obtuse cones and gram matrices with nonnegative inverse abstract we study some properties of gram matrices with nonnegative inverse which lead to the constructions of obtuse cones these constructions can find applications in optimization problems e g in methods of convex feasibility problems or of convex minimization problems

how to project onto an isotone projection cone abstract the solution of the complementarity problem defined by a mapping f r n r n and a cone k r n consists of finding the fixed points of the operator p k i f where p k is the projection onto the cone k and i stands for the identity mapping for the class of isotone projection cones cones admitting projections isotone with respect to the order relation they generate and f satisfying certain monotonicity properties the solution can be obtained by iterative processes see g isac a b nemeth projection methods isotone projection cones and the complementarity problem j math anal appl 153 1 1990 258 275 and s z nemeth iterative methods for nonlinear complementarity problems on isotone projection cones j math anal appl 350 1 2009 340 347 these algorithms require computing at each step the projection onto the cone k in general computing the projection mapping onto a cone k is a difficult and computationally expensive problem in this note it is shown that the projection of an arbitrary point onto an isotone projection cone in r n can be obtained by projecting recursively at most n 1 times into subspaces of decreasing dimension this emphasizes the efficiency of the algorithms mentioned above and furnishes a handy tool for some problems involving special isotone projection cones as for example the non negative monotone cones occurring in reconstruction problems see e g section 5 13 in j dattorro convex optimization and euclidean distance geometry meboo 2005 v2009 04 11

an adaptive level set method for nondifferentiable constrained image recovery the formulation of a wide variety of image recovery problems leads to the minimization of a convex objective over a convex set representing the constraints derived from a priori knowledge and consistency with the observed signals in previous years nondifferentiable objectives have become popular due in part to their ability to capture certain features such as sharp edges they also arise naturally in minimax inconsistent set theoretic recovery problems at the same time the issue of developing reliable numerical algorithms to solve such convex programs in the context of image recovery applications has received little attention we address this issue and propose an adaptive level set method for nondifferentiable constrained image recovery the asymptotic properties of the method are analyzed and its implementation is discussed numerical experiments illustrate applications to total variation and minimax set theoretic image restoration and denoising problems

eigenvalue accumulation for singular sturm liouville problems nonlinear in the spectral parameter abstract for certain singular sturm liouville equations whose coefficients depend continuously on the spectral parameter in an interval it is shown that accumulation nonaccumulation of eigenvalues at an endpoint of is essentially determined by oscillatory properties of the equation at the boundary as applications new results are obtained for the radial dirac operator and the klein gordon equation three other physical applications are also considered

eigenvalue accumulation for dirac operators with spherically symmetric potential we consider dirac operators h in with spherically symmetric potentials the main result is a criterion for eigenvalue accumulation and non accumulation at the endpoints 1 and 1 of the essential spectrum under rather weak assumptions on the potential this result is proved by showing an analogous criterion for the associated radial dirac operators h and by proving that for sufficiently large each h does not have any eigenvalues in the interval 1 0 and 0 1 respectively of the gap 1 1 of the essential spectrum for the latter properties of solutions of certain riccati differential equations depending on the parameter and the spectral parameter are used

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

the efficiency of ballstep subgradient level methods for convex optimization 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 establish convergence and efficiency estimates for simple ballstep level controls without requiring that the feasible set be compact our framework may handle accelerations based on cheap projections surrogate constraints and conjugate subgradient techniques

real versus complex robustness margin continuity as a smooth versus holomorphic singularity problem abstract it is well known that the real robustness margin can be discontinuous while the complex robustness margin is always continuous relative to problem data using some concepts from set valued analysis continuity of can be viewed as structural stability of the neutral stability region from this point of view the crucial issue is whether 0 j0 is a critical value of the return difference map this paper shows that the discrepancy between real and complex cases is due to the additional holomorphic property of the nyquist return difference mapping of the complex function the critical points of the nyquist map in the complex case are at most finite in number in contrast the critical points of the nyquist map of the real smooth case form generically a curve furthermore and more importantly in the complex case even when 0 j0 is critical the stability crossover is continuously deformed under the variation of certain parameters while in the real case the crossover could sustain a catastrophic change

computational geometry of positive definiteness abstract in matrix computations such as in factoring matrices hermitian and preferably positive definite elements are occasionally required related problems can often be cast as those of existence of respective elements in a matrix subspace for two dimensional matrix subspaces first results in this regard are due to finsler to assess positive definiteness in larger dimensional cases the task becomes computational geometric for the joint numerical range in a natural way the hermitian element of the frobenius norm one with the maximal least eigenvalue is found to this end extreme eigenvalue computations are combined with ellipsoid and perceptron algorithms

a spectral theory for a λ rational sturm liouville problem abstract we consider the regular sturm liouville problem y py q u y 0 which contains the eigenvalue parameter rationally under certain assumptions on p q and u it is shown that the spectrum of the problem consists of a continuous component the range of the function u discrete eigenvalues and possibly a finite number of embedded eigenvalues in the considered situation the continuous spectrum is absolutely continuous and explicit formulas for the spectral density and the corresponding fourier transform are given

