metric subregularity and the proximal point method we examine the linear convergence rates of variants of the proximal point method for finding zeros of maximal monotone operators we begin by showing how metric subregularity is sufficient for local linear convergence to a zero of a maximal monotone operator this result is then generalized to obtain convergence rates for the problem of finding a common zero of multiple monotone operators by considering randomized and averaged proximal methods

a generalization of polyak s convergence result for subgradient optimization this paper generalizes a practical convergence result first presented by polyak this new result presents a theoretical justification for the step size which has been successfully used in several specialized algorithms which incorporate the subgradient optimization approach

compositions and averages of two resolvents relative geometry of fixed points sets and a partial answer to a question by c byrne abstract we show that the set of fixed points of the average of two resolvents can be found from the set of fixed points for compositions of two resolvents associated with scaled monotone operators recently the proximal average has attracted considerable attention in convex analysis our results imply that the minimizers of proximal average functions can be found from the set of fixed points for compositions of two proximal mappings associated with scaled convex functions when both convex functions in the proximal average are indicator functions of convex sets least squares solutions can be completely recovered from the limiting cycles given by compositions of two projection mappings this provides a partial answer to a question posed by c byrne a novelty of our approach is to use the notion of resolvent average and proximal average

the asymptotic behavior of the composition of two resolvents abstract the asymptotic behavior of the composition of two resolvents in a hilbert space is investigated connections are made between the solutions of associated monotone inclusion problems and their dual versions the applications provided include a study of an alternating minimization procedure and a new proof of von neumann s classical result on the method of alternating projections

high dimensional correlation matrices the central limit theorem and its applications summary statistical inferences for sample correlation matrices are important in high dimensional data analysis motivated by this the paper establishes a new central limit theorem for a linear spectral statistic of high dimensional sample correlation matrices for the case where the dimension p and the sample size n are comparable this result is of independent interest in large dimensional random matrix theory we also further investigate the sample correlation matrices of a high dimensional vector whose elements have a special correlated structure and the corresponding central limit theorem is developed meanwhile we apply the linear spectral statistic to an independence test for p random variables and then an equivalence test for p factor loadings and n factors in a factor model the finite sample performance of the test proposed shows its applicability and effectiveness in practice an empirical application to test the independence of household incomes from various cities in china is also conducted

linear and strong convergence of algorithms involving averaged nonexpansive operators abstract we introduce regularity notions for averaged nonexpansive operators combined with regularity notions of their fixed point sets we obtain linear and strong convergence results for quasicyclic cyclic and random iterations new convergence results on the borwein tam method btm and on the cyclically anchored douglas rachford algorithm cadra are also presented finally we provide a numerical comparison of btm cadra and the classical method of cyclic projections for solving convex feasibility problems

determining the number of factors in the general dynamic factor model this article develops an information criterion for determining the number q of common shocks in the general dynamic factor model developed by forni et al as opposed to the restricted dynamic model considered by bai and ng and by amengual and watson our criterion is based on the fact that this number q is also the number of diverging eigenvalues of the spectral density matrix of the observations as the number n of series goes to infinity we provide sufficient conditions for consistency of the criterion for large n and t where t is the series length we show how the method can be implemented and provide simulations and empirics illustrating its very good finite sample performance application to real data adds a new empirical facet to an ongoing debate on the number of factors driving the u s economy

modelling multiple time series via common factors we propose a new method for estimating common factors of multiple time series one distinctive feature of the new approach is that it is applicable to some nonstationary time series the unobservable nonstationary factors are identified by expanding the white noise space step by step thereby solving a high dimensional optimization problem by several low dimensional sub problems asymptotic properties of the estimation are investigated the proposed methodology is illustrated with both simulated and real datasets

