on greedy randomized kaczmarz method for solving large sparse linear systems for solving large scale systems of linear equations by iteration methods we introduce an effective probability criterion for selecting the working rows from the coefficient matrix and construct a

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

multiplicativity of lp norms for matrices abstract the l p norm and the l p operator norm of an m n complex matrix a ij are given by a p i j ij p 1 p and a p max ax p xe c n x p 1 respectively the main purpose of this paper is to investigate the multipicativity of the l p norms and their relation to the l p operator norms

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

linear convergence of subgradient algorithm for convex feasibility on riemannian manifolds we study the convergence issue of the subgradient algorithm for solving the convex feasibility problems in riemannian manifolds which was first proposed and analyzed by bento and melo j optim theory appl 152 2012 pp 773 785 the linear convergence property about the subgradient algorithm for solving the convex feasibility problems with the slater condition in riemannian manifolds are established and some step sizes rules are suggested for finite convergence purposes which are motivated by the work due to de pierro iusem appl math optim 17 1988 pp 225 235 as a by product the convergence result of this algorithm is obtained for the convex feasibility problem without the slater condition assumption these results extend and or improve the corresponding known ones in both the euclidean space and riemannian manifolds

relaxation methods for problems with strictly convex separable costs and linear constraints we consider the minimization problem with strictly convex possibly nondifferentiable separable cost and linear constraints the dual of this problem is an unconstrained minimization problem with differentiable cost which is well suited for solution by parallel methods based on gauss seidel relaxation we show that these methods yield the optimal primal solution and under additional assumptions an optimal dual solution to do this it is necessary to extend the classical gauss seidel convergence results because the dual cost may not be strictly convex and may have unbounded level sets

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

a method for approximating the solution set of a system of convex inequalities by polytopes abstract in this note a method for computing approximating by polytopes of the solution set q of a system of convex inequalities is presented it is shown that such approximations can be determined by an algorithm which converges in finitely many steps when the solution set of the given system of inequalities is bounded in this case the algorithm generates inner and outer approximations having the hausdorff distance to each other and to the set q no greater than an a priori fixed and having their extreme points in q and in the relative exterior of q respectively

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

optimization of log x entropy over linear equality constraints in this paper we develop a special purpose iterative algorithm of the row action type for solving the problem of maximizing the log x entropy functional over linear equality constraints the algorithm employs projections onto hyperplanes which we call log x entropy projections a complete proof of convergence is given

operator norms multiplicativity factors and c numerical radii abstract let v be a normed vector space over c let b v denote the algebra of linear bounded operators on v and let n be an arbitrary seminorm or norm on b v in this paper we discuss multiplicativity factors for n i e constants 0 for which n n is submultiplicative we find that while in the finite dimensional case nontrivial indefinite seminorms have no multiplicativity factors and norms do have multiplicativity factors in the infinite dimensional case n may or may not have such factors our results are then applied in order to compute multiplicativity factors for certain generalizations of the classical numerical radius called c numerical radii this is done with the help of a combinatorial inequality which seems to be of independent interest

first and second order sensitivity analysis of linear and nonlinear structures this paper employs the principle of virtual work to derive sensitivity derivatives of structural response with respect to stiffness parameters using both direct and adjoint approaches the computations required are based on additional load conditions characterized by imposed initial strains body forces or surface tractions as such they are equally applicable to numerical or analytical solution techniques the relative efficiency of various approaches for calculating first and second derivatives is assessed it is shown that for the evaluation of second derivatives the most efficient approach is one that makes use of both the first order sensitivities and adjoint vectors two example problems are used for demonstrating the various approaches

optimal design of rib stiffeners in disks and plates abstract disks and plates subjected to stretching and flexure are stiffened by curvilinear ribstiffeners whose shape and cross sectional area are subject to optimization sensitivity analysis is first discussed for an arbitrary integral functional of generalized stress strain and displacement and boundary domains the optimality conditions are next derived the general theory is illustrated by three examples of optimal design of ring stiffeners in circular disks and plates and of rib stiffeners in a rectangular plate

