dimension reduction and coefficient estimation in multivariate linear regression summary we introduce a general formulation for dimension reduction and coefficient estimation in the multivariate linear model we argue that many of the existing methods that are commonly used in practice can be formulated in this framework and have various restrictions we continue to propose a new method that is more flexible and more generally applicable the method proposed can be formulated as a novel penalized least squares estimate the penalty that we employ is the coefficient matrix s ky fan norm such a penalty encourages the sparsity among singular values and at the same time gives shrinkage coefficient estimates and thus conducts dimension reduction and coefficient estimation simultaneously in the multivariate linear model we also propose a generalized cross validation type of criterion for the selection of the tuning parameter in the penalized least squares simulations and an application in financial econometrics demonstrate competitive performance of the new method an extension to the non parametric factor model is also discussed

iterative approaches to convex feasibility problems in banach spaces abstract the convex feasibility problem cfp of finding a point in the nonempty intersection i 1 n c i is considered where n 1 is an integer and each c i is assumed to be the fixed point set of a nonexpansive mapping t i x x with x a banach space it is shown that the iterative scheme x n 1 n 1 y 1 n 1 t n 1 x n is strongly convergent to a solution of cfp provided the banach space x either is uniformly smooth or is reflexive and has a weakly continuous duality map and provided the sequence n satisfies certain conditions the limit of x n is located as q y where q is the sunny nonexpansive retraction from x onto the common fixed point set of the t i s

convergence of the modified mann s iteration method for asymptotically strict pseudo contractions abstract let c be a closed convex subset of a real hilbert space h and assume that t is an asymptotically strict pseudo contraction on c with a fixed point for some 0 1 given an initial guess x 0 c and given also a real sequence n in 0 1 the modified mann s algorithm generates a sequence x n via the formula x n 1 n x n 1 n t n x n n 0 it is proved that if the control sequence n is chosen so that n 1 for some 0 1 then x n converges weakly to a fixed point of t we also modify this iteration method by applying projections onto suitably constructed closed convex sets to get an algorithm which generates a strongly convergent sequence

a self adaptive projection and contraction algorithm for the traffic assignment problem with path specific costs abstract this paper considers solving a special case of the nonadditive traffic equilibrium problem presented by gabriel and bernstein transportation science 31 4 1997 337 348 in which the cost incurred on each path is made up of the sum of the arc travel times plus a path specific cost for traveling on that path a self adaptive projection and contraction method is suggested to solve the path specific cost traffic equilibrium problem which is formulated as a nonlinear complementarity problem ncp the computational effort required per iteration is very modest it consists of only two function evaluations and a simple projection on the nonnegative orthant a self adaptive technique is embedded in the projection and contraction method to find suitable scaling factor without the need to do a line search the method is simple and has the ability to handle a general monotone mapping f numerical results are provided to demonstrate the features of the projection and contraction method

on projection algorithms for solving convex feasibility problems due to their extraordinary utility and broad applicability in many areas of classical mathematics and modern physical sciences most notably computerized tomography algorithms for solving convex feasibility problems continue to receive great attention to unify generalize and review some of these algorithms a very broad and flexible framework is investigated several crucial new concepts which allow a systematic discussion of questions on behaviour in general hilbert spaces and on the quality of convergence are brought out numerous examples are given

convergence theorems for common fixed points for finite families of nonexpansive mappings in reflexive banach spaces let e be a real reflexive banach space with a uniformly g teaux differentiable norm let k be a nonempty closed convex subset of e suppose that every nonempty closed convex bounded subset of k has the fixed point property for nonexpansive mappings let t1 t2 tn be a family of nonexpansive self mappings of k with f i 1nfix ti 0 f fix tntn 1 t1 fix t1tn t2 fix tn 1tn 2 t1tn let n be a sequence in 0 1 satisfying the following conditions c1 lim n 0 c2 n for a fixed 0 1 define sn k k by snx 1 x tnx x k where tn tnmodn for an arbitrary fixed u x0 k let b x k tntn 1 t1x x 1 u for some 1 be bounded and let the sequence xn be defined iteratively by xn 1 n 1u 1 n 1 sn 1xn for n 0 assume that limn tnxn tn 1xn 0 then xn converges strongly to a common fixed point of the family t1 t2 tn this convergence theorem is also proved for non self maps

a new method for a class of linear variational inequalities in this paper we introduce a new iterative scheme for the numerical solution of a class of linear variational inequalities each iteration of the method consists essentially only of a projection to a closed convex set and two matrix vector multiplications both the method and the convergence proof are very simple