on the r superlinear convergence of the kkt residuals generated by the augmented lagrangian method for convex composite conic programming due to the possible lack of primal dual type error bounds it was not clear whether the karush kuhn tucker kkt residuals of the sequence generated by the augmented lagrangian method alm for solving convex composite conic programming cccp problems converge superlinearly in this paper we resolve this issue by establishing the r superlinear convergence of the kkt residuals generated by the alm under only a mild quadratic growth condition on the dual of cccp with easy to implement stopping criteria for the augmented lagrangian subproblems this discovery may help to explain the good numerical performance of several recently developed semismooth newton cg based alm solvers for linear and convex quadratic semidefinite programming

block iterative surrogate projection methods for convex feasibility problems abstract a unified framework is presented for studying the convergence of projection methods for finding a common point of finitely many closed convex sets in r n every iteration approximates each set by a half space given either by an approximate projection of the current iterate or by an aggregate inequality derived from the convex inequalities describing this set the next iterate is found by projecting the current one on a surrogate half space formed by taking a convex combination of the half space inequalities convergence to a solution is established under weak conditions that allow various acceleration techniques and choices of aggregating weights the resulting methods are block iterative and hence lend themselves to parallel implementation we show that the idea of forming cut maps via surrogate inequalities encompasses many classical as well as recently proposed methods for set intersection problems and convex feasibility problems with nondifferentiable inequalities and linear equations and inequalities

determining the number of factors when the number of factors can increase with sample size abstract correctly specifying the number of factors r is a fundamental issue for the application of factor models in this paper we develop an econometric method to estimate the number of factors in factor models of large dimensions where the number of factors is allowed to increase as the two dimensions cross section size n and time period t increase using similar information criteria as proposed by bai and ng 2002 we show that the number of factors can be consistently estimated using the criteria we propose a new procedure that avoids over estimating the number of factors while allowing for one to search for possible number of factors over a wide range of positive integers so that it also avoids underestimation of the number of factors we conduct monte carlo simulation to investigate the finite sample properties of the proposed approach

prox penalization and splitting methods for constrained variational problems this paper is concerned with the study of a class of prox penalization methods for solving variational inequalities of the form ax n c x ni0 where mathcal h is a real hilbert space a mathcal h rightrightarrows mathcal h is a maximal monotone operator and n c is the outward normal cone to a closed convex set c subset mathcal h given psi mathcal h to mathbb r cup infty which acts as a penalization function with respect to the constraint x in c and a penalization parameter beta n we consider a diagonal proximal algorithm of the form x n i lambda n a beta n partial psi 1 x n 1 and an algorithm which alternates proximal steps with respect to a and penalization steps with respect to c and reads as x n i lambda n beta n partial psi 1 i lambda na 1 x n 1 we obtain weak ergodic convergence for a general maximal monotone operator a and weak convergence of the whole sequence x n when a is the subdifferential of a proper lower semicontinuous convex function mixing with passty s idea we can extend the ergodic convergence theorem thus obtaining the convergence of a prox penalization splitting algorithm for constrained variational inequalities governed by the sum of several maximal monotone operators our results are applied to an optimal control problem where the state variable and the control are coupled by an elliptic equation we also establish robustness and stability results that account for numerical approximation errors

parallel projected aggregation methods for solving the convex feasibility problem convergence conditions are established for new sequential and parallel projected aggregation methods pams that find a feasible point of a large system of convex inequalities and linear equations

canonical coset parametrization and the bures metric of the three level quantum systems an explicit parametrization for the state space of an n level density matrix is given the parametrization is based on the canonical coset decomposition of unitary matrices we also compute explicitly the bures metric tensor over the state space of two and three level quantum systems

the rate of convergence for the method of alternating projections ii abstract the purpose of the paper is threefold 1 to develop a useful error bound for the method of alternating projections which is relatively easy to compute and remember 2 to exhibit a counterexample to a conjecture of kayalar and weinert 3 to show that in the case of at least three subspaces any error bound which only depends on the angles between the various subspaces involved can never be sharp

