vector nonsmooth variational like inequalities and optimization problems abstract in this paper we first extend the concepts of clarke s generalized derivation and subdifferential for a function whose domain is a subset of a metrizable topological vector space we also introduce and consider a new class of variational like inequalities which are called the vector nonsmooth variational like inequalities we also establish the relationship between the nonsmooth variational like inequalities and vector optimization problems under some suitable conditions in particular our results extend and generalize the results of mishra and wang s k mishra s y wang vector variational like inequalities and nonsmooth vector optimization problems nonlinear anal 64 2006 1939 1945 and mishra and noor s k mishra m aslam noor on vector variational like inequality problems j math anal appl 311 2005 69 75

an iterative algorithm for solving fixed point problems variational inequality problems and mixed equilibrium problems abstract in this paper we introduce an iterative algorithm for finding a common element of the set of solutions of a mixed equilibrium problem the set of fixed points of an infinite family of nonexpansive mappings and the set of solutions of some variational inequality furthermore we prove that the proposed iterative algorithm converges strongly to a common element of the above three sets our results extend and improve the corresponding results of zeng and yao l c zeng and j c yao a hybrid iterative scheme for mixed equilibrium problems and fixed point problems j comput appl math 214 2008 186 201 takahashi and takahashi s takahashi and w takahashi viscosity approximation methods for equilibrium problems and fixed point problems in hilbert spaces j math anal appl 331 2007 506 515 and many others

perturbed proximal point algorithms for general quasi variational like inclusions in this paper we introduce two new concepts of subdifferential and proximal mappings of proper functionals on hilbert spaces the existence and lipschitz continuity of proximal mapping of a proper functional are proved by applying these concepts we introduce and study a class of general quasi variational like inclusions and develop some new perturbed proximal point algorithms of mann and ishikawa type for finding the approximate solutions of the general quasi variational like inclusions the convergence criteria of the sequences of approximate solutions generated by these new algorithms are also discussed our algorithms and results improve and generalize many known results in the literature

on a class of nonconvex equilibrium problems in this paper we introduce a new class of equilibrium problems known as nonconvex equilibrium problems we suggest some algorithms for solving nonconvex equilibrium problems by using the auxiliary principle technique the convergence of the proposed methods either requires partially relaxed strongly monotonicity or pseudomononicity we also introduce the concept of well posedness for the nonconvex equilibrium problems as special cases we obtain a number of known and new results for solving various classes of equilibrium and variational inequality problems our results can be considered a significant improvement of the previously known results

on a predictor corrector method for solving invex equilibrium problems abstract in this paper we use the auxiliary principle technique in conjunction with the bregman function to suggest and analyze a predictor corrector method for solving invex equilibrium problems we also study the convergence criteria of this new method under some mild conditions we obtain as special cases new and known methods for solving variational like inequalities and equilibrium problems

a variational like inequality for multifunctions with applications abstract we study a variational like inequality problem for a multifunction a certain relevant complementarity problem is defined we establish some existence theorems for the above two problems an application to convex mathematical programming is also shown

invex equilibrium problems in this paper we introduce a new class of equilibrium problems known as invex equilibrium problems in the setting of invexity this class of equilibrium problems includes equilibrium problems variational inequalities and variational like inequalities as special cases we use the auxiliary principle technique to suggest and analyze some iterative schemes for solving invex equilibrium problems and study the convergence criteria of these methods under some mild conditions we also consider the concept of well posedness for invex equilibrium problems our results represent significant and important refinements of the previously known results

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

auxiliary principle and iterative algorithms for generalized set valued strongly nonlinear mixed variational like inequalities in this paper we extend the auxiliary principle technique to study a class of generalized set valued strongly nonlinear mixed variational like inequalities we prove the existence of a solution of the auxiliary problem for the generalized set valued strongly nonlinear mixed variational like inequalities construct the iterative algorithm for the generalized set valued strongly nonlinear mixed variational like inequalities and show the existence of a solution of the generalized set valued strongly nonlinear mixed variational like inequality by using the auxiliary principle technique we also prove the convergence of iterative sequences generated by the algorithm

a trace minimization algorithm for the generalized eigenvalue problem an algorithm for computing a few of the smallest or largest eigenvalues and associated eigenvectors of the large sparse generalized eigenvalue problem ax lambda bx is presented the matrices a and b are assumed to be symmetric and haphazardly sparse with b being positive definite the problem is treated as one of constrained optimization and an inverse iteration is developed which requires the solution of linear algebraic systems only to the accuracy demanded by a given subspace the rate of convergence of the method is established and a technique for improving it is discussed numerical experiments and comparisons with other methods are presented

generalized mixed quasi variational like inequalities in this paper we use the auxiliary principle technique to suggest and analyze some iterative methods for solving generalized mixed quasi variational like inequalities we show that the convergence of the proposed method requires either pseudomonotonicity or partially relaxed strongly monotonicity under some mild condition as special cases one can obtain a number of new and known algorithms for solving variational inequalities and complementarity problems our results represent significant and important refinements of the previously known results

on vector variational like inequality problems abstract in this paper we establish some relationships between vector variational like inequality and vector optimization problems under the assumptions of invex functions we identify the vector critical points the weakly efficient points and the solutions of the weak vector variational like inequality problems under pseudo invexity assumptions these conditions are more general than those of existing ones in the literature in particular this work extends the earlier work of ruiz garzon et al g ruiz garzon r osuna gomez a rufian lizan relationships between vector variational like inequality and optimization problems european j oper res 157 2004 113 119 to a wider class of functions namely the pseudo invex functions studied in a recent work of noor m a noor on generalized preinvex functions and monotonicities j inequal pure appl math 5 2004 1 9