approximation methods for common fixed points for a countable family of nonexpansive mappings abstract let e l p or l p space 1 p let k be a closed convex and nonempty subset of e let t i i 1 be a family of nonexpansive self mappings of k for arbitrary fixed 0 1 define a family of nonexpansive maps s i i 1 by s i 1 i t i where i is the identity map of k let f i 1 f t i 0 it is proved that the iterative sequence x n defined by x 0 k x n 1 n u i 1 i t n s i x n n 0 converges strongly to a common fixed point of the family t i i 1 where n and i t n are sequences in 0 1 satisfying appropriate conditions in each of the following cases a e l p 1 p and b e l p 1 p and at least one of the maps t i s is demicompact our theorems extend the results of p mainge approximation methods for common fixed points of nonexpansive mappings in hilbert space j math anal appl 325 2007 469 479 from hilbert spaces to l p spaces 1 p

gibbs posterior for variable selection in high dimensional classification and data mining in the popular approach of bayesian variable selection bvs one uses prior and posterior distributions to select a subset of candidate variables to enter the model a completely new direction will be considered here to study bvs with a gibbs posterior originating in statistical mechanics the gibbs posterior is constructed from a risk function of practical interest such as the classification error and aims at minimizing a risk function without modeling the data probabilistically this can improve the performance over the usual bayesian approach which depends on a probability model which may be misspecified conditions will be provided to achieve good risk performance even in the presence of high dimensionality when the number of candidate variables k can be much larger than the sample size n in addition we develop a convenient markov chain monte carlo algorithm to implement bvs with the gibbs posterior

viscosity approximation to common fixed points of a nonexpansive semigroup with a generalized contraction mapping abstract the purpose of this paper is to introduce and construct the implicit and explicit viscosity iterative processes by a generalized contraction mapping f and a nonexpansive semigroup t t t 0 and to prove that under suitable conditions these iterative processes converge strongly to a unique common fixed point of t t t 0 in reflexive banach spaces which admits a weakly sequentially continuous duality mapping

projection methods algorithms and a new system of nonlinear variational inequalities abstract the convergence of projection methods is based on a new iterative algorithm for the approximation solvability of the following system of nonlinear variational inequalities snvi determine elements x y k such that t y x y x x 0 for all x k and for 0 and t x y x x y 0 for all x k and for 0 where t k h is a mapping from a nonempty closed convex subset k of a real hilbert space h into h this new class of generalized nonlinear variational inequalities reduces to standard class of nonlinear variational inequalities which are widely studied and applied to various problems arising from mathematical sciences optimization and control theory and other related fields

a class of projection and contraction methods for asymmetric linear variational inequalities and their relations to fukushima s descent method for solving asymmetric linear variational inequalities we present a class of projection and contraction methods under the general g norm the search direction of our methods is just a convex combination of two descent directions of fukushima s merit function however we use the direction to reduce the distance function 12 u u 2g where is a solution point of the problem finally we report some numerical results for spatial price equilibrium problems by using the presented methods

a general iterative method for an infinite family of nonexpansive mappings abstract let h be a real hilbert space consider the iterative sequence x n 1 n f x n n x n 1 n i n a w n x n where 0 is some constant f h h is a given contractive mapping a is a strongly positive bounded linear operator on h and w n is the w mapping generated by an infinite countable family of nonexpansive mappings t 1 t 2 t n and 1 2 n such that the common fixed points set f n 1 fix t n 0 under very mild conditions on the parameters we prove that x n converges strongly to p f where p is the unique solution in f of the following variational inequality a f p p x 0 for all x f which is the optimality condition for the minimization problem min x f 1 2 a x x h x

weak and strong convergence theorems for strict pseudo contractions in hilbert spaces abstract let c be a closed convex subset of a real hilbert space h and assume that t is a strict pseudo contraction on c with a fixed point for some 0 1 given an initial guess x 0 c and given also a real sequence n in 0 1 the mann s algorithm generates a sequence x n by the formula x n 1 n x n 1 n t x n n 0 it is proved that if the control sequence n is chosen so that n 1 and n 0 n 1 n then x n converges weakly to a fixed point of t however this convergence is in general not strong we then modify mann s algorithm by applying projections onto suitably constructed closed convex sets to get an algorithm which generates a strong convergent sequence this result extends a recent result of nakajo and takahashi k nakajo w takahashi strong convergence theorems for nonexpansive mappings and nonexpansive semigroups j math anal appl 279 2003 372 379 from nonexpansive mappings to strict pseudo contractions

