random products of contractions in banach spaces we show that the random product of a finite number of w contractions converges weakly in all smooth reflexive banach spaces if one of the contractions is compact then the convergence is uniform let x 1 be a real banach space and t x x a linear operator recall that t is called a contraction if i tvi i v i for all vectors v in x we say that a contraction t satisfies condition w if whenever v is bounded and ivni itvni 0 it follows that the weak limn o vn tvn 0 the algebraic semigroup s generated by a possibly infinite set of contractions will be said to satisfy condition w if for any bounded sequence of vectors vn c x and a sequence of words wn from s such that i v n wnvni 0 the weak limn oo vn wnvn 0 now let t1 t2 be a sequence of contractions that satisfy condition w and let r be a self mapping of the set of natural numbers a random product of these contractions is the sequence sn n 1 2 defined by sn tr n tr n 1 tro 1 our purpose in this paper is to study the convergence properties of such products such a study is not only of intrinsic interest but is also motivated by applications to the numerical solution of partial differential equations 2 14 and linear inequalities 19 approximation theory 16 9 and computer tomography 18 11 we begin by noting several properties of contractions which satisfy condition w we shall call such contractions w contractions for short we then consider the algebraic semigroup generated by a finite number of w contractions and establish a weak convergence theorem for random products of these contractions in all smooth reflexive banach spaces theorem 1 this seems to be the first such result outside hilbert space it includes the hilbert space theorem due to amemiya and ando 1 we continue with a weak convergence result for semigroups generated by an infinite sequence of contractions theorem 2 finally we obtain a uniform convergence theorem for random received by the editors october 24 1988 and in revised form april 4 1989 1980 mathematics subject classification 1985 revision primary 47a05 secondary 47b05 65j10

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

there is no variational characterization of the cycles in the method of periodic projections the method of periodic projections consists in iterating projections onto m closed convex subsets of a hilbert space according to a periodic sweeping strategy in the presence of m 3 sets a long standing question going back to the 1960s is whether the limit cycles obtained by such a process can be characterized as the minimizers of a certain functional in this paper we answer this question in the negative projection algorithms for minimizing smooth convex functions over a product of convex sets are also discussed

volterra series arising from the discrete schrodinger wave equation in l2 a b space the wave solution of the stationary schrodinger wave equation has been studied extensively however after removing restriction on potential function the problem of obtaining the wave solution becomes considerably difficult methods available in the literature for the closed form wave solution are applicable to specific potential for this reason it is important to strive for the wave solution with a general class of potential functions this paper introduces the volterra series framework to address this limitation the volterra system is a natural extension of linear systems such systems take into account quadratic cubic and more generally polynomial kind of non linearities in the systems the volterra system is a non linear system with the structure of the volterra series here we decompose first a wave function in an appropriate function space and subsequently the wave equation is discretized as a result of this we obtain a bilinear system treating potential as input and wave function as output such an analysis is based on inherent bilinear character of the schrodinger equation in which potential is bilinearly coupled to the wave function this paper discusses state space methodology to recover the volterra series from the bilinear systems the volterra series is attractive from the system theoretic point of view since it describes the output of a class of non linear systems explicitly in terms of the input rather than involving coupling terms and allows substantial simplifications for the numerical simulation a simulation result is introduced to demonstrate the efficacy of the proposed analysis

convex set theoretic image recovery by extrapolated iterations of parallel subgradient projections solving a convex set theoretic image recovery problem amounts to finding a point in the intersection of closed and convex sets in a hilbert space the projection onto convex sets pocs algorithm in which an initial estimate is sequentially projected onto the individual sets according to a periodic schedule has been the most prevalent tool to solve such problems nonetheless pocs has several shortcomings it converges slowly it is ill suited for implementation on parallel processors and it requires the computation of exact projections at each iteration we propose a general parallel projection method emopsp that overcomes these shortcomings at each iteration of emopsp a convex combination of subgradient projections onto some of the sets is formed and the update is obtained via relaxation the relaxation parameter may vary over an iteration dependent extrapolated range that extends beyond the interval 0 2 used in conventional projection methods emopsp not only generalizes existing projection based schemes but it also converges very efficiently thanks to its extrapolated relaxations theoretical convergence results are presented as well as numerical simulations