a general iterative method for equilibrium problems and fixed point problems in hilbert spaces abstract in this paper we introduce two iterative schemes by the general iterative method for finding a common element of the set of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a hilbert space then we prove two strong convergence theorems for nonexpansive mappings to solve a unique solution of the variational inequality which is the optimality condition for the minimization problem these results extended and improved the corresponding results of marino and xu g marino h k xu a general iterative method for nonexpansive mapping in hilbert spaces j math anal appl 318 2006 43 52 s takahashi and w takahashi s takahashi w takahashi viscosity approximation methods for equilibrium problems and fixed point problems in hilbert spaces j math anal appl 331 1 2007 506 515 and many others

algorithms for approximating minimization problems in hilbert spaces in this paper we study the following minimization problem minx fix s w m2 12 x 2 h x where b is a bounded linear operator m 0 is some constant h is a potential function for c f fix t is the set of fixed points of nonexpansive mapping s and w is the solution set of an equilibrium problem this paper introduces two new algorithms one implicit and one explicit that can be used to find the solution of the above minimization problem

on iterative methods for equilibrium problems abstract in this paper we introduce a hybrid iterative scheme for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of finitely many nonexpansive mappings we prove that the approximate solution converges strongly to a solution of a class of variational inequalities under some mild conditions which is the optimality condition for some minimization problem we also give some comments on the results of plubtieng and punpaeng s plubtieng r punpaeng a general iterative method for equilibrium problems and fixed point problems in hilbert spaces j math anal appl 336 2007 455 469 results obtained in this paper may be viewed as an improvement and refinement of the previously known results in this area

wielandt and ky fan theorem for matrix pairs the generalization of wielandt and ky fan theorem is given for hermitian matrix pairs and some new eigenvalue perturbation estimates are obtained an application is made on a class of quadratic matrix pencils

accelerated simultaneous iterations for large finite element eigenproblems abstract an accelerated simultaneous iteration method is presented for the solution of the generalized eigenproblem ax bx where a and b are real sparse symmetric positive definite matrices the approach is well suited for the determination of the leftmost eigenpairs of problems with large size n the procedure relies on the optimization of the rayleigh quotient over a subspace of orthogonal vectors by a conjugate gradient technique effectively preconditioned with the pointwise incomplete cholesky factorization the method is applied to the evaluation of the smallest 15 eigenpairs of finite element models with size ranging between 150 and 2300 the numerical experiments show that while the simultaneous conjugate gradient scheme fails to converge the accelerated iterations yield accurate results in a number of steps which is much smaller than n the new approach does not require the a priori estimate of any empirical parameter and appears to be a robust reliable and efficient tool for the partial eigensolution of large finite element problems

a general iterative method for nonexpansive mappings in hilbert spaces let h be a real hilbert space consider on h a nonexpansive mapping t with a fixed point a contraction f with coefficient 0 0 let 0

existence theorems for generalized noncoercive equilibrium problems the quasi convex case we provide a characterization of the nonemptiness of the solution set for generalized noncoercive equilibrium problems an extension of generalized quasi variational inequalities defined in reflexive banach spaces in the quasi convex case in addition several necessary and sufficient conditions for the set of solutions to these problems to be nonempty and bounded are also given our approach is based on recession notions which proved to be very useful in the study of noncoercive minimization problems in fact we find some particular cones as estimates for the recession cone of the solution set these cones for the ones containing the latter set are proved to be sharp enough to encompass several special situations found in the literature

strong convergence of krasnoselskii and mann s type sequences for one parameter nonexpansive semigroups without bochner integrals abstract in this paper we prove krasnoselskii and mann s type convergence theorems for nonexpansive semigroups without using bochner integral and without assuming the strict convexity of banach spaces one of our main results is the following let c be a compact convex subset of a banach space e and let t t t 0 be a one parameter strongly continuous semigroup of nonexpansive mappings on c let t n be a sequence in 0 satisfying lim inf n t n lim sup n t n and lim n t n 1 t n 0 let 0 1 define a sequence x n in c by x 1 c and x n 1 t t n x n 1 x n for n n then x n converges strongly to a common fixed point of t t t 0

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

parameter selection for total variation based image restoration using discrepancy principle there are two key issues in successfully solving the image restoration problem 1 estimation of the regularization parameter that balances data fidelity with the regularity of the solution and 2 development of efficient numerical techniques for computing the solution in this paper we derive a fast algorithm that simultaneously estimates the regularization parameter and restores the image the new approach is based on the total variation tv regularized strategy and morozov s discrepancy principle the tv norm is represented by the dual formulation that changes the minimization problem into a minimax problem a proximal point method is developed to compute the saddle point of the minimax problem by adjusting the regularization parameter adaptively in each iteration the solution is guaranteed to satisfy the discrepancy principle we will give the convergence proof of our algorithm and numerically show that it is better than some state of the art methods in terms of both speed and accuracy