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

iterative selection methods for common fixed point problems many problems encountered in applied mathematics can be recast as the problem of selecting a particular common fixed point of a countable family of quasi nonexpansive operators in a hilbert space we propose two iterative methods to solve such problems our convergence analysis is shown to cover a variety of existing results in this area applications to solving monotone inclusion and equilibrium problems are considered

a simultaneous projections method for linear inequalities abstract an iterative method for solving general systems of linear inequalities is considered the method a relaxed generalization of cimmino s scheme for solving linear systems was first suggested by censor and elfving each iterate is obtained as a convex combination of the orthogonal projections of the previous iterate on the half spaces defined by the linear inequalities the algorithm is particularly suitable for implementation on computers with parallel processors we prove convergence from any starting point for both consistent and nonconsistent systems to a feasible point in the first case and to a weighted least squares type solutions in the second

surrogate projection methods for finding fixed points of firmly nonexpansive mappings we present methods for finding common fixed points of finitely many firmly nonexpansive mappings on a hilbert space at every iteration an approximation to each mapping generates a halfspace containing its set of fixed points the next iterate is found by projecting the current iterate on a surrogate halfspace formed by taking a convex combination of the halfspace inequalities this acceleration technique extends one for convex feasibility problems cfps since projection operators onto closed convex sets are firmly nonexpansive the resulting methods are block iterative and hence lend themselves to parallel implementation we extend to accelerated methods some recent results of bauschke and borwein siam rev 38 1996 pp 367 426 on the convergence of projection methods

asymptotic convergence analysis of the proximal point algorithm the asymptotic convergence of the proximal point algorithm ppa for the solution of equations of type 0 in tz where t is a multivalued maximal monotone operator in a real hilbert space is analyzed when 0 in tz has a nonempty solution set bar z convergence rates are shown to depend on how rapidly t 1 grows away from bar z in a neighbourhood of 0 when this growth is bounded by a power function with exponent s then for a sequence z k generated by the ppa z k bar z converges to zero like o k s 2 linearly superlinearly or in a finite number of steps according to whether s in 0 1 s 1 s in 1 infty or s infty

convergence rate analysis for averaged fixed point iterations in common fixed point problems in this paper we establish sublinear and linear convergence of fixed point iterations generated by averaged operators in a hilbert space our results are achieved under a bounded holder regularity

faster convergence rates of relaxed peaceman rachford and admm under regularity assumptions in this paper we provide a comprehensive convergence rate analysis of the douglas rachford splitting drs peaceman rachford splitting prs and alternating direction method of multipliers admm algorithms under various regularity assumptions including strong convexity lipschitz differentiability and bounded linear regularity the main consequence of this work is that relaxed prs and admm automatically adapt to the regularity of the problem and achieve convergence rates that improve upon the tight worst case rates that hold in the absence of such regularity all of the results are obtained using simple techniques

block iterative projection methods for parallel computation of solutions to convex feasibility problems abstract an iterative method is proposed for solving convex feasibility problems each iteration is a convex combination of projections onto the given convex sets where the weights of the combination may vary from step to step it is shown that any sequence of iterations generated by the algorithm converges if the intersection of the given family of convex sets is nonempty and that the limit point of the sequence belongs to this intersection under mild conditions on the sequence of weight functions special cases are block iterative processes where in each iterative step a certain subfamily of the given family of convex sets is used in particular a block iterative version of the agmon motzkin schoenberg relaxation method for solving systems of linear inequalities is derived such processes lend themselves to parallel implementation and will be useful in various areas of applications including image reconstruction from projections image restoration and other fully discretized inversion problems

random products of contractions in metric and banach spaces abstract suppose x d is a metric space and t0 tn is a family of quasinonexpansive self mappings on x we give conditions sufficient to guarantee that every possible iteration of mappings drawn from t0 tn converges as a consequence if c0 cn are closed convex subsets of a hilbert space with nonempty intersection one of which is compact and the proximity mappings are iterated in any order provided only that each is used infinitely often then the resulting sequence converges strongly to a point of the common intersection