acceleration method for convex optimization over the fixed point set of a nonexpansive mapping the existing algorithms for solving the convex minimization problem over the fixed point set of a nonexpansive mapping on a hilbert space are based on algorithmic methods such as the steepest descent method and conjugate gradient methods for finding a minimizer of the objective function over the whole space and attach importance to minimizing the objective function as quickly as possible meanwhile it is of practical importance to devise algorithms which converge in the fixed point set quickly because the fixed point set is the set with the constraint conditions that must be satisfied in the problem this paper proposes an algorithm which not only minimizes the objective function quickly but also converges in the fixed point set much faster than the existing algorithms and proves that the algorithm with diminishing step size sequences strongly converges to the solution to the convex minimization problem we also analyze the proposed algorithm with each of the fletcher reeves polak ribiere polyak hestenes stiefel and dai yuan formulas used in the conventional conjugate gradient methods and show that there is an inconvenient possibility that their algorithms may not converge to the solution to the convex minimization problem we numerically compare the proposed algorithm with the existing algorithms and show its effectiveness and fast convergence

volterra series for solving weakly non linear partial differential equations application to a dissipative burgers equation a method to solve weakly non linear partial differential equations with volterra series is presented in the context of single input systems the solution x z t is represented as the output of a z parameterized volterra system where z denotes the space variable but z could also have a different meaning or be a vector in place of deriving the kernels from purely algebraic equations as for the standard case of ordinary differential systems the problem turns into solving linear differential equations this paper introduces the method on an example a dissipative burgers equation which models the acoustic propagation and accounts for the dominant effects involved in brass musical instruments the kernels are computed analytically in the laplace domain as a new result writing the volterra expansion for periodic inputs leads to the analytic resolution of the harmonic balance method which is frequently used in acoustics furthermore the ability of the volterra system to treat other signals constitutes an improvement for the sound synthesis it allows the simulation for any regime including attacks and transients numerical simulations are presented and their validity are discussed

computation of convergence bounds for volterra series of linear analytic single input systems in this paper the volterra series decomposition of a class of single input time invariant systems analytic in state and affine in input is analyzed input to state convergence results are obtained for several typical norms l 0 t l r as well as exponentially weighted norms from the standard recursive construction of volterra kernels new estimates of the kernel norms are derived the singular inversion theorem is then used to obtain the main result of the paper namely an easily computable bound of the convergence radius guaranteed error bounds for the truncated series are also provided the relevance of the method is illustrated in several examples

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

all schatten spaces endowed with the schur product are q algebras we prove that the banach algebra formed by the space of compact operators on a hilbert space endowed with the schur product is a quotient of a uniform algebra also known as a q algebra together with a similar result of perez garcia for the trace class this completes the answer to a long standing question of varopoulos

hierarchical convex optimization with primal dual splitting this paper addresses the selection of a desirable solution among all the solutions of a convex optimization problem referred to as the first stage problem mainly for inverse problems in signal processing this is realized in the framework of hierarchical convex optimization i e minimizing another convex function over the solution set of the first stage problem hierarchical convex optimization is an ideal strategy when the first stage problem has infinitely many solutions because of the non strict convexity of its objective function which could arise in various scenarios e g convex feasibility problems to this end first the fixed point set characterization behind a primal dual splitting type method is incorporated into the framework of hierarchical convex optimization which enables the framework to cover a broad class of first stage problem formulations then a pair of efficient algorithmic solutions to the hierarchical convex optimization problem as certain realizations of the hybrid steepest descent method are provided with guaranteed convergence we also present a specialized form of the proposed framework to focus on a typical scenario of inverse problems and show its application to signal interpolation

adaptive constrained learning in reproducing kernel hilbert spaces the robust beamforming case this paper establishes a new paradigm for convexly constrained adaptive learning in reproducing kernel hilbert spaces rkhs although the technique is of a general nature we present it in the context of the beamforming problem a priori knowledge like beampattern specifications and constraints concerning robustness against steering vector errors takes the form of multiple closed convex sets in a high possibly infinite dimensional rkhs every robustness constraint is shown to be equivalent to a min max optimization task formed by means of the robust statistics epsiv insensitive loss function such a multiplicity of specifications turns out to obtain a simple expression by using the rich frame of fixed point sets of certain mappings defined in a hilbert space moreover the cost function that the final solution has to optimize is expressed as an infinite sequence of convex nondifferentiable loss functions springing from the sequence of the incoming training data a novel adaptive beamforming design of linear complexity with respect to the number of unknown parameters to such a constrained nonlinear learning problem is derived by employing a very recently developed version of the adaptive projected subgradient method apsm the method produces a sequence that under mild conditions exhibits properties like the strong convergence to a beamformer that satisfies all of the imposed constraints and in the meantime asymptotically minimizes the sequence of the loss functions imposed by the training data the numerical examples demonstrate that the proposed method displays increased resolution in cases where the classical linear beamforming solutions collapse moreover it leads to solutions which are in agreement with the imposed a priori knowledge as opposed to unconstrained online kernel regression techniques