the choice of variables in multivariate regression a non conjugate bayesian decision theory approach we consider the choice of explanatory variables in multivariate linear regression our approach balances prediction accuracy against costs attached to variables in a multivariate version of a decision theory approach pioneered by lindley 1968 we also employ a non conjugate proper prior distribution for the parameters of the regression model extending the standard normal inverse wishart by adding a component of error which is unexplainable by any number of predictor variables thus avoiding the determinism identified by dawid 1988 simulated annealing and fast updating algorithms are used to search for good subsets when there are very many regressors the technique is illustrated on a near infrared spectroscopy example involving 39 observations and 300 explanatory variables this demonstrates the effectiveness of multivariate regression as opposed to separate univariate regressions it also emphasises that within a bayesian framework more variables than observations can be utilised

solving a class of matrix minimization problems by linear variational inequality approaches abstract a class of matrix optimization problems can be formulated as a linear variational inequalities with special structures for solving such problems the projection and contraction method pc method is extended to variational inequalities with matrix variables then the main costly computational load in pc method is to make a projection onto the semi definite cone exploiting the special structures of the relevant variational inequalities the levenberg marquardt type projection and contraction method is advantageous preliminary numerical tests up to 1000 1000 matrices indicate that the suggested approach is promising

a neural network model for monotone linear asymmetric variational inequalities a linear variational inequality is a uniform approach for some important problems in optimization and equilibrium problems we give a neural network model for solving asymmetric linear variational inequalities the model is based on a simple projection and contraction method computer simulation is performed for linear programming lp and linear complementarity problems lcp the test results for the lp problem demonstrate that our model converges significantly faster than the three existing neural network models examined in a comparative study paper

prediction via orthogonalized model mixing abstract we introduce an approach and algorithms for model mixing in large prediction problems with correlated predictors we focus on the choice of predictors in linear models and mix over possible subsets of candidate predictors our approach is based on expressing the space of models in terms of an orthogonalization of the design matrix advantages are both statistical and computational statistically orthogonalization often leads to a reduction in the number of competing models by eliminating correlations computationally large model spaces cannot be enumerated recent approaches are based on sampling models with high posterior probability via markov chains based on orthogonalization of the space of candidate predictors we can approximate the posterior probabilities of models by products of predictor specific terms this leads to an importance sampling function for sampling directly from the joint distribution over the model space without resorting to markov chains compared to the latter ortho

a new polynomial time method for a linear complementarity problem the purpose of this paper is to present a new polynomial time method for a linear complementarity problem with a positive semi definite matrix the method follows a sequence of points if we generate the sequence on a path we can construct a path following method and if we generate the sequence based on a potential function we can construct a potential reduction method the method has the advantage that it requires at most o sqrt n l iterations for any initial feasible point whose components lie between 2 o l and 2o l

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

bayes model averaging with selection of regressors when a number of distinct models contend for use in prediction the choice of a single model can offer rather unstable predictions in regression stochastic search variable selection with bayesian model averaging offers a cure for this robustness issue but at the expense of requiring very many predictors here we look at bayes model averaging incorporating variable selection for prediction this offers similar mean square errors of prediction but with a vastly reduced predictor space this can greatly aid the interpretation of the model it also reduces the cost if measured variables have costs the development here uses decision theory in the context of the multivariate general linear model in passing this reduced predictor space bayes model averaging is contrasted with single model approximations a fast algorithm for updating regressions in the markov chain monte carlo searches for posterior inference is developed allowing many more variables than observations to be contemplated we discuss the merits of absolute rather than proportionate shrinkage in regression especially when there are more variables than observations the methodology is illustrated on a set of spectroscopic data used for measuring the amounts of different sugars in an aqueous solution

a proximal gradient algorithm for decentralized composite optimization this paper proposes a decentralized algorithm for solving a consensus optimization problem defined in a static networked multi agent system where the local objective functions have the smooth nonsmooth composite form examples of such problems include decentralized constrained quadratic programming and compressed sensing problems as well as many regularization problems arising in inverse problems signal processing and machine learning which have decentralized applications this paper addresses the need for efficient decentralized algorithms that take advantages of proximal operations for the nonsmooth terms we propose a proximal gradient exact first order algorithm pg extra that utilizes the composite structure and has the best known convergence rate it is a nontrivial extension to the recent algorithm extra at each iteration each agent locally computes a gradient of the smooth part of its objective and a proximal map of the nonsmooth part as well as exchanges information with its neighbors the algorithm is exact in the sense that an exact consensus minimizer can be obtained with a fixed step size whereas most previous methods must use diminishing step sizes when the smooth part has lipschitz gradients pg extra has an ergodic convergence rate of o left 1 over k right in terms of the first order optimality residual when the smooth part vanishes pg extra reduces to p extra an algorithm without the gradients so no g in the name which has a slightly improved convergence rate at o left 1 over k right in a standard non ergodic sense numerical experiments demonstrate effectiveness of pg extra and validate our convergence results