multiplicative noise removal using variable splitting and constrained optimization multiplicative noise also known as speckle noise models are central to the study of coherent imaging systems such as synthetic aperture radar and sonar and ultrasound and laser imaging these models introduce two additional layers of difficulties with respect to the standard gaussian additive noise scenario 1 the noise is multiplied by rather than added to the original image 2 the noise is not gaussian with rayleigh and gamma being commonly used densities these two features of multiplicative noise models preclude the direct application of most state of the art algorithms which are designed for solving unconstrained optimization problems where the objective has two terms a quadratic data term log likelihood reflecting the additive and gaussian nature of the noise plus a convex possibly nonsmooth regularizer e g a total variation or wavelet based regularizer prior in this paper we address these difficulties by 1 converting the multiplicative model into an additive one by taking logarithms as proposed by some other authors 2 using variable splitting to obtain an equivalent constrained problem and 3 dealing with this optimization problem using the augmented lagrangian framework a set of experiments shows that the proposed method which we name midal multiplicative image denoising by augmented lagrangian yields state of the art results both in terms of speed and denoising performance

total nuclear variation and jacobian extensions of total variation for vector fields we explore a class of vectorial total variation vtv measures formed as the spatial sum of a pixel wise matrix norm of the jacobian of a vector field we give a theoretical treatment that indicates that while color smearing and affine coupling bias often reported as gray scale bias are typically cited as drawbacks for vtv these are actually fundamental to smoothing vector direction i e smoothing hue and saturation in color images additionally we show that encouraging different vector channels to share a common gradient direction is equivalent to minimizing jacobian rank we thus propose total nuclear variation tnv and since nuclear norm is the convex envelope of matrix rank we argue that tnv is the optimal convex regularizer for enforcing shared directions we also propose extended jacobians which use larger neighborhoods than the conventional finite difference operator and we discuss efficient vtv optimization algorithms in simple color image denoising experiments tnv outperformed other common vtv regularizers and was further improved by using extended jacobians tnv was also competitive with the method of non local means often outperforming it by 0 25 to 2 db when using extended jacobians

a projected preconditioned conjugate gradient algorithm for computing many extreme eigenpairs of a hermitian matrix we present an iterative algorithm for computing an invariant subspace associated with the algebraically smallest eigenvalues of a large sparse or structured hermitian matrix a we are interested in the case in which the dimension of the invariant subspace is large e g over several hundreds or thousands even though it may still be small relative to the dimension of a these problems arise from for example density functional theory dft based electronic structure calculations for complex materials the key feature of our algorithm is that it performs fewer rayleigh ritz calculations compared to existing algorithms such as the locally optimal block preconditioned conjugate gradient or the davidson algorithm it is a block algorithm and hence can take advantage of efficient blas3 operations and be implemented with multiple levels of concurrency we discuss a number of practical issues that must be addressed in order to implement the algorithm efficiently on a high performance computer

hemiequilibrium like problems abstract in this paper we introduce and consider a new class of equilibrium problems known as hemiequilibrium like problems this new class includes hemiequilibrium equilibrium like problems and several classes of variational inequalities as special cases a number of iterative methods for solving hemiequilibrium like problems are suggested and analyzed by using the auxiliary principle technique we also study the convergence analysis of these iterative methods under some mild conditions the results obtained in this paper can be considered as a novel application of the auxiliary principle technique

some characterizations of strongly preinvex functions in this paper a new class of generalized convex function is introduced which is called the strongly preinvex function we study some properties of strongly preinvex function in particular we establish the equivalence among the strongly preinvex functions strongly invex functions and strongly monotonicity under some suitable conditions as special cases one can obtain several new and previously known results for preinvex invex functions

viscosity approximation methods for nonexpansive mappings abstract viscosity approximation methods for nonexpansive mappings are studied consider a nonexpansive self mapping t of a closed convex subset c of a banach space x suppose that the set fix t of fixed points of t is nonempty for a contraction f on c and t 0 1 let x t c be the unique fixed point of the contraction x tf x 1 t tx consider also the iteration process x n where x 0 c is arbitrary and x n 1 n f x n 1 n tx n for n 1 where n 0 1 if x is either hilbert or uniformly smooth then it is shown that x t and under certain appropriate conditions on n x n converge strongly to a fixed point of t which solves some variational inequality

variational restoration of nonflat image features models and algorithms we develop both mathematical models and computational algorithms for variational denoising and restoration of nonflat image features nonflat image features are those that live on riemannian manifolds instead of on the euclidean spaces familiar examples include the orientation feature from optical flows or gradient flows that lives on the unit circle s1 the alignment feature from fingerprint waves or certain texture images that lives on the real projective line mathbb rp 1 and the chromaticity feature from color images that lives on the unit sphere s2 in this paper we apply the variational method to denoise and restore general nonflat image features mathematical models for both continuous image domains and discrete domains or graphs are constructed riemannian objects such as metric distance and levi civita connection play important roles in the models computational algorithms are also developed for the resulting nonlinear equations the mathematical framework can be applied to res

an efficient iterative method for the generalized stokes problem the generalized stokes problem which arises frequently in the simulation of time dependent navier stokes equations for incompressible fluid flow gives rise to symmetric linear systems of equations these systems are indefinite due to a set of linear constraints on the velocity causing difficulty for most preconditioners and iterative methods this paper presents a novel method to obtain a preconditioned linear system from the original one which is then solved by an iterative method this new method generates a basis for the velocity space and solves a reduced system which is symmetric and positive definite numerical experiments indicating superior convergence compared to existing methods are presented a natural extension of this method to elliptic problems is also proposed along with theoretical bounds on the rate of convergence and results of experiments demonstrating robust and effective preconditioning