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

robust minimum variance beamforming this paper introduces an extension of minimum variance beamforming that explicitly takes into account variation or uncertainty in the array response sources of this uncertainty include imprecise knowledge of the angle of arrival and uncertainty in the array manifold in our method uncertainty in the array manifold is explicitly modeled via an ellipsoid that gives the possible values of the array for a particular look direction we choose weights that minimize the total weighted power output of the array subject to the constraint that the gain should exceed unity for all array responses in this ellipsoid the robust weight selection process can be cast as a second order cone program that can be solved efficiently using lagrange multiplier techniques if the ellipsoid reduces to a single point the method coincides with capon s method we describe in detail several methods that can be used to derive an appropriate uncertainty ellipsoid for the array response we form separate uncertainty ellipsoids for each component in the signal path e g antenna electronics and then determine an aggregate uncertainty ellipsoid from these we give new results for modeling the element wise products of ellipsoids we demonstrate the robust beamforming and the ellipsoidal modeling methods with several numerical examples

convergence of non cyclic infinite products of operators using a geometric approach in particular angles between subspaces we establish useful estimates regarding convergence of infinite products involving orthogonal projections and other possibly nonlinear nonexpansive operators in hilbert space

three term conjugate gradient method for the convex optimization problem over the fixed point set of a nonexpansive mapping many constrained sets in problems such as signal processing and optimal control can be represented as a fixed point set of a certain nonexpansive mapping and a number of iterative algorithms have been presented for solving a convex optimization problem over a fixed point set this paper presents a novel gradient method with a three term conjugate gradient direction that is used to accelerate conjugate gradient methods for solving unconstrained optimization problems it is guaranteed that the algorithm strongly converges to the solution to the problem under the standard assumptions numerical comparisons with the existing gradient methods demonstrate the effectiveness and fast convergence of this algorithm

a parallel splitting method for coupled monotone inclusions a parallel splitting method is proposed for solving systems of coupled monotone inclusions in hilbert spaces and its convergence is established under the assumption that solutions exist unlike existing alternating algorithms which are limited to two variables and linear coupling our parallel method can handle an arbitrary number of variables as well as nonlinear coupling schemes the breadth and flexibility of the proposed framework is illustrated through applications in the areas of evolution inclusions variational problems best approximation and network flows

spokes mirrors and alternating projections abstract we give an elementary iterative construction of an example similar to that of hundal of a cone and a half space in the hilbert space l 2 which intersect at the origin but such that the corresponding sequence of alternating nearest point mappings although weakly convergent does not converge in norm

image recovery by convex combinations of projections the problem of image recovery in a hilbert space setting by using convex projections by projection we always mean minimum distance projection onto a closed convex set may be stated as follows the original unknown image f is known a priori to belong to the intersection co of r well defined dosed convex sets c 1 c r in a hilbert space h hence f e co n 1 ci given only the projection operators pi onto the individual sets ci i 1 r recover f by an iterative scheme practically the image f is a function of two real variables and the hilbert space h is the real or complex space l2 q where q c r2 since the sets ci are only assumed to be convex the projections pi are in general nonlinear iterative methods of finding a common point of sets by means of contraction like operators can be found in 1 2 5j these methods have been applied to image processing first by youla and webb 11 a procedure with essentially the same ideas has been used in 7j for the extrapolation of bandlimited functions in the methods of image recovery used until now 9 12 7j from the projections pi i 1 r r operators ti are constructed and from these the composition operator t tr tr 1 t2 t1 starting from an arbitrary element x of h the sequence tnx 0 is shown to converge at least weakly to an element of co two cases are important the case where each ti equals pi hence t prpr 1 p2p1 and the case where each ti is given by ti l oi pi l with 1 the identity operator on h and ai a relaxation parameter with 0 ai 2 when oi 1 for all i this reduces to the foregoing case so at each iteration step the computation has to be done sequentially when a parallel computer is available the computing process could be speeded up a lot if at each iteration step the contribution of the different 413