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

a note on riesz bases of eigenvectors of certain holomorphic operator functions abstract operator valued functions of the form a a q with q a 1 compact valued and holomorphic on certain domains c are considered in separable hilbert space assuming that the resolvent of a is compact its eigenvalues are simple and the corresponding eigenvectors form a riesz basis for h of finite defect it is shown that under certain growth conditions on q a 1 the eigenvectors of a corresponding to a part of its spectrum also form a riesz basis of finite defect applications are given to operator valued functions of the form a a b d 1c and to spectral problems in l2 0 1 of the form f x p x f x q x f x f x with for example dirichlet boundary conditions

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

new variants of bundle methods in this paper we describe a number of new variants of bundle methods for nonsmooth unconstrained and constrained convex optimization convex concave games and variational inequalities we outline the ideas underlying these methods and present rate of convergence estimates

on essential spectra of operator matrices and their feshbach maps abstract a connection between the essential spectrum of certain operator matrices and essential spectra of the corresponding feshbach maps is discussed and applied to some concrete rational operator valued functions

variable target value subgradient method polyak s subgradient algorithm for nondifferentiable optimization problems requires prior knowledge of the optimal value of the objective function to find an optimal solution in this paper we extend the convergence properties of the polyak s subgradient algorithm with a fixed target value to a more general case with variable target values then a target value updating scheme is provided which finds an optimal solution without prior knowledge of the optimal objective value the convergence proof of the scheme is provided and computational results of the scheme are reported

a method of projection onto an acute cone with level control in convex minimization we study a projection method for convex minimization problems in each step we construct a certain polyhedral function that approximates the objective function from below the affine pieces of the approximating function are linearizations of the objective function obtained in previous steps the linearization are determined by a linearly independent system of subgradients which generates an obtuse cone the current approximation of the solution is projected onto a sublevel set of the polyhedral function with level which estimates the optimal value and is updated in each iteration the projection can be easily obtained by solving a system of linear equations and is actually the projection onto a translated acute cone dual to the constructed obtuse cone the construction ensures long steps between succesive approximations of the solution such long steps are important for the good behavior of the method

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

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

m form numerical range and the computation of the structured singular value an equivalent expressions for the structures singular value is proposed leading to an alternative algorithm for its computation the approach is based to the geometric properties of the m form numerical range of certain tuples of matrices similar to previously considered schemes the algorithm proposed is proved to give the correct value for block structures of size up to three for larger sizes insight is gained into the question of the possible gap between the structured singular value and its known upper bound

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 survey on the continuous nonlinear resource allocation problem our problem of interest consists of minimizing a separable convex and differentiable function over a convex set defined by bounds on the variables and an explicit constraint described by a separable convex function applications are abundant and vary from equilibrium problems in the engineering and economic sciences through resource allocation and balancing problems in manufacturing statistics military operations research and production and financial economics to subproblems in algorithms for a variety of more complex optimization models this paper surveys the history and applications of the problem as well as algorithmic approaches to its solution the most common techniques are based on finding the optimal value of the lagrange multiplier for the explicit constraint most often through the use of a type of line search procedure we analyze the most relevant references especially regarding their originality and numerical findings summarizing with remarks on possible extensions and future research 2006 elsevier b v all rights reserved

generalized inverses of partitioned matrices in banachiewicz schur form abstract the problem of developing conditions under which generalized inverses of a partitioned matrix can be expressed in the so called banachiewicz schur form is reconsidered theorem of marsaglia and styan sankhy ser a 36 1974 437 concerning the class of all generalized inverses the class of reflexive generalized inverses and the moore penrose inverse is strengthened and new results are established for the classes of outer inverses least squares generalized inverses and minimum norm generalized inverses

