self adaptive methods for mixed quasi variational inequalities it is well known that the variational inequalities involving the nonlinear term are equivalent to the fixed point problems and the resolvent equations in this paper we use these alternative equivalent formulations to suggest and analyze some new self adaptive iterative methods for solving mixed quasi variational inequalities our results can be viewed as significant extensions of the previously known results for mixed quasi variational inequalities an example is given to illustrate the efficiency of the proposed method

a semi smooth newton method for projection equations and linear complementarity problems with respect to the second order cone abstract in this paper a special semi smooth equation associated to the second order cone is studied it is shown that under mild assumptions the semi smooth newton method applied to this equation is well defined and the generated sequence is globally and q linearly convergent to a solution as an application the obtained results are used to study the linear second order cone complementarity problem with special emphasis on the particular case of positive definite matrices moreover some computational experiments designed to investigate the practical viability of the method are presented

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

a krylov subspace method for large scale second order cone linear complementarity problem in this paper we first show that the second order cone linear complementarity problem soclcp can be solved by finding a positive zero s in mathbb r of a particular rational function h s and we then propose a krylov subspace method to reduce h s to h ell s as in the model reduction the zero s of h s can be accurately approximated by that of h ell s 0 which itself can be cast as a small eigenvalue problem the new method is made possible by a complete characterization of the curve of h s and it has several advantages over the bisection newton sf bn iteration recently proposed by l h zhang and w h yang math comp 83 2013 pp 1701 1720 and shown to be very efficient for small to medium size problems the method is tested and compared against the sf bn iteration and two other state of the art packages sf sdpt3 and sf sedumi our numerical results show that the method is very efficient for both small to medium dense problems and large scale ones

a new smoothing newton type method for second order cone programming problems a new smoothing function of the well known fischer burmeister function is given based on this new function a smoothing newton type method is proposed for solving second order cone programming at each iteration the proposed algorithm solves only one system of linear equations and performs only one line search this algorithm can start from an arbitrary point and it is q quadratically convergent under a mild assumption numerical results demonstrate the effectiveness of the algorithm

a regularized smoothing newton method for solving soccps based on a new smoothing c function abstract in this paper an soc complementarity function is constructed based on this function a regularized smoothing newton method is proposed for solving monotone second order cone complementarity problems denoted by soccps the proposed algorithm is proved to be globally and quadratically convergent under mild conditions unlike some existing smoothing newton type methods we take the regularization parameter the same as the smoothing parameter and treat the parameter as independent variables in our algorithm some numerical results are reported and indicate that the proposed method is quite effective

three steps iterative algorithms for mixed variational inequalities it is well known that the mixed variational inequalities are equivalent to the fixed point problems and the resolvent equations using this equivalence we suggest and consider a new three step iterative method for solving mixed variational inequalities the new iterative method is obtained by using three steps under suitable conditions we prove that the new method is globally convergent our results can be viewed as significant extensions of the previously known results for mixed variational inequalities since mixed variational inequalities include variational inequalities as special cases our method appears to be a new one for solving variational inequalities preliminary numerical experiments are included to illustrate the advantage and efficiency of the proposed method

on the identification of degenerate indices in the nonlinear complementarity problem with the proximal point algorithm in this paper we focus on the problem of identifying the index sets p x i xi 0 n x i fi x 0 and c x i xi fi x 0 for a solution x of the monotone nonlinear complementarity problem ncp f the correct identification of these sets is important from both theoretical and practical points of view such an identification enables us to remove complementarity conditions from the ncp and locally reduce the ncp to a system which can be dealt with more easily we present a new technique that utilizes a sequence generated by the proximal point algorithm ppa using the superlinear convergence property of ppa we show that the proposed technique can identify the correct index sets without assuming the nondegeneracy and the local uniqueness of the solution

on the quadratic eigenvalue complementarity problem over a general convex cone the solution of the conic quadratic eigenvalue complementarity problem cqeicp is first investigated without assuming symmetry on the matrices defining the problem a new sufficient condition for existence of solutions of cqeicp is presented extending to arbitrary pointed closed and convex cones a condition known to hold when the cone is the nonnegative orthant we also address the symmetric cqeicp where all its defining matrices are symmetric we show that assuming that two of its defining matrices are positive definite this symmetric cqeicp reduces to the computation of a stationary point of an appropriate merit function on a convex set furthermore we discuss the use of the so called spectral projected gradient spg algorithm for solving cqeicp when the cone of interest is the second order cone socqeicp a new algorithm is designed for the computation of the projections required by the spg method to deal with socqeicp numerical results are included to illustrate the efficiency of the spg method and the new projection technique in practice