multivalued general equilibrium problems in this paper we use the auxiliary principle technique to suggest some new classes of iterative algorithms for solving multivalued equilibrium problems the convergence of the proposed methods either requires partially relaxed strongly monotonicity or pseudomonotonicity as special cases we obtain a number of known and new results for solving various classes of equilibrium and variational inequality problems since multivalued equilibrium problems include equilibrium variational inequality and complementarity problems as specials cases our results continue to hold for these problems

an improved iterative optimization technique for the leftmost eigenpairs of large symmetric matrices abstract an accelerated optimization technique combined with a stepwise deflation procedure is presented for the efficient evaluation of the p p 20 leftmost eigenvalues and eigenvectors of finite element symmetric positive definite p d matrices of very large size the optimization is performed on the rayleigh quotient of the deflated matrices by the aid of a conjugate gradient cg scheme effectively preconditioned with the incomplete cholesky factorization no a priori estimate of acceleration parameters is required numerical experiments on large arbitrarily sparse problems taken from the engineering finite elements f e practice show a very fast convergence rate for any value of p within the explored interval and particularly so for the minimal eigenpair in this case the number of iterations needed to achieve an accurate solution may be as much as 2 orders of magnitude smaller than the problem size several results concerning the spectral behavior of the cg preconditioning matrices are also given and discussed

a bayesian approach for envelope models the envelope model is a new paradigm to address estimation and prediction in multivariate analysis using sufficient dimension reduction techniques it has the potential to achieve substantial efficiency gains compared to standard models this model was first introduced by statist sinica 20 2010 927 960 for multivariate linear regression and has since been adapted to many other contexts however a bayesian approach for analyzing envelope models has not yet been investigated in the literature in this paper we develop a comprehensive bayesian framework for estimation and model selection in envelope models in the context of multivariate linear regression our framework has the following attractive features first we use the matrix bingham distribution to construct a prior on the orthogonal basis matrix of the envelope subspace this prior respects the manifold structure of the envelope model and can directly incorporate prior information about the envelope subspace through the specification of hyperparamaters this feature has potential applications in the broader bayesian sufficient dimension reduction area second sampling from the resulting posterior distribution can be achieved by using a block gibbs sampler with standard associated conditionals this in turn facilitates computationally efficient estimation and model selection third unlike the current frequentist approach our approach can accommodate situations where the sample size is smaller than the number of responses lastly the bayesian approach inherently offers comprehensive uncertainty characterization through the posterior distribution we illustrate the utility of our approach on simulated and real datasets

descent methods for equilibriumproblems in a banach space in this paper we consider equilibrium problems with differentiable bifunctions in abanach space setting and investigate properties of gap functions for such problems we suggest a derivative free descent method and give conditions which provide strong convergence of the method

strong convergence theorems for a new iterative method of k strictly pseudo contractive mappings in hilbert spaces in this paper we introduce a new iterative method of a k strictly pseudo contractive mapping for some email protected k 1 and prove that the sequence x n converges strongly to a fixed point of t which solves a variational inequality related to the linear operator a our results have extended and improved the corresponding results of y j cho s m kang and x qin some results on k strictly pseudo contractive mappings in hilbert spaces nonlinear anal 70 2008 1956 1964 and many others

hardy littlewood polya type theorems for invex functions in this paper the hardy littlewood polya theorem on majorization is extended from convex functions to invex ones some variants for pseudo invex and quasi invex functions are also considered the framework used is that of similarly separable vectors the results obtained are illustrated for monotonic monotonic in mean and star shaped vectors respectively applications to relative invexity are given

block minimum perturbation algorithm based on block arnoldi process for nonsymmetric linear systems with multiple right hand sides abstract in many applications in scientific computing and engineering one has to solve huge sparse linear systems of equations with several right hand sides customarily one uses the residual error as a stopping condition but small residue does not imply accurate approximate solutions therefore minimizing quantities other than the residual norm may be more adequate in this context based on the above consideration a block minimum perturbation krylov subspace method bminpet for the solution xm satisfying a a x m b b at each step of nonsymmetric linear systems a x b is shown such that the normwise backward error meets some optimality condition with minimizing the block joint backward perturbation norm a b f of the matrix a b and a stopping criterion is given for consistent system with multiple right hand sides to get a more robust way to stop the iterations this process is a generalization of minimum perturbation algorithm minpert by kasenally and simoncini 1997 analysis of a minimum perturbation algorithm for nonsymmetric linear systems siam j numer anal 34 48 66 to multiple right hand sides for the benefit of the amount of calculation we give lower and upper bounds about min a b f that are easier to compute as a by product we come up with a conception of function f g which is a generalization of function that is a symmetric gauge sg function the algorithm is used to analyze the performance of bgmres method by evaluating the proximity of their solutions to block joint backward perturbation optimality the relationships between bminpet method and relative methods are investigated numerical examples show us that bminpert shows its superiority compared with bfgmres s m pf gsgmres bl bicg rq bgmres and barnoldi in solving the large sparse ill conditioned problems

trace minimization and definiteness of symmetric pencils abstract a symmetric matrix pencil a b of order n is called positive definite if there is a such that the matrix a b is positive definite we consider the case with b nonsingular and show that the definiteness is closely related to the existence of min tr x t ax under the condition x t bx j 1 where j 1 is a given diagonal matrix of order n and j 2 1 i we also prove an analog of the cauchy interlacing theorem for some such pencils