optimal bayesian design by inhomogeneous markov chain simulation we consider decision problems defined by a utility function and an underlying probability model for all unknowns the utility function quantifies the decision maker s preferences over consequences the optimal decision maximizes the expected utility function where the expectation is taken with respect to all unknowns that is future data and parameters in many problems the solution is not analytically tractable for example the utility function might involve moments that can be computed only by numerical integration or simulation also the nature of the decision space i e the set of all possible actions might have a shape or dimension that complicates the maximization the motivating application for this discussion is the choice of a monitoring network when the optimization is performed over the high dimensional set of all possible locations of monitoring stations possibly including choice of the number of locations we propose an approach to optimal bayesian design based on inhomogeneous markov

solving a class of asymmetric variational inequalities by a new alternating direction method abstract the augmented lagrangian method also referred to as an alternating direction method solves a class of variational inequalities vi via solving a series of sub vi problems the method is effective whenever the subproblems can be solved efficiently however the subproblem to be solved in each iteration of the augmented lagrangian method itself is still a vi problem it is essentially as difficult as the original one the only difference is that the dimension of the subproblems is lower in this paper we propose a new alternating direction method for solving a class of monotone variational inequalities in each iteration the method solves a convex quadratic programming with simple constrains and a well conditioned system of nonlinear equations for such easier subproblems existing efficient numerical softwares are applicable the effectiveness of the proposed method is demonstrated with an illustrative example

strong convergence to common fixed points of families of nonexpansive mappings abstract we introduce iteration schemes for families of nonexpansive mappings in hilbert spaces and prove that the iterates converge strongly to common fixed points of the mappings

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

bayesian variable selection in linear regression abstract this article is concerned with the selection of subsets of predictor variables in a linear regression model for the prediction of a dependent variable it is based on a bayesian approach intended to be as objective as possible a probability distribution is first assigned to the dependent variable through the specification of a family of prior distributions for the unknown parameters in the regression model the method is not fully bayesian however because the ultimate choice of prior distribution from this family is affected by the data it is assumed that the predictors represent distinct observables the corresponding regression coefficients are assigned independent prior distributions for each regression coefficient subject to deletion from the model the prior distribution is a mixture of a point mass at 0 and a diffuse uniform distribution elsewhere that is a spike and slab distribution the random error component is assigned a normal distribution with mean 0 and standard deviation

solving the bicriteria traffic equilibrium problem with variable demand and nonlinear path costs in this paper we present an algorithm for solving the bicriteria traffic equilibrium problem with variable demand and nonlinear path costs the path cost function considered is comprised of two attributes travel time and toll that are combined into a nonlinear generalized cost travel demand is determined endogenously according to a travel disutility function travelers choose routes with the minimum overall generalized costs the algorithm involves two components a bicriteria shortest path routine to implicitly generate the set of non dominated paths and a projection and contraction method to solve the nonlinear complementarity problem ncp describing the traffic equilibrium problem numerical experiments are conducted to demonstrate the feasibility of the algorithm to this class of traffic equilibrium problems

a new merit function for nonlinear complementarity problems and a related algorithm we investigate the properties of a new merit function which allows us to reduce a nonlinear complementarity problem to an unconstrained global minimization one assuming that the complementarity problem is defined by a p 0 function we prove that every stationary point of the unconstrained problem is a global solution furthermore if the complementarity problem is defined by a uniform p function the level sets of the merit function are bounded the properties of the new merit function are compared with those of mangasarian solodov s implicit lagrangian and fukushima s regularized gap function we also introduce a new simple active set local method for the solution of complementarity problems and show how this local algorithm can be made globally convergent by using the new merit function

coefficient sign prediction methods for model selection summary we consider a bayesian model selection strategy based on predicting the signs of the coefficients in a regression model i e we consider identification of coefficients in a full or encompassing model for which we can confidently predict whether they are positive or negative this is useful when our main purpose in doing model selection is interpretation since the sign of a coefficient is often of primary importance for this task in the case of a linear model with standard non informative prior we connect our sign coefficient prediction approach to the classical zheng loh procedure for model selection one advantage of our approach is that only specification of a prior on the full model is required unlike standard bayesian variable selection approaches which require specification of prior distributions on parameters in all submodels and specification of a prior on the model itself we consider applying our method with proper hierarchical shrinkage priors which makes the procedure more useful in large p small n regression problems with more predictors than observations and in problems involving multicollinearity in these problems we may wish to do prediction by using shrinkage methods in the full model but interpreting which variables are important is also of interest we compare selection by using our coefficient sign prediction approach with the recently proposed elastic net procedure of zou and hastie and observe that our method shares some of the features of the elastic net such as a group selection property the method can be extended to more complex model selection problems such as selection on variance components in random effects models for selection on variance components where the parameter of interest is non negative and hence prediction of the sign of the parameter not the appropriate way to proceed we consider instead prediction of the sign of the score component for the parameter at zero obtaining a method that is related to classical score tests on variance components