a projection algorithm for general variational inequalities with perturbed constraint sets we consider the problem of general variational inequalities gvi with nonmonotone operator in a finite dimensional space we propose a method to solve gvi that at each iteration considers only one projection on an easy approximation of the constraint set which is important from a practical point of view we analyse the convergence of the algorithm under a weak cocoercivity condition using variational metric analysis computational experience is reported and comparative analysis with other two algorithms is also given for the monotone case

global and superlinear convergence of the smoothing newton method and its application to general box constrained variational inequalities the smoothing newton method for solving a system of nonsmooth equations f x 0 which may arise from the nonlinear complementarity problem the variational inequality problem or other problems can be regarded as a variant of the smoothing method at the kth step the nonsmooth function f is approximated by a smooth function f and the derivative of f at x k is used as the newton iterative matrix the merits of smoothing methods and smoothing newton methods are global convergence and convenience in handling in this paper we show that the smoothing newton method is also superlinearly convergent if f is semismooth at the solution and f satisfies a jacobian consistency property we show that most common smooth functions such as the gabriel more function have this property as an application we show that for box constrained variational inequalities if the involved function is p uniform the iteration sequence generated by the smoothing newton method will converge to the unique solution of the problem globally and superlinearly quadratically

modified newton methods for solving a semismooth reformulation of monotone complementarity problems in this paper we propose a newton type method for solving a semismooth reformulation of monotone complementarity problems in this method a direction finding subproblem which is a system of linear equations is uniquely solvable at each iteration moreover the obtained search direction always affords a direction of sufficient decrease for the merit function defined as the squared residual for the semismooth equation equivalent to the complementarity problem we show that the algorithm is globally convergent under some mild assumptions next by slightly modifying the direction finding problem we propose another newton type method which may be considered a restricted version of the first algorithm we show that this algorithm has a superlinear or possibly quadratic rate of convergence under suitable assumptions finally some numerical results are presented

merit functions for semi definite complementarity problems merit functions such as the gap function the regularized gap function the implicit lagrangian and the norm squared of the fischer burmeister function have played an important role in the solution of complementarity problems defined over the cone of nonnegative real vectors we study the extension of these merit functions to complementarity problems defined over the cone of block diagonal symmetric positive semi definite real matrices the extension suggests new solution methods for the latter problems

convexity and differentiability properties of spectral functions and spectral mappings on euclidean jordan algebras abstract we study in this paper several properties of the eigenvalues function of a euclidean jordan algebra extending several known results in the framework of symmetric matrices in particular we give a concise form for the directional differential of a single eigenvalue we especially focus on spectral functions f on euclidean jordan algebras which are the composition of a symmetric real valued function f with the eigenvalues function we explore several properties of f that are transferred to f in particular convexity strong convexity and differentiability spectral mappings are also considered a special case of which is the gradient mapping of a spectral function answering a problem proposed by h sendov we give a formula for the jacobian of these functions

semismooth homeomorphisms and strong stability of semidefinite and lorentz complementarity problems based on an inverse function theorem for a system of semismooth equations this paper establishes several necessary and sufficient conditions for an isolated solution of a complementarity problem defined on the cone of symmetric positive semidefinite matrices to be strongly regular stable we show further that for a parametric complementarity problem of this kind if a solution corresponding to a base parameter is strongly stable then a semismooth implicit solution function exists whose directional derivatives can be computed by solving certain affine problems on the critical cone at the base solution similar results are also derived for a complementarity problem defined on the lorentz cone the analysis relies on some new properties of the directional derivatives of the projector onto the semidefinite cone and the lorentz cone

an inexact implicit method for general mixed variational inequalities in the present paper we present an inexact implicit method with a variable parameter for general mixed variational inequalities we use a self adaptive technique to adjust parameter r at each iteration the main advantage of this technique is that the method can adjust the parameter automatically and the numbers of iteration are not very sensitive to different initial parameter r 0

semismooth karush kuhn tucker equations and convergence analysis of newton and quasi newton methods for solving these equations there are several forms of systems of nonsmooth equations which are equivalent to the karush kuhn tucker kkt system of a nonlinearly constrained optimization problem nlp if the nlp is twice continuously differentiable and the hessian functions of its objective and constraint functions are locally lipschitzian then these kkt equations are strongly semismooth if furthermore the linear independence condition and the strong second order sufficiency condition are satisfied at a kkt point then the generalized jacobians of these kkt equations are nonsingular at that point and the sequence generated by the generalized newton method converges to this point q quadratically however direct application of quasi newton methods cannot guarantee q superlinear convergence we present a mixed quasi newton method which converges q superlinearly with common symmetrical updating rules under the above conditions for the generalized newton method superlinear convergence of the primal variables and global convergence are also discussed