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

strong convergence for an iterative method for the triple hierarchical constrained optimization problem abstract the variational inequality problem for a monotone operator over the fixed point set of a nonexpansive mapping is connected with many signal processing problems and such problems have hierarchical structure for example the convex optimization problem over the solution set of the variational inequality problem over the fixed point set has triple hierarchical structure in this paper we present an iterative algorithm for this problem the strong convergence for the proposed algorithm to the solution is guaranteed under some assumptions

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

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

fixed point optimization algorithm and its application to network bandwidth allocation a convex optimization problem for a strictly convex objective function over the fixed point set of a nonexpansive mapping includes a network bandwidth allocation problem which is one of the central issues in modern communication networks we devised an iterative algorithm called a fixed point optimization algorithm for solving the convex optimization problem and conducted a convergence analysis on the algorithm the analysis guarantees that the algorithm with slowly diminishing step size sequences weakly converges to a unique solution to the problem moreover we apply the proposed algorithm to a network bandwidth allocation problem and show its effectiveness

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

inconsistent signal feasibility problems least squares solutions in a product space presents parallel projection methods to find least squares solutions to inconsistent convex set theoretic signal synthesis problems the problem of finding a signal that minimizes a weighted average of the squares of the distances to constraint sets is reformulated in a product space where it is equivalent to that of finding a point that lies in a particular subspace and at minimum distance from the cartesian product of the original sets a solution is obtained in the product space via methods of alternating projections which naturally lead to methods of parallel projections in the original space the convergence properties of the proposed methods are analyzed and signal synthesis applications are demonstrated

strong asymptotic convergence of evolution equations governed by maximal monotone operators with tikhonov regularization abstract we consider the tikhonov like dynamics u t a u t e t u t where a is a maximal monotone operator on a hilbert space and the parameter function e t tends to 0 as t with 0 e t d t when a is the subdifferential of a closed proper convex function f we establish strong convergence of u t towards the least norm minimizer of f in the general case we prove strong convergence towards the least norm point in a 1 0 provided that the function e t has bounded variation and provide a counterexample when this property fails

on the computation of the eigenproblems of hydrogen helium in strong magnetic and electric fields with the sparse grid combination technique abstract we introduce the combination technique for the numerical solution of d dimensional eigenproblems on sparse grids here o d log n d 1 different problems each of size o n have to be solved independently this is in contrast to the one problem of size o n d for a conventional finite element discretization where n denotes the number of grid points in one coordinate direction therefore also higher dimensional eigenvalue problems can be treated by our sparse grid combination approach we apply this method to solve the three dimensional schrodinger equation for hydrogen one electron problem and the six dimensional schrodinger equation for helium two electron problem in strong magnetic and electric fields

alternating projections in cat 0 spaces abstract by using recently developed theory which extends the idea of weak convergence into cat 0 space we prove the convergence of the alternating projection method for convex closed subsets of a cat 0 space given the right notion of weak convergence it turns out that the generalization of the well known results in hilbert spaces is straightforward and allows the use of the method in a nonlinear setting as an application we use the alternating projection method to minimize convex functionals on a cat 0 space

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

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

sound synthesis of a nonlinear string using volterra series this paper proposes to solve and simulate various kirchhoff models of nonlinear strings using volterra series two nonlinearities are studied the string tension is supposed to depend either on the global elongation of the string first model or on the local strain located at x second and more precise model the boundary conditions are simple dirichlet homogeneous ones or general dynamic conditions allowing the string to be connected to any system typically a bridge for each model a volterra series is used to represent the displacement as a functional of excitation forces the volterra kernels are solved using a modal decomposition the first kernel of the series yields the standard modes of the linearized problem while the next kernels introduce the nonlinear dynamics as a last step systematic identification of the kernels lead to a structure composed of linear filters sums and products which are well suited to the sound synthesis using standard signal processing techniques the nonlinear dynamic introduced through this simulation is significant and perceptible in sound results for sufficiently large excitations