strong conical hull intersection property bounded linear regularity jameson s property g and error bounds in convex optimization the strong conical hull intersection property and bounded linear regularity are properties of a collection of finitely many closed convex intersecting sets in euclidean space these fundamental notions occur in various branches of convex optimization constrained approximation convex feasibility problems linear inequalities for instance it is shown that the standard constraint qualification from convex analysis implies bounded linear regularity which in turn yields the strong conical hull intersection property jameson s duality for two cones which relates bounded linear regularity to property g is re derived and refined for polyhedral cones a statement dual to hoffman s error bound result is obtained a sharpening of a result on error bounds for convex inequalities by auslender and crouzeix is presented finally for two subspaces property g is quantified by the angle between the subspaces

new douglas rachford algorithmic structures and their convergence analyses in this paper we study new algorithmic structures with douglas rachford dr operators to solve convex feasibility problems we propose to embed the basic two set dr algorithmic operator into the string averaging projections and into the block iterative projection algorithmic structures thereby creating new dr algorithmic schemes that include the recently proposed cyclic dr algorithm and the averaged dr algorithm as special cases we further propose and investigate a new multiple set dr algorithmic operator convergence of all these algorithmic schemes is studied by using properties of strongly quasi nonexpansive operators and firmly nonexpansive operators

on the convergence of the products of firmly nonexpansive mappings consider a finite collection of firmly nonexpansive self mappings on a hilbert space whose fixed point sets intersect it is shown that in the finite dimensional case any iteration of mappings drawn from this collection converges this resolves for the finite dimensional case at least a popular conjecture concerning the convergence of the successive projection method in the infinite dimensional case it is shown that if the mappings are drawn according to a certain order called the quasi cyclic order then the iteration converges weakly in a sense the quasi cyclic order may be viewed as an extension of the well known cyclic order in which the lengths of the cycles are permitted to grow without bound

erratum differential geometry on su 3 with applications to three state systems j math phys 39 6125 6136 1998 the first equation on page 6127 just above eq 2 is the equation for the right invariant vector fields not the left invariant vector fields this makes several sign changes in the appendices that we present here

a rank test for the number of factors with high frequency data abstract in the literature consistency of the estimates of the number of factors for large dimensional factor models had been extensively studied recently but the second order property of the estimator has long been unsolved due to lack of limiting distribution of the estimators in this paper we propose a rank test of the number of factors using large panel high frequency data contaminated with microstructure noise the rank test is realized by forming a fixed number of portfolios which reduce the dimension to a finite number in the process of constructing portfolios the number of factors is equal to the rank of the volatility matrix of the diversified portfolios asymptotically via estimating the volatility rank of a low dimensional price dynamics of the portfolios we establish a central limit theorem of the estimated factor number we then apply the asymptotic normality to testing on the number of factors numerical experiments including the monte carlo simulations and real data analysis justify our theory

the efficiency of subgradient projection methods for convex optimization part i general level methods we study subgradient methods for convex optimization that use projections onto successive approximations of level sets of the objective corresponding to estimates of the optimal value we present several variants and show that they enjoy almost optimal efficiency estimates in another paper we discuss possible implementations of such methods in particular their projection subproblems may be solved inexactly via relaxation methods thus opening the way for parallel implementations they can also exploit accelerations of relaxation methods based on simultaneous projections surrogate constraints and conjugate and projected conditional subgradient techniques

the rank of demand systems theory and nonparametric estimation w m gorman s 1981 concept of engel curve rank is extended to apply to any demand system rank is shown to have implications for specification separability and aggregation of demands a simple nonparametric test of rank using engel curve data is described and applied to u s and u k consumer survey data the test employs a new general method for testing the rank of estimated matrices the results are used to assess theoretical and empirical aggregation error in representative consumer models and to explain a representative consumer paradox copyright 1991 by the econometric society