differentiable exact penalty functions for nonlinear second order cone programs we propose a method for solving nonlinear second order cone programs socps based on a continuously differentiable exact penalty function the construction of the penalty function is given by incorporating a multipliers estimate in the augmented lagrangian for socps under the nondegeneracy assumption and the strong second order sufficient condition we show that a generalized newton method has global and superlinear convergence we also present some preliminary numerical experiments

generalized viscosity approximation methods for mixed equilibrium problems and fixed point problems in this paper we present a new iterative method based on the hybrid viscosity approximation method and the hybrid steepest descent method for finding a common element of the set of solutions of generalized mixed equilibrium problems and the set of common fixed points of a finite family of nonexpansive mappings in hilbert spaces furthermore we prove that the proposed iterative method has strong convergence under some mild conditions imposed on algorithm parameters the results presented in this paper improve and extend the corresponding results reported by some authors recently

analysis of nonsmooth vector valued functions associated with infinite dimensional second order cones abstract given a hilbert space h the infinite dimensional lorentz second order cone k is introduced for any x h a spectral decomposition is introduced and for any function f r r we define a corresponding vector valued function f h x on hilbert space h by applying f to the spectral values of the spectral decomposition of x h with respect to k we show that this vector valued function inherits from f the properties of continuity lipschitz continuity differentiability smoothness as well as s semismoothness these results can be helpful for designing and analyzing solution methods for solving infinite dimensional second order cone programs and complementarity problems

the gus property of second order cone linear complementarity problems the globally uniquely solvable gus property of the linear transformation of the linear complementarity problems over symmetric cones has been studied recently by gowda et al via the approach of euclidean jordan algebra in this paper we contribute a new approach to characterizing the gus property of the linear transformation of the second order cone linear complementarity problems soclcp via some basic linear algebra properties of the involved matrix of soclcp some more concrete and checkable sufficient and necessary conditions for the gus property are thus derived

a generalized newton method for absolute value equations associated with second order cones in this paper we introduce the absolute value equations associated with second order cones socave in short which is a generalization of the absolute value equations discussed recently in the literature it is proved that the socave is equivalent to a class of second order cone linear complementarity problems soclcp in short in particular we propose a generalized newton method for solving the socave and show that the proposed method is globally linearly and locally quadratically convergent under suitable assumptions we also report some preliminary numerical results of the proposed method for solving the socave and the soclcp which show the efficiency of the proposed method

an inexact smoothing method for soccps based on a one parametric class of smoothing function in this paper we introduce a one parametric class of smoothing functions which contains the fischer burmeister smoothing function as special case based on this class of smoothing functions an inexact smoothing method for solving second order cone complementarity problems soccps is proposed in each iteration the corresponding linear system is solved only approximately under mild assumptions it is proved that the proposed method has global convergence and local superlinear convergence properties preliminary numerical results indicate that the method is effective for large scale problems

macdonald s formula for kac moody groups over local fields for an almost split kac moody group g over a local non archimedean field the last two authors constructed a spherical hecke algebra h over the complex numbers c say and its satake isomorphism with the commutative algebra of weyl invariant elements in some formal series algebra c y in this article we prove a macdonald s formula i e an explicit formula for the image of a basis element of h the proof involves geometric arguments in the masure associated to g and algebraic tools including the cherednik s representation of the bernstein lusztig hecke algebra introduced in a previous article and the cherednik s identity between some symmetrizers

a new iterative method for variational inequalities it is well known that the variational inequalities are equivalent to the fixed point problems using this equivalence we suggest and consider a new three step iterative method for solving variational inequalities the new iterative method is obtained by using three steps under suitable conditions we prove that the new method is globally convergent our results can be viewed as significant extensions of the previously known results for variational inequalities preliminary numerical experiments are included to illustrate the advantage and efficiency of the proposed method

a self adaptive method for solving general mixed variational inequalities abstract the general mixed variational inequality containing a nonlinear term is a useful and an important generalization of variational inequalities the projection method cannot be applied to solve this problem due to the presence of the nonlinear term to overcome this disadvantage noor m a noor pseudomonotone general mixed variational inequalities appl math comput 141 2003 529 540 used the resolvent equations technique to suggest and analyze an iterative method for solving general mixed variational inequalities in this paper we present a new self adaptive iterative method which can be viewed as a refinement and improvement of the method of noor global convergence of the new method is proved under the same assumptions as noor s method some preliminary computational results are given