two grid method for two dimensional nonlinear schrodinger equation by mixed finite element method abstract a conservative two grid mixed finite element scheme is presented for two dimensional nonlinear schrodinger equation one newton iteration is applied on the fine grid to linearize the fully discrete problem using the coarse grid solution as the initial guess moreover error estimates are conducted for the two grid method it is shown that the coarse space can be extremely coarse with no loss in the order of accuracy and still achieve the asymptotically optimal approximation as long as the mesh sizes satisfy h o h 1 2 in the two grid method the numerical results show that this method is very effective

steered sequential projections for the inconsistent convex feasibility problem we study a steered sequential gradient algorithm which minimizes the sum of convex functions by proceeding cyclically in the directions of the negative gradients of the functions and using steered step sizes this algorithm is applied to the convex feasibility problem by minimizing a proximity function which measures the sum of the bregman distances to the members of the family of convex sets the resulting algorithm is a new steered sequential bregman projection method

a volterra series representation for a class of nonlinear infinite dimensional systems with periodic boundary conditions abstract this paper proves the existence of a volterra series representation for the mild solutions of a class of nonlinear infinite dimensional systems more specifically given the evolutionary system operator u t s 0 s t associated with a semilinear evolution equation u t 2 u x 2 f u u 0 u 0 x with periodic boundary conditions it is proved that under suitable conditions the unique mild solution u t u t 0 u 0 t 0 can be expanded by a volterra series a recursive algorithm is given to construct the volterra kernels series terms and a nonlinear heat equation is discussed to illustrate the proposed method

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

an alternating projection that does not converge in norm this paper proves that the method of alternating projections between two closed convex intersecting sets does not always converge in norm weak convergence was established by bregman soviet math dokl 6 1965 688 but the status of norm convergence was undetermined an explicit counterexample is provided

diffusive approximation of a time fractional burger s equation in nonlinear acoustics a fractional time derivative is introduced into burger s equation to model losses of nonlinear waves this term amounts to a time convolution product which greatly penalizes the numerical modeling a diffusive representation of the fractional derivative is adopted here replacing this nonlocal operator by a continuum of memory variables that satisfy local in time ordinary differential equations then a quadrature formula yields a system of local partial differential equations well suited to numerical integration the determination of the quadrature coefficients is crucial to ensure both the well posedness of the system and the computational efficiency of the diffusive approximation for this purpose optimization with constraint is shown to be a very efficient strategy strang splitting is used to solve successively the hyperbolic part by a shock capturing scheme and the diffusive part exactly numerical experiments are proposed to assess the efficiency of the numerical modeling and to illustrate the effect of the fractional attenuation on the wave propagation

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

projection and proximal point methods convergence results and counterexamples recently hundal has constructed a hyperplane h a cone k and a starting point y0 in 2 such that the sequence of alternating projections pk ph n y0 n n converges weakly to some point in h k but not in norm we show how this construction results in a counterexample to norm convergence for iterates of averaged projections hence we give an a rmative answer to a question raised by reich two decades ago furthermore new counterexamples to norm convergence for iterates of rmly nonexpansive maps a la genel and lindenstrauss and for the proximal point algorithm al a g uler are provided we also present a counterexample along with some weak and norm convergence results for the new framework of string averaging projection methods introduced by censor et al extensions to banach spaces and the situation for the hilbert ball are discussed as well 2003 elsevier ltd all rights reserved msc 47h09 47j25 90c25

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

antenna array pattern synthesis via convex optimization we show that a variety of antenna array pattern synthesis problems can be expressed as convex optimization problems which can be numerically solved with great efficiency by recently developed interior point methods the synthesis problems involve arrays with arbitrary geometry and element directivity constraints on far and near field patterns over narrow or broad frequency bandwidth and some important robustness constraints we show several numerical simulations for the particular problem of constraining the beampattern level of a simple array for adaptive and broadband arrays

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

signal recovery with certain involved convex data fidelity constraints this paper proposes an optimization framework that can efficiently deal with convex data fidelity constraints onto which the metric projections are difficult to compute although such an involved data fidelity constraint is expected to play an important role in signal recovery under non gaussian noise contamination the said difficulty precludes existing algorithms from solving convex optimization problems with the constraint to resolve this dilemma we introduce a fixed point set characterization of involved data fidelity constraints based on a certain computable quasi nonexpansive mapping this characterization enables us to mobilize the hybrid steepest descent method to solve convex optimization problems with such a constraint the proposed framework can handle a variety of involved data fidelity constraints in a unified manner without geometric approximation to them in addition it requires no computationally expensive procedure such as operator inversion and inner loop as applications of the proposed framework we provide image restoration under several types of non gaussian noise contamination with illustrative examples