matrices vector spaces and information retrieval the evolution of digital libraries and the internet has dramatically transformed the processing storage and retrieval of information efforts to digitize text images video and audio now consume a substantial portion of both academic and industrial activity even when there is no shortage of textual materials on a particular topic procedures for indexing or extracting the knowledge or conceptual information contained in them can be lacking recently developed information retrieval technologies are based on the concept of a vector space data are modeled as a matrix and a user s query of the database is represented as a vector relevant documents in the database are then identified via simple vector operations orthogonal factorizations of the matrix provide mechanisms for handling uncertainty in the database itself the purpose of this paper is to show how such fundamental mathematical concepts from linear algebra can be used to manage and index large text collections

anisotropic diffusion of multivalued images with applications to color filtering a general framework for anisotropic diffusion of multivalued images is presented we propose an evolution equation where at each point in time the directions and magnitudes of the maximal and minimal rate of change in the vector image are first evaluated these are given by eigenvectors and eigenvalues of the first fundamental form in the given image metric then the image diffuses via a system of coupled differential equations in the direction of minimal change the diffusion strength is controlled by a function that measures the degree of dissimilarity between the eigenvalues we apply the proposed framework to the filtering of color images represented in cie l a b space

generalized quasi variational like inclusions with nonconvex functionals in this paper two new concepts of h subdifferential and h proximal mappings of a proper functional is introduced in hilbert spaces the existence and lipschitz continuity of h proximal mapping of a proper functional are proved by applying these concepts we introduce and study a class of generalized quasi variational like inclusions with a nonconvex functional a new iterative algorithms for finding the approximate solutions and the convergence criteria of the iterative sequences generated by the algorithm are also given these algorithm and existence result generalize many known results in the literature

numerical methods for electronic structure calculations of materials the goal of this article is to give an overview of numerical problems encountered when determining the electronic structure of materials and the rich variety of techniques used to solve these problems the paper is intended for a diverse scientific computing audience for this reason we assume the reader does not have an extensive background in the related physics our overview focuses on the nature of the numerical problems to be solved their origin and the methods used to solve the resulting linear algebra or nonlinear optimization problems it is common knowledge that the behavior of matter at the nanoscale is in principle entirely determined by the schrodinger equation in practice this equation in its original form is not tractable successful but approximate versions of this equation which allow one to study nontrivial systems took about five or six decades to develop in particular the last two decades saw a flurry of activity in developing effective software one of the main practical variants of the schrodinger equation is based on what is referred to as density functional theory dft the combination of dft with pseudopotentials allows one to obtain in an efficient way the ground state configuration for many materials this article will emphasize pseudopotential density functional theory but other techniques will be discussed as well

unconstrained energy functionals for electronic structure calculations the performance of conjugate gradient schemes for minimizing unconstrained energy functionals in the context of condensed matter electronic structure density functional calculations is studied the unconstrained functionals allow a straightforward application of conjugate gradients by removing the explicit orthonormality constraints on the quantum mechanical wave functions however the removal of the constraints can lead to slow convergence in particular when preconditioning is used the convergence properties of two previously suggested energy functionals are analyzed and a new functional is proposed which unifies some of the advantages of the other functionals a numerical example derived from a diamond crystal confirms the analysis

viscosity approximation methods for equilibrium problems and fixed point problems in hilbert spaces abstract in this paper we introduce an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a hilbert space then we prove a strong convergence theorem which is connected with combettes and hirstoaga s result p l combettes s a hirstoaga equilibrium programming in hilbert spaces j nonlinear convex anal 6 2005 117 136 and wittmann s result r wittmann approximation of fixed points of nonexpansive mappings arch math 58 1992 486 491 using this result we obtain two corollaries which improve and extend their results

viscosity approximation methods for generalized equilibrium problems and fixed point problems with applications abstract in this paper we introduce a general iterative scheme for finding a common element of the set of common solutions of generalized equilibrium problems the set of common fixed points of a family of infinite non expansive mappings strong convergence theorems are established in a real hilbert space under suitable conditions as some applications we consider convex feasibility problems and equilibrium problems the results presented improve and extend the corresponding results of many others

diffusion of general data on non flat manifolds viaharmonic maps theory the direction diffusion case in a number of disciplines directional data provides a fundamental source of information a novel framework for isotropic and anisotropic diffusion of directions is presented in this paper the framework can be applied both to denoise directional data and to obtain multiscale representations of it the basic idea is to apply and extend results from the theory of harmonic maps and in particular harmonic maps in liquid crystals this theory deals with the regularization of vectorial data while satisfying the intrinsic unit norm constraint of directional data we show the corresponding variational and partial differential equations formulations for isotropic diffusion obtained from an l 2 norm and edge preserving diffusion obtained from an l norm in general and an l 1 norm in particular in contrast with previous approaches the framework is valid for directions in any dimensions supports non smooth data and gives both isotropic and anisotropic formulations in addition the framework of harmonic maps here described can be used to diffuse and analyze general image data defined on general non flat manifolds that is functions between two general manifolds we present a number of theoretical results open questions and examples for gradient vectors optical flow and color images