on an iterative algorithm for general variational inequalities in this paper we suggest and analyze a new three step iterative algorithm for solving the general variational inequalities we also study the global convergence of the proposed method under some mild conditions since the general variational inequalities include variational inequalities quasi variational inequalities and quasi complementarity problems as special cases one can deduce similar results for these problems our results can be viewed as an improvement and refinement of the previously known and new results preliminary numerical experiments are included to illustrate the advantage and efficiency of the proposed method

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

the strong macdonald conjecture and hodge theory on the loop grassmannian we prove the strong macdonald conjecture of hanlon and feigin for reductive groups g in a geometric reformulation we show that the dolbeault cohomology hq x op of the loop grassmannian x is freely generated by de rham s forms on the disk coupled to the indecomposables of h bg equating the two euler characteristics gives an identity independently known to macdonald m which generalises ramanujan s 1 1 sum for simply laced root systems at level 1 we also find a strong form of bailey s 4 4 sum failure of hodge decomposition implies the singularity of x and of the algebraic loop groups some of our results were announced in t2

a globally convergent newton method for solving strongly monotone variational inequalities variational inequality problems have been used to formulate and study equilibrium problems which arise in many fields including economics operations research and regional sciences for solving variational inequality problems various iterative methods such as projection methods and the nonlinear jacobi method have been developed these methods are convergent to a solution under certain conditions but their rates of convergence are typically linear in this paper we propose to modify the newton method for variational inequality problems by using a certain differentiable merit function to determine a suitable step length the purpose of introducing this merit function is to provide some measure of the discrepancy between the solution and the current iterate it is then shown that under the strong monotonicity assumption the method is globally convergent and under some additional assumptions the rate of convergence is quadratic limited computational experience indicates the high efficiency of the proposed method

two unconstrained optimization approaches for the euclidean κ centrum location problem abstract consider the single facility euclidean centrum location problem in r n this problem is a generalization of the classical euclidean 1 median problem and 1 center problem in this paper we develop two efficient algorithms that are particularly suitable for problems where n is large by using unconstrained optimization techniques the first algorithm is based on the neural networks smooth approximation for the plus function and reduces the problem to an unconstrained smooth convex minimization problem the second algorithm is based on the fischer burmeister merit function for the second order cone complementarity problem and transforms the kkt system of the second order cone programming reformulation for the problem into an unconstrained smooth minimization problem our computational experiments indicate that both methods are extremely efficient for large problems and the first algorithm is able to solve problems of dimension n up to 10 000 efficiently

a power penalty method for second order cone nonlinear complementarity problems a power penalty method for solving nonlinear second order cone complementarity problems soccps is proposed by using this method the nonlinear soccp is converted to asymptotic nonlinear equations the merit of this method shows that the solution sequence of the asymptotic nonlinear equations converges to the solution of the nonlinear soccp at an exponential rate when the penalty parameter tends to positive infinity under mild assumptions an algorithm is constructed and numerical examples indicate the feasibility of our method

a regularized explicit exchange method for semi infinite programs with an infinite number of conic constraints the semi infinite program sip is normally represented with infinitely many inequality constraints and has been studied extensively so far however there have been very few studies on the sip involving conic constraints even though it has important applications such as chebyshev like approximation filter design and so on in this paper we focus on the sip with infinitely many conic constraints called an sicp for short we show that under the robinson constraint qualification a local optimum of the sicp satisfies the kkt conditions that can be represented only with a finite subset of the conic constraints we also introduce two exchange type algorithms for solving the convex sicp we first provide an explicit exchange method and show that it has global convergence under the strict convexity assumption on the objective function we then propose an algorithm combining a regularization method with the explicit exchange method and establish global convergence of the hybrid algorithm without the strict c

pseudomonotone general mixed variational inequalities in this paper we use the resolvent equations and the auxiliary principle techniques to suggest a class of iterative resolvent methods for solving general mixed variational inequalities the convergence of the proposed methods only requires the pseudomonotonicity of the operator which is weaker than monotonicity as special cases we obtain various known and new results for solving various classes of variational inequalities and related problems