eigenvalue ratio test for the number of factors this paper proposes two new estimators for determining the number of factors r in static approximate factor models we exploit the well known fact that the r largest eigenvalues of the variance matrix of n response variables grow unboundedly as n increases while the other eigenvalues remain bounded the new estimators are obtained simply by maximizing the ratio of two adjacent eigenvalues our simulation results provide promising evidence for the two estimators

convergence rates with inexact non expansive operators in this paper we present a convergence rate analysis for the inexact krasnosel ski mann iteration built from non expansive operators the presented results include two main parts we first establish the global pointwise and ergodic iteration complexity bounds then under a metric sub regularity assumption we establish a local linear convergence for the distance of the iterates to the set of fixed points the obtained results can be applied to analyze the convergence rate of various monotone operator splitting methods in the literature including the forward backward splitting the generalized forward backward the douglas rachford splitting alternating direction method of multipliers and primal dual splitting methods for these methods we also develop easily verifiable termination criteria for finding an approximate solution which can be seen as a generalization of the termination criterion for the classical gradient descent method we finally develop a parallel analysis for the non stationary krasnosel ski mann iteration

determining the number of factors in approximate factor models in this paper we develop some econometric theory for factor models of large dimensions the focus is the determination of the number of factors r which is an unresolved issue in the rapidly growing literature on multifactor models we first establish the convergence rate for the factor estimates that will allow for consistent estimation of r we then propose some panel criteria and show that the number of factors can be consistently estimated using the criteria the theory is developed under the framework of large cross sections n and large time dimensions t no restriction is imposed on the relation between n and t simulations show that the proposed criteria have good finite sample properties in many configurations of the panel data encountered in practice

on time varying factor models estimation and testing abstract conventional factor models assume that factor loadings are fixed over a long horizon of time which appears overly restrictive and unrealistic in applications in this paper we introduce a time varying factor model where factor loadings are allowed to change smoothly over time we propose a local version of the principal component method to estimate the latent factors and time varying factor loadings simultaneously we establish the limiting distributions and uniform convergence of the estimated factors and factor loadings in the standard large n and large t framework we also propose a bic type information criterion to determine the number of factors which can be used in models with either time varying or time invariant factor models based on the comparison between the estimates of the common components under the null hypothesis of no structural changes and those under the alternative we propose a consistent test for structural changes in factor loadings we establish the null distribution the asymptotic local power property and the consistency of our test simulations are conducted to evaluate both our nonparametric estimates and test statistic we also apply our test to investigate stock and watson s 2009 u s macroeconomic data set and find strong evidence of structural changes in the factor loadings

a norm convergence result on random products of relaxed projections in hilbert space suppose x is a hilbert space and c 1 ldots c n are closed convex intersecting subsets with projections p 1 ldots p n suppose further r is a mapping from bbb n onto 1 ldots n that assumes every value infinitely often we prove a more general version of the following result begin quote if the n tuple c 1 ldots c n is innately boundedly regular then the sequence x n defined by x 0 in x mbox arbitrary x n 1 p r n x n mbox for all n geq 0 converges in norm to some point in bigcap i 1 n c i end quote examples without the usual assumptions on compactness are given methods of this type have been used in areas like computerized tomography and signal processing

on the hamiltonian reduction of geodesic motion on su 3 to su 3 su 2 the reduced hamiltonian system on t su 3 su 2 is derived from a riemannian geodesic motion on the su 3 group manifold parametrized by the generalized euler angles and endowed with a bi invariant metric our calculations show that the metric defined by the derived reduced hamiltonian flow on the orbit space su 3 su 2 s5 is not isometric or even geodesically equivalent to the standard riemannian metric on the five sphere s5 embedded into r6

parallel application of block iterative methods in medical imaging and radiation therapy some row action algorithms which exploit special objective function and constraints structure have proven advantageous for solving huge and sparse feasibility or optimization problems recently developed block iterative versions of such special purpose methods enable parallel computation when the underlying problem is appropriately decomposed this opens the door for parallel computation in image reconstruction problems of computerized tomography and in the inverse problem of radiation therapy treatment planning all in their fully discretized modelling approach since there is more than one way of deriving block iterative versions of any row action method the choice has to be made with reference to the underlying real world problem