the projection method for solving nonlinear systems of equations under the most violated constraint control abstract we extend a block version of kaczmarz s method with the most violated constraint control to the case of a square nonlinear system of equations without convexity assumptions we prove a local linear convergence theorem and show some numerical experiences

a computational solution of the inverse problem in radiation therapy treatment planning radiation therapy concerns the delivery of the proper dose of radiation to a tumor volume without causing irreparable damage to healthy tissue and critical organs the forward problem refers to calculating the dose distribution that would be delivered to a measured patient cross section when the radiation field generated by the beam sources is specified the inverse problem refers to calculating the radiation field that will provide a specified dose distribution in the patient the forward and inverse problems of radiation therapy treatment planning are first formulated in their continuous versions and the point is made that in this field of application the inverse problem calls for the inversion of an operator for which no analytic closed form mathematical representation exists to attack the inverse problem under such circumstances a discretized model is set up in which both patient section and radiation field are finely discretized this leads to a linear feasibility problem which is solved by a relaxation method the paper includes full details of this new approach with discussion of technical aspects such as dose apportionment software development and experimental results consequences and limitations as well as a precise comparison with other methodologies are also discussed

variational approach by means of adjoint systems to structural optimization and sensitivity analysis ii structure shape variation abstract for a linear elastic structure the first variation of an arbitrary stress strain and displacement functionals corresponding to variation of shape of external boundaries or interfaces is derived by using the solutions for primary and adjoint systems the application to optimal design is next presented and the relevant optimally conditions are derived from general expressions the path independent integrals used in fracture mechanics are rederived as a particular case of general expressions

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

multiplicativity factors for c numerical radii abstract for matrices a c c n x n the c numerical radius of a is the nonnegative quantity r c a max tr cu au uunitary this generalizes the classical numerical radius r a it is known that r c constitutes a norm on c n x n if and only if c is nonscalar and tr c 0 for all such c we obtain multiplicativity factors for r c i e constant 0 for which r c is submultiplicative on c n x n

mixed multiplicativity and lp norms for matrices abstract let cm n denote the class of m n complex matrices and let n1 n2 and n3 be arbitrary norms on cm n cm k and ck n respectively in this paper we discuss the best least positive constant minwhich satisfies n 1 ab min n 2 a n 3 b a c m k b c k n in particular for 1 p let a p i 1 m j 1 n ij p 1 p be the lp norm of a matrix a ij cm n then for arbitrary p q r such that 1 p q r we determine explicitly the best constant min for which ab p min a q b r a c m k b c k n

maximum likelihood reconstruction for emission tomography previous models for emission tomography et do not distinguish the physics of et from that of transmission tomography we give a more accurate general mathematical model for et where an unknown emission density x y z generates and is to be reconstructed from the number of counts n d in each of d detector units d within the model we give an algorithm for determining an estimate of which maximizes the probability p n of observing the actual detector count data n over all possible densities let independent poisson variables n b with unknown means b b 1 b represent the number of unobserved emissions in each of b boxes pixels partitioning an object containing an emitter suppose each emission in box b is detected in detector unit d with probability p b d d 1 d with p b d a one step transition matrix assumed known we observe the total number n n d of emissions in each detector unit d and want to estimate the unknown b b 1 b for each the observed data n has probability or likelihood p n the em algorithm of mathematical statistics starts with an initial estimate 0 and gives the following simple iterative procedure for obtaining a new estimate new from an old estimate old to obtain k k 1 2 new b old b dd 1 n d p b d old b p b d b 1 b

design sensitivity metric for structural dynamic response an approach to study the sensitivity of dynamic responses for structural systems with respect to a set of design parameters is presented the sensitivity is evaluated by considering the behavior of the system response when the design parameters vary within a given region of the design space the sensitivity is computed by means of approximate responses which are evaluated using approximation concepts the approximation is based on modal analysis and it is valid for general underdamped linear systems intermediate design variable and intermediate response quantity concepts are used to enhance the accuracy of the approximation in this approach modal energies and mode shapes are chosen as intermediate response quantities and they are approximated in terms of selected intermediate system parameters the approximation of these quantities requires a standard eigenvalue and eigenvector sensitivity analysis numerical results that illustrate the usefulness and effectiveness of the method are presented great insight into the behavior of the system can be gained using this methodology