a one parametric class of merit functions for the symmetric cone complementarity problem abstract in this paper we extend the one parametric class of merit functions proposed by kanzow and kleinmichel c kanzow h kleinmichel a new class of semismooth newton type methods for nonlinear complementarity problems comput optim appl 11 1998 227 251 for the nonnegative orthant complementarity problem to the general symmetric cone complementarity problem sccp we show that the class of merit functions is continuously differentiable everywhere and has a globally lipschitz continuous gradient mapping from this we particularly obtain the smoothness of the fischer burmeister merit function associated with symmetric cones and the lipschitz continuity of its gradient in addition we also consider a regularized formulation for the class of merit functions which is actually an extension of one of the ncp function classes studied by c kanzow y yamashita m fukushima new ncp functions and their properties j optim theory appl 97 1997 115 135 to the sccp by exploiting the cartesian p properties for a nonlinear transformation we show that the class of regularized merit functions provides a global error bound for the solution of the sccp and moreover has bounded level sets under a rather weak condition which can be satisfied by the monotone sccp with a strictly feasible point or the sccp with the joint cartesian r 02 property all of these results generalize some recent important works in j s chen p tseng an unconstrained smooth minimization reformulation of the second order cone complementarity problem math program 104 2005 293 327 c k sim j sun d ralph a note on the lipschitz continuity of the gradient of the squared norm of the matrix valued fischer burmeister function math program 107 2006 547 553 p tseng merit function for semidefinite complementarity problems math program 83 1998 159 185 under a unified framework

self adaptive methods for general variational inequalities abstract it is well known that the general variational inequalities are equivalent to the fixed point problems and the wiener hopf equations in this paper we use these alternative equivalent formulations to suggest and analyze some new self adaptive iterative methods for solving the general variational inequalities our results can be viewed as a significant extension of the previously known results for variational inequalities an example is given to illustrate the efficiency of the proposed method

polynomial convergence of primal dual algorithms for the second order cone program based on the mz family of directions in this paper we study primal dual path following algorithms for the second order cone programming socp based on a family of directions that is a natural extension of the monteiro zhang mz family for semidefinite programming we show that the polynomial iteration complexity bounds of two well known algorithms for linear programming namely the short step path following algorithm of kojima et al and monteiro and adler and the predictor corrector algorithm of mizuno et al carry over to the context of socp that is they have an o loge 1 iteration complexity to reduce the duality gap by a factor of e where n is the number of second order cones since the mz type family studied in this paper includes an analogue of the alizadeh haeberly and overton pure newton direction we establish for the first time the polynomial convergence of primal dual algorithms for socp based on this search direction

applications of second order cone programming in a second order cone program socp a linear function is minimized over the intersection of an affine set and the product of second order quadratic cones socps are nonlinear convex problems that include linear and convex quadratic programs as special cases but are less general than semidefinite programs sdps several efficient primaldual interior point methods for socp have been developed in the last few years after reviewing the basic theory of socps we describe general families of problems that tan be recast as socps these include robust linear programming and robust leastsquares problems problems involving sums or maxima of norms or with convex hyperbolic constraints we discuss a variety of engineering applications such as filter design antenna array weight design truss design and grasping forte optimization in robotics we describe an efficient primaldual interior point method for solving socps which shares many of the features of primaldual interior point methods for linear program

an explicit exchange algorithm for linear semi infinite programming problems with second order cone constraints in this paper we propose an explicit exchange algorithm for solving the semi infinite programming problem sip with second order cone soc constraints we prove by using the complementarity slackness conditions that the algorithm terminates in a finite number of iterations and the obtained solution sufficiently approximates the original sip solution in existing studies on sips only the nonnegative constraints were considered and hence the complementarity slackness conditions were separable to each scalar component however in our study the existing componentwise analyses are no longer applicable since the complementarity slackness conditions are associated with socs in order to overcome such a difficulty we introduce a certain coordinate system based on the spectral factorization in the numerical experiments we solve some test problems to see the effectiveness of the proposed algorithm first we compare our algorithm with two other algorithms based on the linear sip lsip reformulation and observe that by exploiting the sdpt3 solver for subproblems our algorithm finds the solution much faster than lsip reformulation approaches when the size of the problem is large also we apply the algorithm to some filter design problems and see that those problems can be solved efficiently

solvability of newton equations in smoothing type algorithms for the soccp in this paper we first investigate the invertibility of a class of matrices based on the obtained results we then discuss the solvability of newton equations appearing in the smoothing type algorithm for solving the second order cone complementarity problem soccp a condition ensuring the solvability of such a system of newton equations is given in addition our results also show that the assumption that the jacobian matrix of the function involved in the soccp is a p 0 matrix is not enough for ensuring the solvability of such a system of newton equations which is different from the one of smoothing type algorithms for solving many traditional optimization problems in n