estimating number of factors by adjusted eigenvalues thresholding determining the number of common factors is an important and practical topic in high dimensional factor models the existing literature is mainly based on the eigenvalues of the covariance matrix

a new use of douglas rachford splitting for identifying infeasible unbounded and pathological conic programs in this paper we present a method for identifying infeasible unbounded and pathological conic programs based on douglas rachford splitting when an optimization program is infeasible unbounded or pathological the iterates of douglas rachford splitting diverge somewhat surprisingly such divergent iterates still provide useful information which our method uses for identification in addition for strongly infeasible problems the method produces a separating hyperplane and informs the user on how to minimally modify the given problem to achieve strong feasibility as a first order method the proposed algorithm relies on simple subroutines and therefore is simple to implement and has low per iteration cost

on the behavior of subgradient projections methods for convex feasibility problems in euclidean spaces we study some methods of subgradient projections for solving a convex feasibility problem with general not necessarily hyperplanes or half spaces convex sets in the inconsistent case and propose a strategy that controls the relaxation parameters in a specific self adapting manner this strategy leaves enough user flexibility but gives a mathematical guarantee for the algorithm s behavior in the inconsistent case we present the numerical results of computational experiments that illustrate the computational advantage of the new method

factor profiled sure independence screening we propose a method of factor profiled sure independence screening for ultrahigh dimensional variable selection the objective of this method is to identify nonzero components consistently from a sparse coefficient vector the new method assumes that the correlation structure of the high dimensional data can be well represented by a set of low dimensional latent factors which can be estimated consistently by eigenvalue eigenvector decomposition the estimated latent factors should then be profiled out from both the response and the predictors such an operation referred to as factor profiling produces uncorrelated predictors therefore sure independence screening can be applied subsequently and the resulting screening result is consistent for model selection a major advantage that standard sure independence screening does not share we refer to the new method as factor profiled sure independence screening numerical studies confirm its outstanding performance copyright 2012 oxford university press

warped proximal iterations for monotone inclusions abstract resolvents of set valued operators play a central role in various branches of mathematics and in particular in the design and the analysis of splitting algorithms for solving monotone inclusions we propose a generalization of this notion called warped resolvent which is constructed with the help of an auxiliary operator the properties of warped resolvents are investigated and connections are made with existing notions abstract weak and strong convergence principles based on warped resolvents are proposed and shown to not only provide a synthetic view of splitting algorithms but to also constitute an effective device to produce new solution methods for challenging inclusion problems

convergence of sequential and asynchronous nonlinear paracontractions we establish the convergence of sequential and asynchronous iteration schemes for nonlinear paracontracting operators acting in finite dimensional spaces applications to the solution of linear systems of equations with convex constraints are outlined a first generalization of one of our convergence results to an infinite pool of asymptotically paracontracting operators is also presented

iterating bregman retractions the notion of a bregman retraction of a closed convex set in euclidean space is introduced bregman retractions include backward bregman projections and forward bregman projections as well as their convex combinations and are thus quite flexible the main result on iterating bregman retractions unifies several convergence results on projection methods for solving convex feasibility problems it is also used to construct new sequential and parallel algorithms

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

asymptotic behaviour of almost nonexpansive sequences in a hilbert space abstract the results in 9 11 12 for nonexpansive sequences are generalized to almost nonexpansive sequences in a hilbert space and using this notion a direct proof of a result of h brezis and f e browder 6 is given

new methods for linear inequalities abstract the iterative method of cimmino for solving linear equations is generalized to linear inequalities we also present a richardson type iterative method for solving the inequality problem which includes the generalized cimmino scheme convergence proofs are provided

identifying the number of factors from singular values of a large sample auto covariance matrix session 3b i3 high dimensional statistics challenges and recent developments invited paper session