extensions of hildreth s row action method for quadratic programming an extended version of hildreth s iterative quadratic programming algorithm is presented geometrically interpreted and proved to produce a sequence of iterates that i converges to the solution and ii has an important intermediate optimality property this extended hildreth algorithm is cast into a new form which more pronouncedly brings out its primal dual nature the application of the algorithm may be governed by an index sequence which is more general than a cyclic sequence namely by an almost cyclic control and a sequence of relaxation parameters is incorporated without ruining convergence the algorithm is a row action method which is particularly suitable for handling large or huge and sparse systems

algorithms for the quasiconvex feasibility problem we study the behavior of subgradient projections algorithms for the quasiconvex feasibility problem of finding a point x r n that satisfies the inequalities f 1 x 0 f 2 x 0 f m x 0 where all functions are continuous and quasiconvex we consider the consistent case when the solution set is nonempty since the fenchel moreau subdifferential might be empty we look at different notions of the subdifferential and determine their suitability for our problem we also determine conditions on the functions that are needed for convergence of our algorithms the quasiconvex functions on the left hand side of the inequalities need not be differentiable but have to satisfy a lipschitz or a holder condition

parallel and sequential kaczmarz methods for solving underdetermined nonlinear equations abstract we analyze the convergence of iterative process in r n of the type x k 1 x k w k using this theory we prove the local convergence of a sequential kaczmarz type method and a parallel kaczmarz type method for solving underdetermined systems of nonlinear equations some numerical experiences are presented

on subgradient projectors the subgradient projector is of considerable importance in convex optimization because it plays the key role in polyak s seminal work and the many papers it spawned on subgradient projection algorithms for solving convex feasibility problems in this paper we offer a systematic study of the subgradient projector fundamental properties such as continuity nonexpansiveness and monotonicity are investigated we also discuss the yamagishi yamada operator numerous examples illustrate our results

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

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 higher numerical radius and spectral norm abstract let c n n be the vector space of n n complex matrices and denote by u n the group of n n unitary matrices in c n n for 1 m n set jm im 0m n whereim is the m m identity matrix and 0k denotes a k k zero matrix with this for ain c n n and 1 m n we define r m a max tr j m uau u u n and a m max tr j m uav u v u n as the m numerical radius and m spectral norm of a respectively when m 1 these quantities are reduced to the classical numerical radius and spectral norm in this note we obtain inequalities concerning m and rm for 1 m n examples are given to show that the inequalities obtained cannot be improved we also study multiplicativity factors for rm 1 m n

comparison of local and global approximations for reliability estimation an approach based on approximation concepts to estimate system reliability is presented limit state functions are approximated explicitly in terms of the random variables of the system these approximations are then used in conjunction with monte carlo simulation to estimate system reliability local and global approximations are studied and compared in terms of efficiency and accuracy for their potential use in reliability based optimization example problems are presented to illustrate the ideas set forth

convex feasibility modeling and projection methods for sparse signal recovery a computationally efficient method for recovering sparse signals from a series of noisy observations known as the problem of compressed sensing cs is presented the theory of cs usually leads to a constrained convex minimization problem in this work an alternative outlook is proposed instead of solving the cs problem as an optimization problem it is suggested to transform the optimization problem into a convex feasibility problem cfp and solve it using feasibility seeking sequential and simultaneous subgradient projection methods which are iterative fast robust and convergent schemes for solving cfps as opposed to some of the commonly used cs algorithms such as bayesian cs and gradient projections for sparse reconstruction which become inefficient as the problem dimension and sparseness degree increase the proposed methods exhibit robustness with respect to these parameters moreover it is shown that the cfp based projection methods are superior to some of the state of the art methods in recovering the signal s support numerical experiments show that the cfp based projection methods are viable for solving large scale cs problems with compressible signals