smoothing functions for second order cone complementarity problems smoothing functions have been much studied in the solution of optimization and complementarity problems with nonnegativity constraints in this paper we extend smoothing functions to problems in which the nonnegative orthant is replaced by the direct product of second order cones these smoothing functions include the chen mangasarian class and the smoothed fischer burmeister function we study the lipschitzian and differential properties of these functions and in particular we derive computable formulas for these functions and their jacobians these properties and formulas can then be used to develop and analyze noninterior continuation methods for solving the corresponding optimization and complementarity problems in particular we establish the existence and uniqueness of the newton direction when the underlying mapping is monotone

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

unramified whittaker functions on p adic loop groups we define unramified whittaker functions on the p adic points of an affine kac moody group and establish an analogue of the casselman shalika formula for these functions

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

a globally convergent method based on fischer burmeister operators for solving second order cone constrained variational inequality problems the karush kuhn tucker system of a second order cone constrained variational inequality problem is transformed into a semismooth system of equations with the help of fischer burmeister operators over second order cones the clarke generalized differential of the semismooth mapping is presented a modified newton method with armijo line search is proved to have global convergence with local superlinear rate of convergence under certain assumptions on the variational inequality problem an illustrative example is given to show how the globally convergent method works

some resolvent iterative methods for variational inclusions and nonexpansive mappings in this paper we suggest and analyze three step iterations for finding the common element of the set of fixed points of a nonexpansive mappings and the set of the solutions of the variational inclusions using the resolvent operator technique we also study the convergence criteria of three step iterative method under some mild conditions our results include the previous results of noor m aslam noor general variational inequalities and nonexpansive mappings j math anal appl 331 2007 810 822 and noor and huang m aslam noor zhenyu huang three step methods for nonexpansive mappings and variational inequalities appl math comput in press doi 10 1016 j amc 2006 08 088 as special cases and may be considered as an improvement and refinement of the previously known results

a smoothing method for second order cone complementarity problem in this paper the second order cone complementarity problem is studied based on a perturbed symmetrically smoothing function which has coerciveness under proper conditions we present a smoothing newton method for this problem the boundedness of the level set can be obtained from the coerciveness which plays an important role in the convergence analysis furthermore the proposed algorithm for the reformulation has no restrictions on the starting point and solves only one system of equations preliminary numerical results indicate that the algorithm is effective

a subgradient extragradient algorithm for solving multi valued variational inequality in this paper we propose a subgradient extragradient method for solving multi valued variational inequality it is showed that the method converges globally to a solution of multi valued variational inequality provided the multi valued mapping is pseudomonotone with nonempty compact convex values convergence rate is analyzed preliminary computational experience is also reported

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

expected residual minimization formulation for a class of stochastic linear second order cone complementarity problems abstract this paper considers a class of stochastic linear second order cone complementarity problems slsoccp noticing that the slsoccp does not have a solution suitable to all realizations in general we present a deterministic formulation called the expected residual minimization erm formulation for it the coercive property of the erm problem and the robustness of its solutions are discussed due to the existence of expectation in the erm problem we employ the monte carlo approximation techniques to approximate the erm problem and show that under mild conditions this approximation approach possesses exponential convergence rate then we extend the above results to a general mixed slsoccp furthermore we apply the theoretical results to a stochastic optimal power flow model in radial network and report some numerical dispatching experiments for real world southern california edison 47 bus network

some new extragradient iterative methods for variational inequalities abstract in this paper we suggest and analyze some new extragradient iterative methods for finding the common element of the fixed points of a nonexpansive mapping and the solution set of the variational inequality for an inverse strongly monotone mapping in a hilbert space we also consider the strong convergence of the proposed method under some mild conditions several special cases are also discussed results proved in this paper may be viewed as improvement and refinement of the previously known results

likelihood based inference for regular functions with fractional polynomial approximations abstract this paper shows that regular fractional polynomials can approximate regular cost production and utility functions and their first two derivatives on closed compact subsets of the strictly positive orthant of euclidean space arbitrarily well these functions therefore can provide reliable approximations to demand functions and other economically relevant characteristics of tastes and technology using canonical cost function data it shows that full bayesian inference for these approximations can be implemented using standard markov chain monte carlo methods