numerical shadows measures and densities on the numerical range abstract for any operator m acting on an n dimensional hilbert space h n we introduce its numerical shadow which is a probability measure on the complex plane supported by the numerical range of m the shadow of m at point z is defined as the probability that the inner product mu u is equal to z where u stands for a random complex vector from h n satisfying u 1 in the case of n 2 the numerical shadow of a non normal operator can be interpreted as a shadow of a hollow sphere projected on a plane a similar interpretation is provided also for higher dimensions for a hermitian m its numerical shadow forms a probability distribution on the real axis which is shown to be a one dimensional b spline in the case of a normal m the numerical shadow corresponds to a shadow of a transparent solid simplex in r n 1 onto the complex plane numerical shadow is found explicitly for jordan matrices j n direct sums of matrices and in all cases where the shadow is rotation invariant results concerning the moments of shadow measures play an important role a general technique to study numerical shadow via the cartesian decomposition is described and a link of the numerical shadow of an operator to its higher rank numerical range is emphasized

modifications and implementation of the ellipsoid algorithm for linear programming we give some modifications of the ellipsoid algorithm for linear programming and describe a numerically stable implementation we are concerned with practical problems where user supplied bounds can usually be provided our implementation allows constraint dropping and updates bounds on the optimal value and should be able to terminate with an indication of infeasibility or with a provably good feasible solution in a moderate number of iterations

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

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

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

accumulation of discrete eigenvalues of the radial dirac operator abstract for bounded potentials which behave like cx at infinity we investigate whether discrete eigenvalues of the radial dirac operatorh accumulate at 1 or not it is well known that 2 is the critical exponent we show thatc 1 8 1 2 is the critical coupling constant in the case 2 our approach is to transform the radial dirac equation into a sturm liouville equation nonlinear in the spectral parameter and to apply a new general result on accumulation of eigenvalues of such equations

on flat portions on the boundary of the numerical range abstract the paper is devoted to matrices with flat portions on the boundary of their numerical range a constructive criterion for such portions to exist is obtained in case of tridiagonal matrices and a particular case of continuant matrices is considered as an application the cases of arbitrary 3 3 and 4 4 matrices are treated it is shown in particular that the sharp bound for the number of flat portions on the boundary of the numerical range for 4 4 matrices is four three if the matrices are assumed unitarily irreducible

a parallel subgradient projections method for the convex feasibility problem abstract an iterative method to solve the convex feasibility problem for a finite family of convex sets is presented the algorithm consists in the application of a generalization of an acceleration procedure introduced by de pierro in a parallel version of the subgradient projections method proposed by censor and lent the generated sequence is shown to converge for any starting point some numerical results are presented

joint numerical ranges quantum maps and joint numerical shadows abstract we associate with a k tuple of hermitian n n matrices a probability measure on r k supported on their joint numerical range the joint numerical shadow of these matrices when k 2 we recover the numerical range and the numerical shadow of the complex matrix corresponding to a pair of hermitian matrices we apply this material to the theory of quantum information thus we show that quantum maps on the set of quantum states defined by kraus operators satisfying the identity resolution assumption shrink joint numerical ranges

more on generalized inverses of partitioned matrices with banachiewicz schur forms abstract necessary and sufficient conditions are derived for a 2 by 2 partitioned matrix to have 1 1 2 1 3 1 4 inverses and the moore penrose inverse with banachiewicz schur forms as applications the banachiewicz schur forms of 1 1 2 1 3 1 4 inverses and the moore penrose inverse of a 2 by 2 partitioned hermitian matrix are also 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

numerical measure of a complex matrix we introduce a natural probability measure over the numerical range of a complex matrix a mn input amssym bbb c this numerical measure a can be defined as the law of the random variable ax x input amssym bbb c when the vector x input amssym bbb c n is uniformly distributed on the unit sphere if the matrix a is normal we show that a has a piecewise polynomial density fa which can be identified with a multivariate b spline in the general nonnormal case we relate the radon transform of a to the spectrum of a family of hermitian matrices and we deduce an explicit representation formula for the numerical density that is appropriate for theoretical and computational purposes as an application we show that the density fa is polynomial in some regions of the complex plane that can be characterized geometrically and we recover some known results about lacunae of symmetric hyperbolic systems in 2 1 dimensions finally we prove under general assumptions that the numerical measure of a matrix a mn input amssym bbb c concentrates to a dirac mass as the size n goes to infinity 2011 wiley periodicals inc

on invertibility and positive invertibility of matrices abstract new non singularity and non negative invertibility criteria for matrices are derived they yield new bounds for the spectrum moreover we obtain a bound for the norm of the inverse matrix and new estimates for the spectral radius