an automatic relaxation method for solving interval linear inequalities abstract the problem of solving iteratively a large and possibly sparse system of interval linear inequalities ax is considered an algorithm of the row action type is proposed which in each iterative step effectively takes account of a pair of inequalities describing a single interval inequality the algorithm realizes in an automatic manner a relaxation principle proposed by goffin but also allows further external relaxation parameters

convergence of the cyclical relaxation method for linear inequalities the relaxation method for linear inequalities is studied and new bounds on convergence obtained an asymptotically tight estimate is given for the case when the inequalities are processed in a cyclical order an improvement of the estimate by an order of magnitude takes place if strong underrelaxation is used bounds on convergence usually involve the so called condition number of a system of linear inequalities which we estimate in terms of their coefficient matrix

the dual normal chip and linear regularity for infinite systems of convex sets in banach spaces for an indexed collection of convex sets in a banach space with index set i of cardinality i of i which may be finite or infinite we investigate the interrelationship between various qualification notions including the linear regularity the normal property the dual normal conical hull intersection property and jameson s property g together with their quantitative versions and thereby we extend the well known results of bakan deutsch and li trans amer math soc 357 2005 pp 3831 3863 and bauschke borwein and li math program ser a 86 1999 pp 135 160 from the finite case to the more general case that allows i to be infinite

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

a modified hybrid projection method for solving generalized mixed equilibrium problems and fixed point problems in banach spaces in this paper we introduce a modified new hybrid projection method for finding the set of solutions of the generalized mixed equilibrium problems and the convex feasibility problems for an infinite family of closed and uniformly quasi asymptotically nonexpansive mappings strong convergence theorems are established in a uniformly smooth and strictly convex banach space which also enjoys the kadec klee property our results improve and extend the corresponding results announced by qin et al 2010 and many authors

improved transient response approximation for general damped systems a new approximation for peak transient dynamic displacements is presented this approximation is based on modal analysis and is valid for general damped structures which includes the case of actively controlled structures intermediate design variable and intermediate response quantity concepts are used to enhance the accuracy of the approximation when this high quality approximation is used within the context of the approximation concepts approach to design synthesis the overall efficiency of the process expected to be improved by reducing the number of complete analyses required for convergence numerical results that illustrate the effectiveness of the method are presented

perturbed projections and subgradient projections for the multiple sets split feasibility problem we study the multiple sets split feasibility problem that requires to find a point closest to a family of closed convex sets in one space such that its image under a linear transformation will be closest to another family of closed convex sets in the image space by casting the problem into an equivalent problem in a suitable product space we are able to present a simultaneous subgradients projections algorithm that generates convergent sequences of iterates in the feasible case we further derive and analyze a perturbed projection method for the multiple sets split feasibility problem and additionally furnish alternative proofs to two known results

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

some open problems in matrix theory arising in linear systems and control control theory has long provided a rich source of motivation for developments in matrix theory accordingly we discuss some open problems in matrix theory arising from theoretical and practical issues in linear systems theory and feedback control the problems discussed include robust stability matrix exponent induced norms stabilizability and pole assignability and nonstandard matrix equations a substantial number of references are included to acquaint matrix theorists with problems and trends in this application area feedback control theory has long provided a rich source of motivation for developments in matrix theory the purpose of this paper is to discuss several open problems in matrix theory that arise from theoretical and practical issues in feedback control theory and the associated area of linear systems theory many of these problems are remarkably simple to state are of intense interest in control theory and applications and yet remain unsolved besides leading to the resolution of these problems it is hoped that this paper will help to stimulate increased interaction between matrix and control theorists accord

