotonicity however under strong monotonicity and geometric stability assumptions the set of active constraints at the solution is implicitly identified and quadratic convergence is achieved

projection algorithms for solving nonmonotone equilibrium problems in hilbert space we propose two projection algorithms for solving an equilibrium problem where the bifunction is not required to be satisfied any monotone property under assumptions on the continuity convexity of the bifunction and the nonemptyness of the solution set of the minty equilibrium problem we show that the sequences generated by the proposed algorithms converge weakly and strongly to a solution of the primal equilibrium problem respectively

iterative methods for nonlinear complementarity problems on isotone projection cones in this paper we present a recursion related to a nonlinear complementarity problem defined by a closed convex cone in a hilbert space and a continuous mapping defined on the cone if the recursion is convergent then its limit is a solution of the nonlinear complementarity problem in the case of isotone projection cones sufficient conditions are given for the mapping so that the recursion to be convergent

error bounds of regularized gap functions for nonsmooth variational inequality problems we study the clarke rockafellar directional derivatives of the regularized gap functions and of some modified ones for the variational inequality problem vip defined by a locally lipschitz but not necessarily differentiable function on a closed convex set in an euclidean space as applications we show that under the strong monotonicity assumption the regularized gap functions have fractional exponent error bounds and consequently that the sequences provided by an algorithm of armijo type converge to the solution of the vip

two metric projection methods for constrained optimization this paper is concerned with the problem min f x mid x in x where x is a convex subset of a linear space h and f is a smooth real valued function on h we propose the class of methods x k 1 p x k alpha k g k where p denotes projection on x with respect to a hilbert space norm cdot g k denotes the frechet derivative of f at x k with respect to another hilbert space norm cdot k on h and alpha k is a positive scalar stepsize we thus remove an important restriction in the original proposal of goldstein 1 and levitin and poljak 2 where the norms cdot and cdot k must be the same it is therefore possible to match the norm cdot with the structure of x so that the projection operation is simplified while at the same time reserving the option to choose cdot k on the basis of approximations to the hessian of f so as to attain a typically superlinear rate of convergence the resulting methods are particularly attrac