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

semismooth matrix valued functions matrix valued functions play an important role in the development of algorithms for semidefinite programming problems this paper studies generalized differential properties of such functions related to nonsmooth smoothing newton methods the first part of this paper discusses basic properties such as the generalized derivative rademacher s theorem b derivative directional derivative and semismoothness the second part shows that the matrix absolute value function the matrix semidefinite projection function and the matrix projective residual function are strongly semismooth

a spanning set for c i n f in denotes the banach space of continuous functions on the unit n cube in in rn let a i 0 1 2 be a countable colii lection of n tuples of positive real numbers satisfying liml aj oo for j 1 n we canonically enlarge the family of monomials xa to a family of functions sr a conjecture the linear span of 9 a is dense in in if and only if z 0 lla1l oo for n 1 this is equivalent to the muntz szasz theorem for n 1 we prove the necessity in general and the sufficiency under the additional hypothesis that there exist constants g n 1 such that aii 0 be dense in the space of continuous functions on the unit n cube in 0 1 n an essentially equivalent problem is the question of characterizing discrete sets of uniqueness for functions of several variables bounded in a cone interesting results on this pro blem have been found by korevaar hellerstein 7 ronkin 11 12 and berndtsson 1 for example in 1 the following result is proved see also 1 1 12 received by the editors november 11 1988 1980 mathematics subject classification 1985 revision primary 32e30 secondary 41a63

the proximal point algorithm with genuine superlinear convergence for the monotone complementarity problem in this paper we consider a proximal point algorithm ppa for solving monotone nonlinear complementarity problems ncp ppa generates a sequence by solving subproblems that are regularizations of the original problem it is known that ppa has global and superlinear convergence properties under appropriate criteria for approximate solutions of subproblems however it is not always easy to solve subproblems or to check those criteria in this paper we adopt the generalized newton method proposed by de luca facchinei and kanzow to solve subproblems and adopt some ncp functions to check the criteria then we show that the ppa converges globally provided that the solution set of the problem is nonempty moreover without assuming the local uniqueness of the solution we show that the rate of convergence is superlinear in a genuine sense provided that the limit point satisfies the strict complementarity condition

correction factors for kac moody groups and t deformed root multiplicities we study a correction factor for kac moody root systems which arises in the theory of p adic kac moody groups in affine type this factor is known and its explicit computation is the content of the macdonald constant term conjecture the data of the correction factor can be encoded as a collection of polynomials m lambda in mathbb z t indexed by positive imaginary roots lambda at t 0 these polynomials evaluate to the root multiplicities so we consider m lambda to be a t deformation of mathrm mult lambda we generalize the peterson algorithm and the berman moody formula for root multiplicities to compute m lambda as a consequence we deduce fundamental properties of m lambda

a generalized newton method for absolute value equations associated with circular cones in this paper we study a class of absolute value equations associated with circular cone ccave for short which is a generalization of the absolute value equations discussed recently in the literature analogously to the fact that circular cone is the very generalization of the second order cone we show that the ccave is equivalent to a class of circular cone linear complementarity problems and hence generalize the well known equivalence between absolute value equations and linear complementarity problems useful properties of the generalized differential of the absolute value function over the circular cone are investigated which helps to propose a generalized newton method for solving the ccave the convergence of this method is established under mild conditions as well as the efficiency of which is illustrated by some preliminary numerical results

the t analog of the basic string function for twisted affine kac moody algebras abstract we study lusztig s t analog of weight multiplicities associated to level one representations of twisted affine kac moody algebras an explicit closed form expression is obtained for the corresponding t string function using constant term identities of macdonald and cherednik the closed form involves the generalized exponents of the graded pieces of the twisted affine algebra considered as modules for the underlying finite dimensional simple lie algebra this extends previous work on level 1 t string functions for the untwisted simply laced affine kac moody algebras

a modified inexact implicit method for mixed variational inequalities in this paper we suggest and analyze an inexact implicit method with a variable parameter for mixed variational inequalities by using a new inexactness restriction under certain conditions the global convergence of the proposed method is proved some preliminary computational results are given to illustrate the efficiency of the new inexactness restriction the results proved in this paper may be viewed as improvement and refinement of the previously known results

on q and r0 properties of a quadratic representation in linear complementarity problems over the second order cone abstract this paper studies the linear complementarity problem lcp m q over the second order lorentz or ice cream cone denoted by n where m is a n n real square matrix and q rn this problem is denoted as solcp m q the study of second order cone programming problems and hence an independent study of solcp is motivated by a number of applications though the second order cone is a special case of the cone of squares symmetric cone in a euclidean jordan algebra the geometry of its faces is much simpler and hence an independent study of lcp over n may yield interesting results in this paper we characterize the r0 property x n m x n and x m x 0 x 0 of a quadratic representation pa x 2a a x a2 x of n for a x n where is a jordan product and show that the r0 property of pa is equivalent to stating that solcp pa q has a solution for all q n