equilibrium programming using proximal like algorithms we compute constrained equilibria satisfying an optimality condition important examples include convex programming saddle problems noncooperative games and variational inequalities under a monotonicity hypothesis we show that equilibrium solutions can be found via iterative convex minimization in the main algorithm each stage of computation requires two proximal steps possibly using bregman functions one step serves to predict the next point the other helps to correct the new prediction to enhance practical applicability we tolerate numerical errors

preconditioning constrained eigenvalue problems abstract the purpose of our paper is to introduce a robust preconditioning scheme for the numerical solution of the leftmost eigenvalues and corresponding eigenvectors of a constrained eigenvalue problem this constrained eigenvalue problem is congruent to a nonsymmetric eigenvalue problem with nontrivial jordan blocks associated with infinite eigenvalues the proposed preconditioning scheme is relevant to the application of krylov subspace methods and preconditioned eigensolvers the two key results are a semi orthogonal decomposition and a transformation process that implicitly combines a preconditioning step followed by abstract projection onto the subspace associated with the finite eigenvalues numerical results demonstrate the effectiveness of the preconditioning scheme

some new algorithms for solving mixed equilibrium problems in this paper we suggest and analyze two projection methods one implicit and one explicit for finding a particular solution of a mixed equilibrium problem in a real hilbert space furthermore we prove that the proposed projection methods converge strongly to a particular solution of the mixed equilibrium problem

a neuro fuzzy system modeling with self constructing rule generationand hybrid svd based learning we propose an approach for neuro fuzzy system modeling a neuro fuzzy system for a given set of input output data is obtained in two steps first the data set is partitioned automatically into a set of clusters based on input similarity and output similarity tests membership functions associated with each cluster are defined according to statistical means and variances of the data points included in the cluster then a fuzzy if then rule is extracted from each cluster to form a fuzzy rule base second a fuzzy neural network is constructed accordingly and parameters are refined to increase the precision of the fuzzy rule base to decrease the size of the search space and to speed up the convergence we develop a hybrid learning algorithm which combines a recursive singular value decomposition based least squares estimator and the gradient descent method the proposed approach has advantages of determining the number of rules automatically and matching membership functions closely with the real distribution of the training data points besides it learns faster consumes less memory and produces lower approximation errors than other methods

convergence theorems of common elements for equilibrium problems and fixed point problems in banach spaces the purpose of this paper is to introduce hybrid projection algorithms for finding a common element of the set of common fixed points of two quasi f nonexpansive mappings and the set of solutions of an equilibrium problem in the framework of banach spaces our results improve and extend the corresponding results announced by many others

mixed equilibrium problems and optimization problems in this paper we introduce and analyze a new hybrid iterative algorithm for finding a common element of the set of solutions of mixed equilibrium problems and the set of fixed points of an infinite family of nonexpansive mappings furthermore we prove some strong convergence theorems for the hybrid iterative algorithm under some mild conditions we also discuss some special cases results obtained in this paper improve the previously known results in this area

on correction equations and domain decomposition for computing invariant subspaces by considering the eigenvalue problem as a system of nonlinear equations it is possible to develop a number of solution schemes which are related to the newton iteration for example to compute eigenvalues and eigenvectors of an n n matrix a the davidson and the jacobi davidson techniques construct good basis vectors by approximately solving a correction equation which provides a correction to be added to the current approximation of the sought eigenvector that equation is a linear system with the residual r of the approximated eigenvector as right hand side one of the goals of this paper is to extend this general technique to the block situation i e the case where a set of p approximate eigenpairs is available in which case the residual r becomes an n p matrix as will be seen solving the correction equation in block form requires solving a sylvester system of equations the paper will define two algorithms based on this approach for symmetric real matrices the first algorithm converges quadratically and the second cubically a second goal of the paper is to consider the class of substructuring methods such as the component mode synthesis cms and the automatic multi level substructuring amls methods and to view them from the angle of the block correction equation in particular this viewpoint allows us to define an iterative version of well known one level substructuring algorithms cms or one level amls experiments are reported to illustrate the convergence behavior of these methods 2006 published by elsevier b v

symmetric convolution and the discrete sine and cosine transforms this paper discusses the use of symmetric convolution and the discrete sine and cosine transforms dsts and dcts for general digital signal processing the operation of symmetric convolution is a formalized approach to convolving symmetrically extended sequences the result is the same as that obtained by taking an inverse discrete trigonometric transform dtt of the product of the forward dtts of those two sequences there are 16 members in the family of dtts each provides a representation for a corresponding distinct type of symmetric periodic sequence the author defines symmetric convolution relates the dsts and dcts to symmetric periodic sequences and then use these principles to develop simple but powerful convolution multiplication properties for the entire family of dsts and dcts symmetric convolution can be used for discrete linear filtering when the filter is symmetric or antisymmetric the filtering will be efficient because fast algorithms exist for all versions of the dtts conventional linear convolution is possible if one first zero pad the input data symmetric convolution and its fast implementation using dtts are now an alternative to circular convolution and the dft

on sufficiency of the kuhn tucker conditions if all the functions fo x f x f x are convex on c then these conditions are also sufficient various classes of functions have been defined for the purpose of weakening this limitation of convexity in mathematical programming mangasarian 11 has speculated that pseudo convexity of x and quasiconvexity off x are the weakest conditions that can be imposed so that the above conditions are sufficient for optimality it will be shown that there are other wide classes of functions for which the conditions are sufficient 545