bayesian wavelet regression on curves with application to a spectroscopic calibration problem motivated by calibration problems in near infrared nir spectroscopy we consider the linear regression setting in which the many predictor variables arise from sampling an essentially continuous curve at equally spaced points and there may be multiple predictands we tackle this regression problem by calculating the wavelet transforms of the discretized curves then applying a bayesian variable selection method using mixture priors to the multivariate regression of predictands on wavelet coefficients for prediction purposes we average over a set of likely models applied to a particular problem in nir spectroscopy this approach was able to find subsets of the wavelet coefficients with overall better predictive performance than the more usual approaches in the application the available predictors are measurements of the nir reflectance spectrum of biscuit dough pieces at 256 equally spaced wavelengths the aim is to predict the composition i e the fat flour sugar and water content of the doug

bayesian model averaging for linear regression models abstract we consider the problem of accounting for model uncertainty in linear regression models conditioning on a single selected model ignores model uncertainty and thus leads to the underestimation of uncertainty when making inferences about quantities of interest a bayesian solution to this problem involves averaging over all possible models i e combinations of predictors when making inferences about quantities of interest this approach is often not practical in this article we offer two alternative approaches first we describe an ad hoc procedure occam s window which indicates a small set of models over which a model average can be computed second we describe a markov chain monte carlo approach that directly approximates the exact solution in the presence of model uncertainty both of these model averaging procedures provide better predictive performance than any single model that might reasonably have been selected in the extreme case where there are many candidate predictors but

approximation methods for common fixed points of infinite countable family of nonexpansive mappings let e be a real strictly convex and reflexive banach space with a uniformly gateaux differentiable norm and c be a nonempty closed convex subset of e consider the iterative sequence x n email protected n 1f x n email protected n email protected l n 1 w nx n where w n is the w mapping generated by an infinite countable family of nonexpansive mappings t n t n 1 t 1 and a n a n 1 a 1 such that the common fixed point sets email protected email protected n 1 f t n email protected and f c c is a given contractive mapping under very mild conditions on the parameters we prove that x n converges strongly to email protected f where p is the unique solution in f to the following variational inequality 0for all x f

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

solving semidefinite programming problems via alternating direction methods in this paper we consider an alternating direction algorithm for the solution of semidefinite programming problems sdp the main idea of our algorithm is that we reformulate the complementary conditions in the primal dual optimality conditions as a projection equation by using this reformulation we only need to make one projection and solve a linear system of equation with reduced dimension in each iterate we prove that the generated sequence converges to the solution of the sdp under weak conditions

viewing parallel projection methods as sequential ones in convex feasibility problems we show that the parallel projection method with variable weights and one variable relaxation coefficient for obtaining a point in the intersection of a finite number of closed convex sets in a given hilbert space may be interpreted as a semi alternating sequential projection method in a suitably newly constructed hilbert space as such convergence results for the parallel projection method may be derived from those which may be constructed in the semi alternating sequential case

solving a class of linear projection equations many interesting and important constrained optimization problems in mathematical programming can be translated into an equivalent linear projection equation u p omega u mu q here p omega cdot is the orthogonal projection on some convex set omega e g omega bbb r n and m is a positive semidefinite matrix this paper presents some new methods for solving a class of linear projection equations the search directions of these methods are straighforward extensions of the directions in traditional methods for unconstrained optimization based on the idea of a projection and contraction method 7 we get a simple step length rule and are able to obtain global linear convergence

relaxed viscosity approximation methods for fixed point problems and variational inequality problems abstract let x be a real strictly convex and reflexive banach space with a uniformly g teaux differentiable norm and c be a nonempty closed convex subset of x let t n n 1 be a sequence of nonexpansive self mappings on c such that the common fixed point set f n 1 f t n 0 and f c c be a given contractive mapping and n be a sequence of nonnegative numbers in 0 1 consider the following relaxed viscosity approximation method x n 1 1 n n x n n f y n n w n y n y n 1 n x n n w n x n n 1 where w n is the w mapping generated by t n t n 1 t 1 and n n 1 1 for each n 1 it is proven that under very mild conditions on the parameters the sequence x n of approximate solutions generated by the proposed method converges strongly to some p f where p is the unique solution in f to the following variational inequality i f p j p x 0 x f