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

on the method of alternating resolvents abstract the work of hundal h hundal an alternating projection that does not converge in norm nonlinear anal 57 1 2004 35 61 has revealed that the sequence generated by the method of alternating projections converges weakly but not strongly in general in this paper we present several algorithms based on alternating resolvents of two maximal monotone operators a and b that can be used to approximate common zeros of a and b in particular we prove that the sequences generated by our algorithms converge strongly a particular case of such algorithms enables one to approximate minimum values of certain convex functionals

fitzpatrick functions cyclic monotonicity and rockafellar s antiderivative several deeper results on maximal monotone operators have recently found simpler proofs using fitzpatrick functions in this paper we study a sequence of fitzpatrick functions associated with a monotone operator the first term of this sequence coincides with the original fitzpatrick function and the other terms turn out to be useful for the identification and characterization of cyclic monotonicity properties it is shown that for any maximal cyclically monotone operator the pointwise supremum of the sequence of fitzpatrick functions is closely related to rockafellar s antiderivative several examples are explicitly computed for the purpose of illustration in contrast to rockafellar s result a maximal 3 cyclically monotone operator need not be maximal monotone a simplified proof of asplund s observation that the rotation in the euclidean plane by n is n cyclically monotone but not n 1 cyclically monotone is provided the fitzpatrick family of the subdifferential operator of a sublinear and of an indicator function is studied in detail we conclude with a new proof of moreau s result concerning the convexity of the set of proximal mappings

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

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

a multigrid conjugate residual method for the numerical solution of the hartree fock equation for diatomic molecules abstract discretization of the hartree fock equations in operational form leads to unsymmetric positive definite and indefinite linear equations to solve these equations a combination of the multigrid method and the orthomin method with gauss seidel relaxation as preconditioner is used the differential equations are approximated to the sixth order and the solution is extrapolated to the eighth order the method is fully parallelized the largest molecule treated is cuh

a general iterative method for a finite family of nonexpansive mappings abstract we study the approximation of common fixed points of a finite family of nonexpansive mappings and suggest a modification of the iterative algorithm without the assumption of any type of commutativity also we show that the convergence of the proposed algorithm can be proved under some types of control conditions

proximal point algorithms for nonsmooth convex optimization with fixed point constraints abstract the problem of minimizing the sum of nonsmooth convex objective functions defined on a real hilbert space over the intersection of fixed point sets of nonexpansive mappings onto which the projections cannot be efficiently computed is considered the use of proximal point algorithms that use the proximity operators of the objective functions and incremental optimization techniques is proposed for solving the problem with the focus on fixed point approximation techniques two algorithms are devised for solving the problem one blends an incremental subgradient method which is a useful algorithm for nonsmooth convex optimization with a halpern type fixed point iteration algorithm the other is based on an incremental subgradient method and the krasnosel ski mann fixed point algorithm it is shown that any weak sequential cluster point of the sequence generated by the halpern type algorithm belongs to the solution set of the problem and that there exists a weak sequential cluster point of the sequence generated by the krasnosel ski mann type algorithm which also belongs to the solution set numerical comparisons of the two proposed algorithms with existing subgradient methods for concrete nonsmooth convex optimization show that the proposed algorithms achieve faster convergence

asymptotic behavior of averaged and firmly nonexpansive mappings in geodesic spaces abstract we further study averaged and firmly nonexpansive mappings in the setting of geodesic spaces with a main focus on the asymptotic behavior of their picard iterates we use methods of proof mining to obtain an explicit quantitative version of a generalization to geodesic spaces of a result on the asymptotic behavior of picard iterates for firmly nonexpansive mappings proved by reich and shafrir from this result we obtain effective uniform bounds on the asymptotic regularity for firmly nonexpansive mappings besides this we derive effective rates of asymptotic regularity for sequences generated by two algorithms used in the study of the convex feasibility problem in a nonlinear setting