a continuous analogue of sturm sequences in the context of sturm liouville equation a shooting method is presented for finding the nth eigenvalue and eigenfunction of a sturm liouville equation in which the eigenvalue occurs nonlinearly the method is analyzed in two ways by applying the sturm comparison and oscillation theorems to the continuous problem and by applying sturm sequences to a discretization the method works for general separated boundary conditions and provides an a posteriors error estimate for the approximate eigenvalue a related method which involves critical lengths in the invariant imbedding method is shown to be incorrect for general boundary conditions

particular formulae for the moore penrose inverse of a columnwise partitioned matrix abstract an essential part of cegielski s obtuse cones and gram matrices with non negative inverse linear algebra appl 335 2001 167 181 considerations of some properties of gram matrices with nonnegative inverses which are pointed out to be crucial in constructing obtuse cones consists in developing some particular formulae for the moore penrose inverse of a columnwise partitioned matrix a a1 a2 under the assumption that it is of full column rank in the present paper these results are generalized and extended the generalization consists in weakening the assumption mentioned above to the requirement that the ranges of a1 and a2 are disjoint while the extension consists in introducing the conditions referring to the class of all generalized inverses of a

particular formulae for the moore penrose inverses of the partitioned bounded linear operators abstract for any two complex hilbert spaces h and k let bl h k be the set of bounded linear operators from h to k and h k be the direct sum of h and k given three hilbert spaces h 1 h 2 h 3 and two operators a 1 bl h 1 h 3 a 2 bl h 2 h 3 a partitioned bounded linear operator a a 1 a 2 bl h 1 h 2 h 3 can be induced where a h 1 h 2 a 1 h 1 a 2 h 2 for h i h i i 1 2 in this note we study the moore penrose inverse a of such a partitioned bounded linear operator a and generalize a recent result of j k baksalary and o m baksalary particular formulae for the moore penrose inverse of a columnwise partitioned matrix linear algebra appl 421 2007 16 23 from finite matrices to hilbert space operators

the moore penrose inverse of a partitioned nonnegative definite matrix abstract consider an arbitrary symmetric nonnegative definite matrix a and its moore penrose inverse a partitioned respectively as a e f f h and a g 1 g 2 g 2 g 4 explicit expressions for g 1 g 2 and g 4 in terms of e f and h are given moreover it is proved that the generalized schur complement a g 4 g 1 g 2 g 4 g 2 is always below the moore penrose inverse a h of the generalized schur complement a h e fh f with respect to the lowner partial ordering

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

a numerically stable dual method for solving strictly convex quadratic programs an efficient and numerically stable dual algorithm for positive definite quadratic programming is described which takes advantage of the fact that the unconstrained minimum of the objective function can be used as a starting point its implementation utilizes the cholesky and qr factorizations and procedures for updating them the performance of the dual algorithm is compared against that of primal algorithms when used to solve randomly generated test problems and quadratic programs generated in the course of solving nonlinear programming problems by a successive quadratic programming code the principal motivation for the development of the algorithm these computational results indicate that the dual algorithm is superior to primal algorithms when a primal feasible point is not readily available the algorithm is also compared theoretically to the modified simplex type dual methods of lemke and van de panne and whinston and it is illustrated by a numerical example

fast image recovery using dynamic load balancing in parallel architectures by means of incomplete projections this paper formulates an incomplete projection algorithm that is applied to the image recovery problem the algorithm allows an easy implementation of dynamic load balancing for parallel architectures furthermore the local computation communication load ratio can be adjusted since each processor performs a finite number of iterations of any projection type technique and this number can be provided as a parameter of the algorithm numerical results compare favorably with those obtained by the extrapolated method of parallel subgradient projections

essential spectra of singular matrix differential operators of mixed order abstract the present paper deals with the essential spectrum of the singular matrix differential operator of mixed order t d p t d q 0 t d h t q 1 t h t d q 1 t d t d d d t on a b a b the essential spectrum of t usually contains two parts a regular part and a singular part the regular part can be approximated by restrictions of t to compact subintervals of a b but the singular part cannot the spectral problem of t is transformed to a spectral problem of a singular hamiltonian differential operator involving the complex parameter of the first spectral problem to determine whether the essential spectrum contains the singular part we apply the classical titchmarsh weyl theory and the perturbation theory to the resulting hamiltonian system the main results cover many of the previous ones in the references