some quantum integral inequalities via preinvex functions in this paper we obtain some new quantum analogues of hermite hadamard and iyengar type inequalities for some classes of preinvex functions some special cases are also discussed

monte carlo sure a black box optimization of regularization parameters for general denoising algorithms we consider the problem of optimizing the parameters of a given denoising algorithm for restoration of a signal corrupted by white gaussian noise to achieve this we propose to minimize stein s unbiased risk estimate sure which provides a means of assessing the true mean squared error mse purely from the measured data without need for any knowledge about the noise free signal specifically we present a novel monte carlo technique which enables the user to calculate sure for an arbitrary denoising algorithm characterized by some specific parameter setting our method is a black box approach which solely uses the response of the denoising operator to additional input noise and does not ask for any information about its functional form this therefore permits the use of sure for optimization of a wide variety of denoising algorithms we justify our claims by presenting experimental results for sure based optimization of a series of popular image denoising algorithms such as total variation denoising wavelet soft thresholding and wiener filtering smoothing splines in the process we also compare the performance of these methods we demonstrate numerically that sure computed using the new approach accurately predicts the true mse for all the considered algorithms we also show that sure uncovers the optimal values of the parameters in all cases

a class of nonconvex functions and pre variational inequalities abstract a class of nonconvex functions is introduced called semi preinvex function which includes the classes of preinvex functions and arc connected convex functions the fritz john conditions of the mathematical programming problem are derived for these kinds of functions the pre variational inequality is given as a necessary condition and also a sufficient condition for a mathematical programming for invex functions the type i function related to unconstrained problems is given as an equivalent form of the pre variational inequality existence theorems for the solution of the pre variational inequality are also proved

viscosity approximation methods for fixed points problems the aim of this work is to propose viscosity approximation methods which amounts to selecting a particular fixed point of a given nonexpansive self mapping

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

absolute and relative weyl theorems for generalized eigenvalue problems abstract weyl type eigenvalue perturbation theories are derived for hermitian definite pencils a b in which b is positive definite the results provide a one to one correspondence between the original and perturbed eigenvalues and give a uniform perturbation bound we give both absolute and relative perturbation results defined in the standard euclidean metric instead of the chordal metric that is often used

predictor corrector methods for multi valued hemiequilibrium problems abstract in this paper we introduce and analyze a new class of equilibrium problems known as multi valued hemiequilibrium problems it is shown that this class includes hemiequilibrium problems hemivariational inequalities and complementarity problems as special cases we use the auxiliary principle techniques to suggest some predictor corrector type iterative methods for solving multi valued hemiequilibrium problems we also analyze the convergence analysis of these new iterative methods under some mild conditions

equilibrium problems with set valued mappings in banach spaces abstract in this paper we consider two classes of equilibrium problems which imply general pseudomonotonicity and general semipseudomonotonicity relative to a set valued map on reflexive and nonreflexive banach spaces respectively using the classical kkm technique we obtain some existence results under boundedness as well as under unboundedness assumptions some examples are given

on invex sets and preinvex functions abstract in this paper we consider the class of the invex function introduced by hanson we show that under certain condition an invex function defined on an invex set a is preinvex on a similarly a quasiinvex function defined on an invex set a is pre quasiinvex

a general iterative method for addressing mixed equilibrium problems and optimization problems abstract in this paper we introduce a new general iterative method for finding a common element of the set of solutions of a mixed equilibrium problem mep the set of fixed points of an infinite family of nonexpansive mappings t n n 1 and the set of solutions of variational inequalities for a inverse strongly monotone mapping in hilbert spaces furthermore we establish the strong convergence theorem for the iterative sequence generated by the proposed iterative algorithm under some suitable conditions which solves some optimization problems our results extend and improve the recent results of yao et al y yao m a noor s zainab y c liou mixed equilibrium problems and optimization problems j math anal appl 354 2009 319 329 y yao m a noor y c liou on iterative methods for equilibrium problems nonlinear anal 70 1 2009 479 509 and many others

restoration of poissonian images using alternating direction optimization much research has been devoted to the problem of restoring poissonian images namely for medical and astronomical applications however the restoration of these images using state of the art regularizers such as those based upon multiscale representations or total variation is still an active research area since the associated optimization problems are quite challenging in this paper we propose an approach to deconvolving poissonian images which is based upon an alternating direction optimization method the standard regularization or maximum a posteriori map restoration criterion which combines the poisson log likelihood with a nonsmooth convex regularizer log prior leads to hard optimization problems the log likelihood is nonquadratic and nonseparable the regularizer is nonsmooth and there is a nonnegativity constraint using standard convex analysis tools we present sufficient conditions for existence and uniqueness of solutions of these optimization problems for several types of regularizers total variation frame based analysis and frame based synthesis we attack these problems with an instance of the alternating direction method of multipliers admm which belongs to the family of augmented lagrangian algorithms we study sufficient conditions for convergence and show that these are satisfied either under total variation or frame based analysis and synthesis regularization the resulting algorithms are shown to outperform alternative state of the art methods both in terms of speed and restoration accuracy