iterative algorithms for large partitioned linear systems with applications to image reconstruction abstract we present a unifying framework for a wide class of iterative methods in numerical linear algebra in particular the class of algorithms contains kaczmarz s and richardson s methods for the regularized weighted least squares problem with weighted norm the convergence theory for this class of algorithms yields as corollaries the usual convergence conditions for kaczmarz s and richardson s methods the algorithms in the class may be characterized as being group iterative and incorporate relaxation matrices as opposed to a single relaxation parameter we show that some well known iterative methods of image reconstruction fall into the class of algorithms under consideration and are thus covered by the convergence theory we also describe a novel application to truly three dimensional image reconstruction

factor modeling for high dimensional time series inference for the number of factors this paper deals with the factor modeling for high dimensional time series based on a dimension reduction viewpoint under stationary settings the inference is simple in the sense that both the number of factors and the factor loadings are estimated in terms of an eigenanalysis for a nonnegative definite matrix and is therefore applicable when the dimension of time series is on the order of a few thousands asymptotic properties of the proposed method are investigated under two settings i the sample size goes to infinity while the dimension of time series is fixed and ii both the sample size and the dimension of time series go to infinity together in particular our estimators for zero eigenvalues enjoy faster convergence or slower divergence rates hence making the estimation for the number of factors easier in particular when the sample size and the dimension of time series go to infinity together the estimators for the eigenvalues are no longer consistent however our estimator for the number of the factors which is based on the ratios of the estimated eigenvalues still works fine furthermore this estimation shows the so called blessing of dimensionality property in the sense that the performance of the estimation may improve when the dimension of time series increases a two step procedure is investigated when the factors are of different degrees of strength numerical illustration with both simulated and real data is also reported

on the range of the douglas rachford operator the problem of finding a minimizer of the sum of two convex functions or more generally that of finding a zero of the sum of two maximally monotone operators is of central importance in variational analysis perhaps the most popular method of solving this problem is the douglas rachford splitting method surprisingly little is known about the range of the douglas rachford operator in this paper we set out to study this range systematically we prove that for 3 monotone operators a very pleasing formula can be found that reveals the range to be nearly equal to a simple set involving the domains and ranges of the underlying operators a similar formula holds for the range of the corresponding displacement mapping we discuss applications to subdifferential operators to the infimal displacement vector and to firmly nonexpansive mappings various examples and counterexamples are presented including some concerning the celebrated brezis haraux theorem

distributed algorithms for computing a common fixed point of a group of nonexpansive operators this paper addresses the problem of seeking a common fixed point for a collection of nonexpansive operators over time varying multi agent networks in real hilbert spaces each operator is only privately and approximately known to each individual agent and all agents need to cooperate to solve this problem by local communications over time varying networks to handle this problem inspired by the centralized inexact krasnosel skii mann ikm iteration we propose a distributed algorithm called distributed inexact averaged operator algorithm d iao it is shown that the d iao algorithm can converge weakly to a common fixed point of the family of nonexpansive operators moreover under the linearly regular assumption it is proved that the d iao algorithm converges with a rate o gamma log 16 4k for some constant gamma in 0 1 where k is the iteration number the same converge speed can be ensured by a novel power regularity condition which is strictly weaker than linear regularity to reduce computational complexity another algorithm called d biao is developed and fully analyzed simulation examples are given to support the theoretical res

bregman monotone optimization algorithms a broad class of optimization algorithms based on bregman distances in banach spaces is unified around the notion of bregman monotonicity a systematic investigation of this notion leads to a simplified analysis of numerous algorithms and to the development of a new class of parallel block iterative surrogate bregman projection schemes another key contribution is the introduction of a class of operators that is shown to be intrinsically tied to the notion of bregman monotonicity and to include the operators commonly found in bregman optimization methods special emphasis is placed on the viability of the algorithms and the importance of legendre functions in this regard various applications are discussed

an explicit computation of the bures metric over the space of n dimensional density matrices the aim of this paper is to provide a method for explicit computation of the bures metric over the space of n level quantum system based on the coset parametrization of density matrices