a new iterative method for equilibrium problems and fixed point problems of infinitely nonexpansive mappings and monotone mappings in this paper we introduce a new iterative scheme for finding the common element of the set of common fixed points of infinitely many nonexpansive mappings the set of solutions of an equilibrium problem and the set of solutions of the variational inequality for a inverse strongly monotone mapping in hilbert spaces we prove that the sequence converges strongly to a common element of the above three sets under some parameters controlling conditions this main result improve and extend plubtieng and punpaeng s corresponding result s plubtieng r punpaeng a new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings applied mathematics and computation 197 2008 548 558 using this theorem we obtain three corollaries

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

a self adaptive projection and contraction method for monotone symmetric linear variational inequalities abstract in this paper we present a self adaptive projection and contraction sapc method for solving symmetric linear variational inequalities preliminary numerical tests show that the proposed method is efficient and effective and depends only slightly on its initial parameter the global convergence of the new method is also addressed

viscosity methods of approximation for a common fixed point of a family of quasi nonexpansive mappings let k be a nonempty closed convex subset of a real reflexive banach space e that has weakly continuous duality mapping j for some gauge let ti k k i 1 2 be a family of quasi nonexpansive mappings with f colon equals i 1 f ti 0 combining long solidus overlay which is a sunny nonexpansive retract of k with q a nonexpansive retraction for given x0 k let xn be generated by the algorithm xn 1 colon equals n f xn 1 n tn xn n 0 where f k k is a contraction mapping and n 0 1 a sequence satisfying certain conditions suppose that xn satisfies condition a then it is proved that xn converges strongly to a common fixed point over x q f over x of a family ti i 1 2 moreover over x is the unique solution in f to a certain variational inequality

viscosity approximation methods for a finite family of nonexpansive mappings in banach spaces abstract by using viscosity approximation methods for a finite family of nonexpansive mappings in banach spaces some sufficient and necessary conditions for the iterative sequence to converging to a common fixed point are obtained the results presented in the paper extend and improve some recent results in h k xu viscosity approximation methods for nonexpansive mappings j math anal appl 298 2004 279 291 h k xu remark on an iterative method for nonexpansive mappings comm appl nonlinear anal 10 2003 67 75 h h bauschke the approximation of fixed points of compositions of nonexpansive mappings in banach spaces j math anal appl 202 1996 150 159 b halpern fixed points of nonexpansive maps bull amer math soc 73 1967 957 961 j s jung iterative approaches to common fixed points of nonexpansive mappings in banach spaces j math anal appl 302 2005 509 520 p l lions approximation de points fixes de contractions c r acad sci paris ser a 284 1977 1357 1359 a moudafi viscosity approximation methods for fixed point problems j math anal appl 241 2000 46 55 s reich strong convergence theorems for resolvents of accretive operators in banach spaces j math anal appl 75 1980 128 292 r wittmann approximation of fixed points of nonexpansive mappings arch math 58 1992 486 491

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

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

nonlinear implicit variational inequalities involving partially relaxed pseudomonotone mappings abstract based on a class of generalized iterative algorithms approximation solvability of a class of nonlinear implicit variational inequalities involving partially relaxed pseudomonotone mappings a mild generalization of the general notion of partial relaxed monotonicity is discussed the partial relaxed monotonicity itself is a widely studied and computation oriented class and is more general than the notions of cocoercivity and strong monotonicity

strong convergence of iteration algorithms for a countable family of nonexpansive mappings abstract for a countable family t n n 1 of nonexpansive mappings a strong convergence of halpern type iteration is shown in order to find a common fixed point of t n n 1 in a reflexive banach space when n 0 1 satisfies the conditions c1 lim n n 0 and c2 n 1 n and several examples satisfying the condition b is given also we also research the strong convergence of the proximal point algorithm finally we apply our results to solve the equilibrium problems and variational inequalities for continuous monotone mappings

a proximal decomposition algorithm for variational inequality problems in this paper we propose a new decomposition algorithm for solving monotone variational inequality problems with linear constraints the algorithm utilizes the problem s structure conductive to decomposition at each iteration the algorithm solves a system of nonlinear equations which is structurally much easier to solve than variational inequality problems the subproblems of classical decomposition methods and then performs a projection step to update the multipliers we allow to solve the subproblems approximately and we prove that under mild assumptions on the problem s data the algorithm is globally convergent we also report some preliminary computational results which show that the algorithm is encouraging