strong underrelaxation in kaczmarz s method for inconsistent systems we investigate the behavior of kaczmarz s method with relaxation for inconsistent systems we show that when the relaxation parameter goes to zero the limits of the cyclic subsequences generated by the method approach a weighted least squares solution of the system this point minimizes the sum of the squares of the euclidean distances to the hyperplanes of the system if the starting point is chosen properly then the limits approach the minimum norm weighted least squares solution the proof is given for a block kaczmarz 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

a simultaneous iterative method for computing projections on polyhedra a simultaneous version of hildreth s iterative algorithm for norm minimization over linear inequalities is presented proofs are given showing that the algorithm converges from any starting point to its projection on the linear constraints set in the feasible case and to the nearest least squares solution in the general case

convergence results for an accelerated nonlinear cimmino algorithm we present an accelerated version of cimmino s algorithm for solving the convex feasibility problem in finite dimension the algorithm is similar to that given by censor and elfving for linear inequalities we show that the nonlinear version converges locally to a weighted least squares solution in the general case and globally to a feasible solution in the consistent case applications to the linear problem are suggested

on ip1 p2 antinorms of nonnegative matrices let 0 p p1 p2 1 x xi rn a aik rm n i e the vector x is elementwise positive and so is the matrix a the lp antinorm of x is defined by x p ixpi1p the lp1 p2 antinorm of a by a p1 p2 k iap1ikp2p11p2 and the lp1 p2 operator antinorm of a by a minp1 p2 infx rn ax p1 x p2 several properties of the antinorms p p1 p2 and minp1 p2 especially the supermultiplicativity of p1 p2 are studied and generalized to hold for nonnegative matrices the results are analogous with the well known submultiplicativity properties of lq1 q2 norms and lq1 q2 operator norms q1 q2 1 a generalization is outlined

multiplicativity factors and mixed multiplicativity abstract a number of authors have discussed multiplicativity factors associated with a single norm on an operator algebra we extend this concept to multiplicativity factors associated with three norms on spaces not necessarily algebras of bounded linear operators we generalize previous results and give several finite and infinite dimensional examples

a row action method for convex programming we present a new method for minimizing a strictly convex function subject to general convex constraints constraints are used one at a time no changes are made in the constraint functions thus the row action nature of the algorithm and at each iteration a subproblem is solved consisting of minimization of the objective function subject to one or two linear equations convergence of the algorithm is established and the method is compared with other row action algorithms for several relevant particular cases

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

sensitivity analysis of discrete structural systems methods for calculating sensitivity derivatives for discrete structural systems are surveyed primarily covering literature published during the past two decades methods are described for calculating derivatives of static displacements and stresses eigenvalues and eigenvectors transient structural response and derivatives of optimum structural designs with respect to problem parameters the survey is focused on publications developed in nonstructural fields such as electronics controls and physical chemistry which are directly applicable to structural problems most notable among the nonstructural based methods are the adjoint variable technique from control theory and the green s function and fast methods from physical chemistry

the adventures of a simple algorithm abstract this paper presents analyzes and tests a row action method for solving the regularized linear programming problem minimize 1 2 x 2 2 c t x subject to a x b where is a given positive constant the proposed method is based on the observation that the dual of this problem has the form maximize b t y 1 2 a t y c 2 2 subject to y 0 and if y solves the dual then the point x a t y c provides the unique solution of the primal maximizing the dual objective function by changing one variable at a time results in an effective row action method which is suitable for solving large sparse problems one aim of this paper is to clarify the convergence properties of the proposed scheme let y k denote the estimated dual solution at the end of the k th iteration and let x k a t y k c denote the corresponding primal estimate it is proved that the sequence x k converges to x while the sequence y k converges to a point y that solves the dual the only assumption which is needed in order to establish these claims is that the feasible region is not empty yet perhaps the more striking features of the algorithm are related to temporary situations in which it attempts to solve an inconsistent system in such cases the sequence y k obeys the rule y k u k k v where u k is a fastly converging sequence and v is a fixed vector that satisfies a t v 0 and b t v 0 so the sequence x k is almost unchanged for many iterations the paper ends with numerical experiments that illustrate the effects of this phenomenon and the close links with kaczmarz s sor method