a projection method for least squares solutions to overdetermined systems of linear inequalities abstract an algorithm previously introduced by the author for finding a feasible point of a system of linear inequalities is further investigated for inconsistent systems it is shown to generate a sequence converging at a linear rate to the set of least squares solutions the algorithm is a projection type method and is a manifestation of the proximal point algorithm

new representations of the moore penrose inverse of 2 2 block matrices abstract this paper presents a full rank factorization of a 2 2 block matrix without any restriction applying this factorization we obtain an explicit representation of the moore penrose inverse in terms of four individual blocks of the partitioned matrix which simplifies results of hung and markham 1975 12 and miao 1991 16 we also derive some important coincidence theorems including in the expressions of the moore penrose inverse of a sum of matrices all these results extend earlier works by various authors

monotone gram matrices and deepest surrogate inequalities in accelerated relaxation methods for convex feasibility problems abstract the relaxation method for linear inequalities iterates by projecting the current point onto the most violated constraint accelerated methods project onto the intersection of several halfspaces or onto a surrogate halfspace corresponding to a nonnegative combination of constraints we extend todd s conditions for finding best surrogate inequalities via the solution of systems of linear equations our techniques may be used for accelerating various methods for convex feasibility and optimization problems

lagrangian relaxation via ballstep subgradient methods we exhibit useful properties of ballstep subgradient methods for convex optimization using level controls for estimating the optimal value augmented with simple averaging schemes they asymptotically find objective and constraint subgradients involved in optimality conditions when applied to lagrangian relaxation of convex programs they find both primal and dual solutions and have practicable stopping criteria up until now similar results have only been known for proximal bundle methods and for subgradient methods with divergent series stepsizes whose convergence can be slow encouraging numerical results are presented for large scale nonlinear multicommodity network flow problems

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

on relaxation methods for systems of linear inequalities abstract in their classical papers agmon and motzkin and schoenberg introduced a relaxation method to find a feasible solution for a system of linear inequalities so far the method was believed to require infinitely many iterations on some problem instances since it could depending on the dimension of the set of feasible soltions converge asymptotically to a feasible solution if one exists hence it could not be used to determine infeasibility using two lemma s basic to khachian s polynomially bounded algorithm we can show that the relaxation method is finite in all cases and thus can handle infeasible systems as well in spite of more refined stopping criteria the worst case behaviour of the relaxation method is not polynomially bounded as examplified by a class of problems constructed here

singular sturm liouville problems whose coefficients depend rationally on the eigenvalue parameter let domega z d q be a differential operator in l 2 0 infinity whose leading coefficient contains the eigenvalue parameter z for the case that omega z has the particular form omega t z p t c t 2 z r t z is an element of c r and the coefficient functions satisfy certain local integrability conditions it is shown that there is an analog for the usual limit point limit circle classification in the limit point case mild sufficient conditions are given so that all but one of the titchmarsh weyl coefficients belong to the so called kac subclass of nevanlinna functions an interpretation of the titchmarsh weyl coefficients is given also in terms of an associated system of differential equations where the eigenvalue parameter appears linearly c 2004 elsevier inc all rights reserved

sturm liouville operators and their spectral functions assume that the differential operator dpd q in l 2 0 infinity has 0 as a regular point and that the limit point case prevails at infinity if p equivalent to 1 and q satisfies some smoothness conditions it was proved by gelfand and levitan that the spectral functions sigma t for the sturm liouville operator corresponding to the boundary conditions pu 0 tauu 0 t e r satisfy the integrability condition integral r dsigma t t 1 infinity the boundary condition u 0 0 is exceptional since the corresponding spectral function does not satisfy such an integrability condition in fact this situation gives an example of a differential operator for which one can construct an analog of the friedrichs extension even though the underlying minimal operator is not semibounded in the present paper it is shown with simple arguments and under mild conditions on the coefficients p and q including the case p equivalent to 1 that there exists an analog of the friedrichs extension for nonsemibounded second order differential operators of the form dpd q by establishing the above mentioned integrability conditions for the underlying spectral functions c 2003 elsevier inc all rights reserved

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

nonnegative moore penrose inverses of gram operators this paper is concerned with necessary and sufficient conditions for the nonnegativity of moore penrose inverses of gram operators between real hilbert spaces these conditions include statements on acuteness or obtuseness of certain closed convex cones the main result generalizes a well known result for inverses in the finite dimensional case over the nonnegative orthant to moore penrose inverses in possibly infinite dimensional hilbert spaces over any general closed convex cone