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

some matrix variate distribution theory notational considerations and a bayesian application summary we introduce and justify a convenient notation for certain matrix variate distributions which by its emphasis on the important underlying parameters and the theory on which it is based eases greatly the task of manipulating such distributions important examples include the matrix variate normal t f and beta and the wishart and inverse wishart distributions the theory is applied to compound matrix distributions and to bayesian prediction in the multivariate linear model

discrimination with many variables abstract many statistical methods for discriminant analysis do not adapt well or easily to situations where the number of variables is large possibly even exceeding the number of cases in the training set we explore a variety of methods for providing robust identification of future samples in this situation we develop a range of flexible bayesian methods and primarily a new hierarchical covariance compromise method akin to regularized discriminant analysis although the methods are much more widely applicable the motivating problem was that of discriminating between groups of samples on the basis of their near infrared spectra here the ability of the bayesian methods to take account of continuity of the spectra may be beneficial the spectra may consist of absorbances or reflectances at as many as 1 000 wavelengths and yet there may be only tens or hundreds of training samples in which both sample spectrum and group identity are known such problems arise in the food and pharmaceutical industries

role of twice frechet differentiable mappings in general auxiliary problem principle abstract the approximation solvability of the following class of nonlinear variational inequality problems nvip based on the general auxiliary problem principle lapp is discussed find an element k such that x x x f x f x o for all x k where t k e is a nonlinear mapping from a nonempty closed convex subset k of a reflexive banach space e into its dual e and f k r is a continuous convex functional on k

iterative methods for common fixed points for a countable family of nonexpansive mappings in uniformly convex spaces abstract let e be a uniformly convex real banach space with a uniformly g teaux differentiable norm let k be a closed convex and nonempty subset of e let t i i 1 be a family of nonexpansive self mappings of k for arbitrary fixed 0 1 define a family of nonexpansive maps s i n 1 by s i 1 i t i where i is the identity map of k let f i 1 f t i 0 it is proved that an iterative sequence x n defined by x 0 k x n 1 n u i 1 i t n s i x n n 0 converges strongly to a common fixed point of the family t i i 1 where n and i t n are sequences in 0 1 satisfying appropriate conditions in each of the following cases a e l p 1 p and b at least one of the t i s is demicompact

image restoration by the method of convex projections part 1ߞtheory a projection operator onto a closed convex set in hilbert space is one of the few examples of a nonlinear map that can be defined in simple abstract terms moreover it minimizes distance and is nonexpansive and therefore shares two of the more important properties of ordinary linear orthogonal projections onto closed linear manifolds in this paper we exploit the properties of these operators to develop several iterative algorithms for image restoration from partial data which permit any number of nonlinear constraints of a certain type to be subsumed automatically their common conceptual basis is as follows every known property of an original image f is envisaged as restricting it to lie in a well defined closed convex set thus m such properties place f in the intersection e0 ei of the corresponding closed convex sets e1 e2 em given only the projection operators pi onto the individual ei s i 1 m we restore f by recursive means clearly in this approach the realization of the pi s in a hilbert space setting is one of the major synthesis problems section i describes the geometrical significance of the three main theorems in considerable detail and most of the underlying ideas are illustrated with the aid of simple diagrams section ii presents rules for the numerical implementation of 11 specific projection operators which are found to occur frequently in many signal processing applications and the appendix contains proofs of all the major results

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

simultaneous variable selection we propose a new method for selecting a common subset of explanatory variables where the aim is to model several response variables the idea is a natural extension of the lasso technique proposed by tibshirani 1996 and is based on the joint residual sum of squares while constraining the parameter estimates to lie within a suitable polyhedral region the properties of the resulting convex programming problem are analyzed for the special case of an orthonormal design for the general case we develop an efficient interior point algorithm the method is illustrated on a dataset with infrared spectrometry measurements on 14 qualitatively different but correlated responses using 770 wavelengths the aim is to select a subset of the wavelengths suitable for use as predictors for as many of the responses as possible

projection and contraction methods for semidefinite programming recently he 1992 1994a b 1996 presented some iterative projection and contraction pc methods for linear variational inequalities its simplicity robustness and ability to handle large problems has attracted a lot of attention this paper extends the pc methods from linear programming to semidefinite programming sdp the sdp problem is transformed into an equivalent projection equation which is solved by pc methods some promising numerical results obtained from a preliminary implementation are included