radiation theraphy treatment planning an l2 approach we formulate the radiation theraphy treatment planning rttp problem as constrained optimization in l 2 this is accomplished by first describing the dose operator as a bounded operator from treatment space into patient space both suitable l 2 spaces within this context we examine two problems of interest in rttp first we take the case where the sources of radiation are fixed a priori and provide algorithms which lead to the optimal choices of beam profiles for these sources secondly we look at the problem of simultaneously choosing the sources and beam profiles and also suggest an algorithm for this case

free steering relaxation methods for problems with strictly convex costs and linear constraints we consider dual coordinate ascent methods for minimizing a strictly convex possibly nondifferentiable function subject to linear constraints such methods are useful in large scale applications e g entropy maximization quadratic programming network flow because they are simply can exploit sparsity and in certain cases are highly parallelizable we establish their global convergence under weak conditions and a free steering order of relaxation previous comparable results were restricted to special problems with separable costs and equality constraints our convergence framework unifies to a certain extent the approaches of bregman censor and lent de pierro and iusem and luo and tseng and complements that of bertsekas and tseng

new approximation method for stress constraints in structural synthesis a new approximation method for dealing with stress constraints in structural synthesis is presented in previous methods it has been common to create an approximation of the actual constraints with respect to the design variables and then use this with optimization to produce a new candidate design in the present method the finite element nodal forces are instead approximated and these are used to create an explicit but often nonlinear approximation to the original problem the principal motivation is to create the best approximation possible in order to reduce the number of detailed finite element analyses needed to reach the optimum the method is shown to be quite simple while providing significant efficiency gains in the overall optimization task also the method produces such high quality approximations to the original problem that the need for move limits during the approximate optimization stage is greatly reduced examples are offered and compared with published results to demonstrate the efficiency and reliability of the proposed method it is concluded that there are considerable gains yet to be achieved through continued careful investigation of approximation techniques

on the numerical radius and its applications abstract we give a brief account of the numerical radius of a linear bounded operator on a hilbert space and some of its better known properties both finite and infinite dimensional aspects are discussed as well as applications to stability theory of finite difference approximations for hyperbolic initial value problems

a statistical model for positron emission tomography abstract positron emission tomography pet still in its research stages is a technique that promises to open new medical frontiers by enabling physicians to study the metabolic activity of the body in a pictorial manner much as in x ray transmission tomography and other modes of computerized tomography the quality of the reconstructed image in pet is very sensitive to the mathematical algorithm to be used for reconstruction in this article we tailor a mathematical model to the physics of positron emissions and we use the model to describe the basic image reconstruction problem of pet as a standard problem in statistical estimation from incomplete data we describe various estimation procedures such as the maximum likelihood ml method using the em algorithm the method of moments and the least squares method a computer simulation of a pet experiment is then used to demonstrate the ml and the least squares reconstructions the main purposes of this article are to report on what we believe is an

on the error estimate of the randomized double block kaczmarz method abstract in this paper we consider the convergence analysis of the randomized double block kaczmarz method and improve the upper bound of the error estimate in expectation of the randomized double block kaczmarz method numerical experiments are given to demonstrate the theoretical results and to show a large gap between the new and the old bound

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

dual ascent methods for problems with strictly convex costs and linear constraints a unified approach consider problems of the form text p qquad min f x ex geqq b where f is a strictly convex possibly nondifferentiable function and e and b are respectively a matrix and a vector a popular method for solving special cases of p e g network flow entropy maximization quadratic program is to dualize the constraints ex geqq b to obtain a differentiable maximization problem and then apply an iterative ascent method to solve it this method is simple and can exploit sparsity thus making it ideal for large scale optimization and in certain cases for parallel computation despite its simplicity however convergence of this method has been shown only under certain very restrictive conditions and only for certain special cases of p in this paper a block coordinate ascent method is presented for solving p that contains as special cases both dual coordinate ascent methods and dual gradient methods it is shown under certain mild assumptions on f and p that this method