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

on the convergence of volterra series of finite dimensional quadratic mimo systems in this paper the volterra series decomposition of a class of quadratic time invariant single input finite dimensional systems is analyzed the kernels are given by a recursive sequence of linear pdes in the time domain and an equivalent algebraic recursion in the laplace domain this is used to prove the convergence of the volterra series to a possibly weak trajectory of the system to provide a practicable value for the radius of convergence of the input in l and to compute a guaranteed error bound in l for the truncated series the result is then extended to mimo systems a numerical simulation is performed on an academic siso example to illustrate how easily the truncated volterra series can be implemented

a family of projective splitting methods for the sum of two maximal monotone operators a splitting method for two monotone operators a and b is an algorithm that attempts to converge to a zero of the sum a b by solving a sequence of subproblems each of which involves only the operator a or only the operator b prior algorithms of this type can all in essence be categorized into three main classes the douglas peaceman rachford class the forward backward class and the little used double backward class through a certain extended solution set in a product space we construct a fundamentally new class of splitting methods for pairs of general maximal monotone operators in hilbert space our algorithms are essentially standard projection methods using splitting decomposition to construct separators we prove convergence through fejer monotonicity techniques but showing fejer convergence of a different sequence to a different set than in earlier splitting methods our projective algorithms converge under more general conditions than prior splitting methods allowing the proximal parameter to vary from iteration to iteration and even from operator to operator while retaining convergence for essentially arbitrary pairs of operators the new projective splitting class also contains noteworthy preexisting methods either as conventional special cases or excluded boundary cases

a contraction proximal point algorithm with two monotone operators abstract it is a known fact that the method of alternating projections introduced long ago by von neumann fails to converge strongly for two arbitrary nonempty closed and convex subsets of a real hilbert space in this paper a new iterative process for finding common zeros of two maximal monotone operators is introduced and strong convergence results associated with it are proved if the two operators are subdifferentials of indicator functions this new algorithm coincides with the old method of alternating projections several other important algorithms such as the contraction proximal point algorithm occur as special cases of our algorithm hence our main results generalize and unify many results that occur in the literature

convex optimization over fixed point sets of quasi nonexpansive and nonexpansive mappings in utility based bandwidth allocation problems with operational constraints network bandwidth allocation is a central issue in modern communication networks the main objective of the bandwidth allocation is to allocate an optimal bandwidth for maximizing a predefined utility over the capacity constraints to traffic sources when a centralized operator which manages all the bandwidth allocations in the network has a certain operational policy the bandwidth allocation reflecting the operational policy should result in the network being more stable and reliable accordingly we need to solve a network bandwidth allocation problem under both capacity constraints and operational constraints to develop a novel algorithm for solving the problem we translate the network bandwidth allocation problem into one of minimizing a convex objective function over the intersection of the fixed point sets of certain quasi nonexpansive and nonexpansive mappings and propose a fixed point optimization algorithm for solving it we numerically compare the proposed algorithm with the existing algorithm for solving a concrete bandwidth allocation problem and show its effectiveness

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

robust wideband beamforming by the hybrid steepest descent method this paper uses the hybrid steepest descent method hsdm to design robust smart antennas several design criteria as well as robustness are mathematically described by a finite collection of closed convex sets in a real euclidean space desirable beamformers are defined as points of the generalized convex feasible set which is well defined even in the case of inconsistent design criteria a quadratic cost function is formed by the correlations of the incoming data and the hsdm constructs a point sequence that strongly converges to the unique minimizer of the cost function over the generalized convex feasible set numerical examples validate the proposed design

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

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

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

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

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

algorithms for zeros of two accretive operators for solving convex minimization problems and its application to image restoration problems abstract in this paper a new viscosity iterative method minimization algorithm for image restoration algorithm is proposed based on a new viscosity iteration the algorithm for finding the common zeros of two accretive operators in the framework of uniformly smooth banach spaces moreover the strong convergence theorems for the iterative algorithms and an example is proposed which shows the validity of main theorem are proved the results of this paper are improved and extended of the corresponding ones announced by many others and we also applied our result to solve a convex minimization problem and g teaux differentiable e p s and variational inequality experiment results show that the proposed algorithms outperform some other methods finally we give the numerical experiments to show the efficiency and advantage of the proposed methods and we also used our proposed algorithm for applying to solve the image deblurring and image recovery problems

operator splittings bregman methods and frame shrinkage in image processing we examine the underlying structure of popular algorithms for variational methods used in image processing we focus here on operator splittings and bregman methods based on a unified approach via fixed point iterations and averaged operators in particular the recently proposed alternating split bregman method can be interpreted from different points of view as a bregman as an augmented lagrangian and as a douglas rachford splitting algorithm which is a classical operator splitting method we also study similarities between this method and the forward backward splitting method when applied to two frequently used models for image denoising which employ a besov norm and a total variation regularization term respectively in the first setting we show that for a discretization based on parseval frames the gradient descent reprojection and the alternating split bregman algorithm are equivalent and turn out to be a frame shrinkage method for the total variation regularizer we also present a numerical comparison with multistep methods