the use of trust regions in kohn sham total energyminimization the self consistent field scf iteration widely used forcomputing the ground state energy and the corresponding single particlewave functions associated with a many electronatomistic system is viewedin this paper as an optimization procedure that minimizes the kohn shamtotal energy indirectly by minimizing a sequence of quadratic surrogatefunctions we point out the similarity and difference between the totalenergy and the surrogate and show how the scf iteration can fail whenthe minimizer of the surrogate produces an increase in the ks totalenergy a trust region technique is introduced as a way to restrict theupdate of the wave functions within a small neighborhood of anapproximate solution at which the gradient of the total energy agreeswith that of the surrogate the use of trust region in scf is not new however it has been observed that directly applying a trust region basedscf trscf to the kohn sham total energy often leads to slowconvergence we propose to use trscf within a direct constrainedminimization dcm algorithm we developed in cite dcm the keyingredients of thedcm algorithm involve projecting the total energyfunction into a sequence of subspaces of small dimensions and seeking theminimizerof the total energy function within each subspace theminimizer of a subspace energy function which is computed by trscf notonly provides amore search direction along which the ks total energy functiondecreases but also gives an optimal step length that yields asufficient decrease in total energy a numerical example is provided todemonstrate that the combination of trscf and dcm is more efficient thanscf less

trace minimization principles for positive semi definite pencils abstract this paper is concerned with x h ax subject to x h bx j for a hermitian matrix pencil a b where j is diagonal and j 2 i the identity matrix of apt size the same problem was investigated earlier by kovac striko and veselic linear algebra appl 216 1995 139 158 for the case in which b is assumed nonsingular but in this paper b is no longer assumed nonsingular and in fact a b is even allowed to be a singular pencil it is proved among others that the infimum is finite if and only if a b is a positive semi definite pencil in the sense that there is a real number 0 such that a 0 b is positive semi definite the infimum when finite can be expressed in terms of the finite eigenvalues of a b sufficient and necessary conditions for the attainability of the infimum are also obtained

pre invex functions in multiple objective optimization and called such functions convexlike if s is a convex set and if f is a convex function then clearly f is convexlike elster and nehse obtained a saddlepoint optimality condition for convexlike mathematical programs hayashi and komiya 9 also considered convexlike functions and developed a gordan type theorem of the alternative involving convexlike functions and in addition considered lagrangian duality for convexlike programs hanson 7 considered differentiable functions f s r for which there exists an n dimensional vector function q q u such that for all x u e s

on the modification of an eigenvalue problem that preserves an eigenspace eigenvalue problems arise in many application areas ranging from computational fluid dynamics to information retrieval in these fields we are often interested in only a few eigenvalues and corresponding eigenvectors of a sparse matrix in this paper we comment on the modifications of the eigenvalue problem that can simplify the computation of those eigenpairs these transformations allow us to avoid difficulties associated with non hermitian eigenvalue problems such as the lack of reliable non hermitian eigenvalue solvers by mapping them into generalized hermitian eigenvalue problems also they allow us to expose and explore parallelism they require knowledge of a selected eigenvalue and preserve its eigenspace the positive definiteness of the hermitian part is inherited by the matrices in the generalized hermitian eigenvalue problem the position of the selected eigenspace in the ordering of the eigenvalues is also preserved under certain conditions the effect of using approximate eigenvalues in the transformation is analyzed and numerical experiments are presented

a variable penalty alternating directions method for convex optimization we study a generalized version of the method of alternating directions as applied to the minimization of the sum of two convex functions subject to linear constraints the method consists of solving consecutively in each iteration two optimization problems which contain in the objective function both lagrangian and proximal terms the minimizers determine the new proximal terms and a simple update of the lagrangian terms follows we prove a convergence theorem which extends existing results by relaxing the assumption of uniqueness of minimizers another novelty is that we allow penalty matrices and these may vary per iteration this can be beneficial in applications since it allows additional tuning of the method to the problem and can lead to faster convergence relative to fixed penalties as an application we derive a decomposition scheme for block angular optimization and present computational results on a class of dual block angular problems

a trust region direct constrained minimization algorithm for the kohn sham equation the self consistent field scf iteration widely used for computing the ground state energy and the corresponding single particle wave functions associated with a many electron atomistic system is viewed in this paper as an optimization procedure that minimizes the kohn sham ks total energy indirectly by minimizing a sequence of quadratic surrogate functions we point out the similarity and difference between the total energy and the surrogate and show how the scf iteration can fail when the minimizer of the surrogate produces an increase in the ks total energy a trust region technique is introduced as a way to restrict the update of the wave functions within a small neighborhood of an approximate solution at which the gradient of the total energy agrees with that of the surrogate the use of trust regions in scf is not new however it has been observed that directly applying a trust region based scf trscf to the ks total energy often leads to slow convergence we propose to use trscf within a direct constrained minimization dcm algorithm we developed in j comput phys 217 2006 pp 709 721 the key ingredients of the dcm algorithm involve projecting the total energy function into a sequence of subspaces of small dimensions and seeking the minimizer of the total energy function within each subspace the minimizer of a subspace energy function which is computed by the trscf not only provides a search direction along which the ks total energy function decreases but also gives an optimal step length that yields a sufficient decrease in total energy a numerical example is provided to demonstrate that the combination of trscf and dcm is more efficient than scf

on strongly α preinvex functions in this paper by means of a series of counterexamples we study in a systematic way the relationships among pseudo quasi preinvexity strict strong pseudo quasi invexity and strict strong pseudo quasi monotonicity results obtained in this paper can be viewed as a refinement and improvement of the results of noor and noor m a noor k i noor some characterizations of strongly preinvex functions j math anal appl 316 2006 697 706