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

singular dirac systems and sturm liouville problems nonlinear in the spectral parameter abstract for the dirac operator with spherically symmetric potential v 0 r we investigate the problem of whether the boundary points of the essential spectrum are accumulation points of discrete eigenvalues or not our main result shows that the accumulation of such eigenvalues is essentially determined by the asymptotic behaviour of v at 0 and we obtain this result by using a levinson type theorem for asymptotically diagonal systems depending on some parameter a comparison theorem for the principal solutions of singular dirac systems and some criteria on the eigenvalue accumulation respectively non accumulation of nonlinear singular sturm liouville problems

multivariable gain margin abstract the gain margin for a multivariable system is defined by examining the positive definite hermitian p d h portion of the polar decomposition of a non singular perturbation a in the feedback path the dual result defining the phase margin for a multivariable system has been derived by bar on and jonckheere 1990 by examining the unitary portion of the polar decomposition for non singular perturbations this paper focuses on the multivariable gain margin the main result is an extension of the classical siso concept of gain margin to mimo systems the definition for gain margin presented encompasses both gain reduction and gain amplification tolerance because the measurement is relative to the identity matrix as opposed to zero closed loop stability is guaranteed for all p d h matrices in the feedback path whose gain relative to the identity matrix is less than the gain margin of the system calculation of the gain margin requires solving a constrained optimization problem which is almost

general expressions for the moore penrose inverse of a 2 2 block matrix abstract the moore penrose inverse of a 2 2 block matrix m m a c b d is discussed general expressions for the moore penrose inverse for the block matrix m in terms of the individual blocks a b c d are delivered without any restrictions imposed under some conditions the moore penrose inverse can be simplified the results extend earlier work by various authors

convexity of the joint numerical range topological and differential geometric viewpoints abstract we investigate the convexity of the joint numerical range of m tuples of n n hermitian matrices the methods come from differential geometry and the differential and algebraic topology our main result is a sufficient condition for convexity of the joint numerical range for arbitrary m and n modulo a mild technical assumption this condition is formulated in terms of the largest eigenvalue of an associated family of hermitian matrices parameterized by the m 1 dimensional unit sphere the condition is that the eigenvalue has constant multiplicity we show that the constant multiplicity condition is in fact a criterion for the stable convexity of numerical ranges as a byproduct of our main result we obtain a new proof of the celebrated toeplitz hausdorff theorem and of its extension the numerical range of any triple of hermitian n n matrices is convex if n 2

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 generalized subgradient method with relaxation step we study conditions for convergence of a generalized subgradient algorithm in which a relaxation step is taken in a direction which is a convex combination of possibly all previously generated subgradients a simple condition for convergence is given and conditions that guarantee a linear convergence rate are also presented we show that choosing the steplength parameter and convex combination of subgradients in a certain sense optimally is equivalent to solving a minimum norm quadratic programming problem it is also shown that if the direction is restricted to be a convex combination of the current subgradient and the previous direction then an optimal choice of stepsize and direction is equivalent to the camerini fratta maffioli modification of the subgradient method

differential topology of numerical range abstract the numerical range of an n n matrix also known as its field of values is reformulated as the image of a smooth quadratic mapping from the n 1 dimensional complex projective space to the complex plane this paper investigates the numerical range from the perspective of differential topology morse theory more specifically the boundary of the range is interpreted as a rank 1 critical value curve and its sharp points are interpreted as rank 0 critical values more importantly the map is shown to have additional critical value curves in the interior of the numerical range these additional curves are shown to have such singularity phenomena as cusps and swallow tails to be the caustic envelopes of families of lines and to exhibit the so called normal bifurcation when an eigenvalue becomes unitarily decoupled

on the convergence of han s method for convex programming with quadratic objective we present an iterative method for minimizing strictly convex quadratic functions over the intersection of a finite number of convex sets the method consists in computing projections onto the individual sets simultaneously and the new iterate is a convex combination of those projections we give convergence proofs even for the inconsistent case i e when the intersection of the sets is empty

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

a subgradient method for vector optimization problems vector optimization problems are a significant extension of scalar optimization and have many real life applications we consider an extension of the projected subgradient method to convex vector optimization which works directly with vector valued functions without using scalar valued objectives we eliminate the scalarization approach a popular strategy for solving vector optimization problems exploring strongly the structure of these kinds of problems under suitable assumptions we show that the sequence generated by the algorithm converges to a weakly efficient optimum point