robust adaptive beamforming for general rank signal models the performance of adaptive beamforming methods is known to degrade severely in the presence of even small mismatches between the actual and presumed array responses to the desired signal such mismatches may frequently occur in practical situations because of violation of underlying assumptions on the environment sources or sensor array this is especially true when the desired signal components are present in the beamformer training data snapshots because in this case the adaptive array performance is very sensitive to array and model imperfections the similar phenomenon of performance degradation can occur even when the array response to the desired signal is known exactly but the training sample size is small we propose a new powerful approach to robust adaptive beamforming in the presence of unknown arbitrary type mismatches of the desired signal array response our approach is developed for the most general case of an arbitrary dimension of the desired signal subspace and is applicable to both the rank one point source and higher rank scattered source fluctuating wavefront desired signal models the proposed robust adaptive beamformers are based on explicit modeling of uncertainties in the desired signal array response and data covariance matrix as well as worst case performance optimization simple closed form solutions to the considered robust adaptive beamforming problems are derived our new beamformers have a computational complexity comparable with that of the traditional adaptive beamforming algorithms while at the same time offer a significantly improved robustness and faster convergence rates

variable metric quasi fejer monotonicity the notion of quasi fejer monotonicity has proven to be an efficient tool to simplify and unify the convergence analysis of various algorithms arising in applied nonlinear analysis in this paper we extend this notion in the context of variable metric algorithms whereby the underlying norm is allowed to vary at each iteration applications to convex optimization and inverse problems are demonstrated

the multigrid method for accelerated solution of the discretized schrodinger equation abstract the use of the new multigrid method for the numerical solution of the discretized schrodinger equation is discussed a simple approach is chosen to circumvent the difficulties associated with the fact that the schrodinger differential operator is not positive definite the scheme is applied to a two dimensional model problem which includes the essential features present in actual scattering problems the results show that significant savings can be achieved both in computational work and core memory requirements the multigrid method should greatly increase the utility of meshing procedures for solving the partial differential equations of quantum mechanics

hard constrained inconsistent signal feasibility problems we consider the problem of synthesizing feasible signals in a hilbert space in the presence of inconsistent convex constraints some of which must imperatively be satisfied this problem is formalized as that of minimizing a convex objective measuring the amount of violation of the soft constraints over the intersection of the sets associated with the hard ones the resulting convex optimization problem is analyzed and numerical solution schemes are presented along with convergence results the proposed formalism and its algorithmic framework unify and extend existing approaches to inconsistent signal feasibility problems an application to signal synthesis is demonstrated

approximation common zero of two accretive operators in banach spaces the purpose of this paper is to introduce a new iterative method that is the combination of the proximal point algorithm viscosity approximation method and alternating resolvent method for finding the common zeros of two accretive operators in banach spaces and we will prove the strong convergence theorems for the iterative algorithms and give the example of the main theorems the results of this paper are improvements and extensions of the corresponding ones announced by many others

monotone operator theory in convex optimization several aspects of the interplay between monotone operator theory and convex optimization are presented the crucial role played by monotone operators in the analysis and the numerical solution of convex minimization problems is emphasized we review the properties of subdifferentials as maximally monotone operators and in tandem investigate those of proximity operators as resolvents in particular we study new transformations which map proximity operators to proximity operators and establish connections with self dual classes of firmly nonexpansive operators in addition new insights and developments are proposed on the algorithmic front

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

set intersection problems supporting hyperplanes and quadratic programming we study how the supporting hyperplanes produced by the projection process can complement the method of alternating projections and its variants for the convex set intersection problem for the problem of finding the closest point in the intersection of closed convex sets we propose an algorithm that like dykstra s algorithm converges strongly in a hilbert space moreover this algorithm converges in finitely many iterations when the closed convex sets are cones in mathbb r n r n satisfying an alignment condition next we propose modifications of the alternating projection algorithm and prove its convergence the algorithm converges superlinearly in mathbb r n r n under some nice conditions under a conical condition the convergence can be finite lastly we discuss the case where the intersection of the sets is empty