double affine hecke algebras and macdonald s conjectures in this paper we prove the main results about the structure of double affine hecke algebras announced in c1 c2 the technique is based on the realization of these algebras in terms of demazure lusztig operators bgg d l2 ls c3 and rather standard facts from the theory of affine weyl groups in particular it completes the proof partially published in c2 of the macdonald scalar product conjecture see ml 12 6 including the famous macdonald constant term conjecture the q t case we mainly follow the opdam paper 0 where the macdonald mehta conjectures in the degenerate differential case were deduced from certain properties of the heckman opdam operators ho and the existence of the shift operators heckman s interpretation of these operators via the so called dunkl operators see he and also c5 was important to our approach we note that the ho operators are closely related to the so called quantum many body problem calogero sutherland moser olshanetsky perelomov the conformal field theory knizhnik zamolodchikov equations the harmonic analysis on symmetric spaces harish chandra helgason etc and last but not the least the classic theory of the hypergeometric functions establishing the connection between the difference counterparts of heckman opdam operators introduced in c4 and the macdonald theory ml m2 including the construction of the difference shift operators is the main thrust of this paper once the connection is established it is not very difficult to calculate the scalar squares of the macdonald polynomials and to prove the constant term conjecture from his fundamental paper m3 to simplify the exposition we discuss the reduced root systems only and impose the relation q tk for k e z to avoid infinite products in the definition of macdonald s pairing the purpose of this work is to present a concrete application of the new technique arbitrary q t can be handled in a

iwahori hecke algebras for p adic loop groups this paper is a continuation of braverman and kazhdan ann math 2 174 3 1603 1642 2011 in which the first two authors have introduced the spherical hecke algebra and the satake isomorphism for an untwisted affine kac moody group over a non archimedian local field in this paper we develop the theory of the iwahori hecke algebra associated to these same groups the resulting algebra is shown to be closely related to cherednik s double affine hecke algebra furthermore using these results we give an explicit description of the affine satake isomorphism generalizing macdonald s formula for the spherical function in the finite dimensional case the results of this paper have been previously announced in braverman and kazhdan european congress of mathematics european mathematical society zurich 2014

exceptional family and solvability of the second order cone complementarity problems abstract in this paper we introduce a concept of exceptional family for second order cone complementarity problems denoted by soccp which is the particular case of the concept of exceptional family of elements introduced by isac and carbone 1999 25 for the nonlinear complementarity problems and the exceptional family for second order cone complementarity problems has some pretty unique properties furthermore we propose a new sufficient existence condition of a solution to soccp and give a particular example to show that the new condition is not stronger than known isac carbone s condition

modified descent projection method for solving variational inequalities in this paper we propose a modified descent projection method for solving variational inequalities the method makes use of a descent direction to produce the new iterate and can be viewed as an improvement of the descent projection method by using a new step size under certain conditions the global convergence of the proposed method is proved in order to demonstrate the efficiency of the proposed method we provide numerical results for a traffic equilibrium problems

smoothing methods for nonsmooth nonconvex minimization we consider a class of smoothing methods for minimization problems where the feasible set is convex but the objective function is not convex not differentiable and perhaps not even locally lipschitz at the solutions such optimization problems arise from wide applications including image restoration signal reconstruction variable selection optimal control stochastic equilibrium and spherical approximations in this paper we focus on smoothing methods for solving such optimization problems which use the structure of the minimization problems and composition of smoothing functions for the plus function x many existing optimization algorithms and codes can be used in the inner iteration of the smoothing methods we present properties of the smoothing functions and the gradient consistency of subdifferential associated with a smoothing function moreover we describe how to update the smoothing parameter in the outer iteration of the smoothing methods to guarantee convergence of the smoothing methods to a stationary point of the original minimization problem

a new non interior continuation method for solving the second order cone complementarity problem abstract in this paper based on a symmetrically perturbed smoothing fischer burmeister function a non interior continuation method is proposed for solving the second order cone complementarity problem soccp the proposed algorithm solves only one linear system of equations and performs only one line search at each iteration under monotonicity it is shown that our algorithm is globally and locally superlinearly convergent without requiring strict complementarity assumption at the soccp solution furthermore the proposed algorithm has local quadratic convergence under mild conditions some numerical results are reported which indicate the effectiveness of the proposed algorithm