row action methods for huge and sparse systems and their applications this paper brings together and discusses theory and applications of methods identified and labelled as row action methods for linear feasibility problems find x in bf r n such that ax leqq b linearly constrained optimization problems minimize f x subject to ax leqq b and some interval convex programming problems minimize f x subject to c leqq ax leqq b the main feature of row action methods is that they are iterative procedures which without making any changes to the original matrix a use the rows of a one row at a time such methods are important and have demonstrated effectiveness for problems with large or huge matrices which do not enjoy any detectable or usable structural pattern apart from a high degree of sparaseness fields of application where row action methods are used in various ways include image reconstruction from projection operations research and game theory learning theory pattern recognition and transportation theory a row action method for the

generalized solutions for the sum of two maximally monotone operators a common theme in mathematics is to define generalized solutions to deal with problems that potentially do not have solutions a classical example is the introduction of least squares solutions via the normal equations associated with a possibly infeasible system of linear equations in this paper we introduce a normal problem associated with finding a zero of the sum of two maximally monotone operators if the original problem admits solutions then the normal problem returns this same set of solutions the normal problem may yield solutions when the original problem does not admit any furthermore it has attractive variational and duality properties several examples illustrate our theory

on the douglas rachford algorithm the douglas rachford algorithm is a very popular splitting technique for finding a zero of the sum of two maximally monotone operators the behaviour of the algorithm remains mysterious in the general inconsistent case i e when the sum problem has no zeros however more than a decade ago it was shown that in the possibly inconsistent convex feasibility setting the shadow sequence remains bounded and its weak cluster points solve a best approximation problem in this paper we advance the understanding of the inconsistent case significantly by providing a complete proof of the full weak convergence in the convex feasibility setting in fact a more general sufficient condition for the weak convergence in the general case is presented our proof relies on a new convergence principle for fejer monotone sequences numerous examples illustrate our results

coincidence points by generalized mann iterates with applications in hilbert spaces this paper deals with the construction of the coincidence points by the generalized mann iterates and their applications in hilbert spaces our results generalize and improve a number of recent results

the efficiency of subgradient projection methods for convex optimization part ii implementations and extensions in the first part of this paper we studied subgradient methods for convex optimization that use projections onto successive approximations of level sets of the objective corresponding to estimates of the optimal value we presented several variants and showed that they enjoy almost optimal efficiency estimates in this part of the paper we discuss possible implementations of such methods in particular their projection subproblems may be solved inexactly via relaxation methods thus opening the way for parallel implementations we discuss accelerations of relaxation methods based on simultaneous projections surrogate constraints and conjugate and projected conditional subgradient techniques

decomposition through formalization in a product space when an optimization problem is posed in a product space it is classical to decompose this problem the goal of this paper is to show how such an approach can be used when the problem to be solved is not naturally posed in a product space by associating systematically to this problem an equivalent one posed in ann fold cartesian product space we obtain by decomposition of the latter both a splitting of operators and a desintegration of constraints for the former applications to three rather classical mathematical programming problems are given

a weak to strong convergence principle for feje monotone methods in hilbert spaces we consider a wide class of iterative methods arising in numerical mathematics and optimization that are known to converge only weakly exploiting an idea originally proposed by haugazeau we present a simple modification of these methods that makes them strongly convergent without additional assumptions several applications are discussed

cyclic subgradient projections a cyclically controlled method of subgradient projections csp for the convex feasibility problem of solving convex inequalities is presented the features of this method make it an efficient tool in handling huge and sparse problems a particular application to an image reconstruction problem of emission computerized tomography is mentioned

generalized mann iterates for constructing fixed points in hilbert spaces the mean iteration scheme originally proposed by mann is extended to a broad class of relaxed inexact fixed point algorithms in hilbert spaces weak and strong convergence results are established under general conditions on the underlying averaging process and the type of operators involved this analysis significantly widens the range of applications of mean iteration methods several examples are given