multivariate bayesian variable selection and prediction the multivariate regression model is considered with p regressors a latent vector with p binary entries serves to identify one of two types of regression coefficients those close to 0 and those not specializing our general distributional setting to the linear model with gaussian errors and using natural conjugate prior distributions we derive the marginal posterior distribution of the binary latent vector fast algorithms aid its direct computation and in high dimensions these are supplemented by a markov chain monte carlo approach to sampling from the known posterior distribution problems with hundreds of regressor variables become quite feasible we give a simple method of assigning the hyperparameters of the prior distribution the posterior predictive distribution is derived and the approach illustrated on compositional analysis of data involving three sugars with 160 near infra red absorbances as regressors

iterative methods for strict pseudo contractions in hilbert spaces abstract let t i i 1 n be n strict pseudo contractions defined on a closed convex subset c of a real hilbert space h consider the problem of finding a common fixed point of these mappings and consider the parallel and cyclic algorithms for solving this problem we will prove the weak convergence of these algorithms moreover by applying additional projections we further prove that these algorithms can be modified to have strong convergence

comparing stochastic optimization methods for variable selection in binary outcome prediction with application to health policy traditional variable selection strategies in generalized linear models glms seek to optimize a measure of predictive accuracy without regard for the cost of data collection when the purpose of such model building is the creation of predictive scales to be used in future studies with constrained budgets the standard approach may not be optimal we propose a bayesian decision theoretic framework for variable selection in binary outcome glms where the budget for data collection is constrained and potential predictors may vary considerably in cost the method is illustrated using data from a large study of quality of hospital care in the u s in the 1980s especially when the number of available predictors p is large it is important to use an appropriate technique for optimization e g in an application presented here where p 83 the space over which we search has 283 1025 elements which is too large to explore using brute force enumeration specifically we investigate simulated annealing sa

iterative approaches to finding nearest common fixed points of nonexpansive mappings in hilbert spaces abstract the iteration scheme xn 1 n 1y 1 n 1 tn 1xn is first considered for infinitely many nonexpansive maps t1 t2 t3 in a hilbert space a result of shimizu and takahashi j math anal appl 211 1997 71 is generalized and it is shown that the sequence of iterates converges to py where p is some projection for this same iteration scheme with finitely many maps t1 t2 tn a complementary result to a result of bauschke j math anal appl 202 1996 150 is proved by introducing a new condition on the sequence of parameters n this condition improves lions condition c r acad sci paris ser a b 284 1977 1357 the iterates converge to py where p is the projection onto the intersection of the fixed point sets of the tis

viscosity approximation to common fixed points of families of nonexpansive mappings with generalized contractions mappings abstract let x be a reflexive and smooth real banach space which has a weakly sequentially continuous duality mapping in this paper we consider the following viscosity approximation scheme x n 1 n 1 f x n 1 n 1 t n 1 x n where f is a generalized contraction mapping for infinitely many nonexpansive self mappings t 1 t 2 t 3 in x we establish a strong convergence result which generalizes some results in the literature

hybrid viscosity approximation schemes for equilibrium problems and fixed point problems of infinitely many nonexpansive mappings abstract recently takahashi and takahashi s takahashi w takahashi viscosity approximation methods for equilibrium problems and fixed point problems in hilbert spaces j math anal appl 2006 doi 10 1016 j jmaa 2006 08 036 suggested and analyzed 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 in this paper we introduce a new 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 common fixed points of infinitely many nonexpansive mappings in a hilbert space then we prove a strong convergence theorem which is the improvements and extension of takahashi and takahashi s 2006 corresponding result using this theorem we obtain two corollaries which improve and extend their corresponding results

methods and criteria for model selection model selection is an important part of any statistical analysis and indeed is central to the pursuit of science in general many authors have examined the question of model selection from both frequentist and bayesian perspectives and many tools for selecting the best model have been suggested in the literature this paper considers the various proposals from a bayesian decision theoretic perspective

the approximation of fixed points of compositions of nonexpansive mappings in hilbert space determining fixed points of nonexpansive mappings is a frequent problem in mathematics and physical sciences an algorithm for finding common fixed points of nonexpansive mappings in hilbert space essentially due to halpern is analyzed the main theorem extends wittmann s recent work and partially generalizes a result by lions algorithms of this kind have been applied to the convex feasibility problem

the common minimal norm fixed point of a finite family of nonexpansive mappings abstract consider a finite family of nonexpansive mappings which are defined on a closed convex subset of a hilbert space h suppose the set of common fixed points of this family is nonempty we then address the problem of finding the minimum norm element of this common fixed point set we introduce both cyclic and parallel iteration methods to find this minimal norm element