a new conjugate gradient method based on quasi newton equation for unconstrained optimization abstract the spectral conjugate gradient method is an effective method for large scale unconstrained optimization problems in this paper based on quasi newton direction and quasi newton equation a new spectral conjugate gradient method is proposed this method is motivated by the three term modified polak ribiyre polyak prp method and spectral parameters the global convergence of algorithm is proved for general functions under a strong wolfe line search numerical results show that the new algorithm is superior to the three term modified prp method

maps preserving the local spectrum of jordan product of matrices abstract let m n c be the algebra of all n n complex matrices and fix a nonzero vector x 0 c n we show that a map from m n c into itself satisfies t s s t x 0 t s s t x 0 t s m n c if and only if there exists an invertible matrix a m n c such that a x 0 x 0 and t a t a 1 for all t m n c

global convergence of a cass of quasi newton methods on convex problems we study the global convergence properties of the restricted broyden class of quasi newton methods when applied to a convex objective function we assume that the line search satisfies a standard sufficient decrease condition and that the initial hessian approximation is any positive definite matrix we show global and superlinear convergence for this class of methods except for dfp this generalizes powell s well known result for the bfgs method the analysis gives us insight into the properties of these algorithms in particular it shows that dfp lacks a very desirable self correcting property possessed by bfgs

surjective linear maps preserving local spectra let b x be the algebra of all bounded linear operators on an infinite dimensional complex banach space x we characterize linear surjective and continuous maps on b x preserving different local spectral quantities at a nonzero fixed vector

a tool for the analysis of quasi newton methods with application to unconstrained minimization the bfgs update formula is shown to have an important property that is inde pendent of the algorithmic context of the update and that is relevant to both constrained and unconstrained optimization the bfgs method for unconstrained optimization using a variety of line searches including backtracking is shown to be globally and superlinearly convergent on uniformly convex problems the analysis is particularly simple due to the use of some new tools introduced in this paper

a new structured quasi newton algorithm using partial information on hessian this paper presents a modified quasi newton method for structured unconstrained optimization the usual sqn equation employs only the gradients but ignores the available function value information several researchers paid attention to other secant conditions to get a better approximation of the hessian matrix of the objective function recently yabe et al 2007 6 proposed the modified secant condition which uses both gradient and function value information in order to get a higher order accuracy in approximating the second curvature of the objective function in this paper we derive a new progressive modified sqn equation with a vector parameter which use both available gradient and function value information that maintains most properties of the usual and modified structured quasi newton methods furthermore local and superlinear convergence of the algorithm is obtained under some reasonable conditions

additive local spectrum compressors let l x be the algebra of all bounded linear operators on an infinite dimensional complex banach space x we characterize additive continuous maps from l x onto itself which compress the local spectrum and the convexified local spectrum at a nonzero fixed vector additive continuous maps from l x onto itself that preserve the local spectral radius at a nonzero fixed vector are also characterized

a modified quasi newton method for nonlinear equations abstract in this paper a modified quasi newton method is proposed for solving the nonlinear equation f x 0 which is based on a new quasi newton approach the usual quasi newton equation is b k 1 s k y k where s k x k 1 x k y k f x k 1 f x k the new quasi newton equation is b k 1 s k y k in which s k is based on the iterates x k 1 x k x k 1 and y k is based on the function values f x k 1 f x k f x k 1 the new quasi newton equation exploits additional information by assuming a quadratic relationship between the information from the last three iterates the modified quasi newton method is based on the new quasi newton equation and possess local superlinear convergence properties numerical experiments show that the modified quasi newton method is promising

a generalized hbv model with diffusion and two delays an hepatitis b virus hbv model with spatial diffusion general incidence rate and time delays subject to homogeneous neumann boundary conditions is formulated the stability of the disease free equilibrium and the chronic infection equilibrium is analyzed by using the linearization method and by constructing appropriate lyapunov functionals furthermore several diffusive hbv models existing in the literature are generalized and extended

new implicit updates in multi step quasi newton methods for unconstrained optimisation multi step quasi newton methods for optimisation using data from more than one previous step to revise the current approximate hessian were introduced by ford and moghrabi in j comput appl math 50 1994 305 where they showed how to construct such methods by means of interpolating curves these methods also utilise standard quasi newton formulae but with the vectors normally employed in the formulae replaced by others determined from a multi step version of the secant equation some methods the accumulative and fixed point approaches for defining the parameter values which correspond to the iterates on the interpolating curve were presented by ford and moghrabi in optim methods software 2 1993 357 both the accumulative and the fixed point methods measure the distances required to parameterise the interpolating polynomials via a norm defined by a positive definite matrix m the fixed point algorithm which takes m to be the current approximate hessian was found experimentally to be the best of the six multi step methods studied in ford and moghrabi 1993 all of which exhibited improved numerical performance by comparison with the standard single step bfgs method to produce a better parameterisation of the interpolation ford comput math appl 42 2001 1083 developed the idea of implicit update methods the fundamental concept here is to determine an improved version of the hessian approximation to be used in computing the metric while avoiding the computational expense of actually calculating the improved version two implicit methods denoted by i2 and i3 were developed from f2 in ford 2001 the method i2 employed parameter values generated from an implicit single step bfgs update while i3 used values from an implicit two step update in this paper we describe the derivation of new implicit updates which are similar to i3 the experimental results we present show that one of the new implicit methods produces markedly better performance than the existing implicit methods particularly as the dimension of the test problem grows

linear preserver problems a brief introduction and some special techniques abstract linear preserver problems concern the characterization of linear operators on matrix spaces that leave certain functions subsets relations etc invariant the earliest papers on linear preserver problems date back to 1897 and a great deal of effort has been devoted to the study of this type of question since then we present a brief picture of the subject aiming at giving a gentle introduction to the reader then we describe some techniques used in our recent papers on this type of problem

maps on matrices compressing the local spectrum in the spectrum abstract let m n c be the algebra of all complex n n matrices let x 0 be a nonzero vector in c n we describe maps on m n c not linear or surjective satisfying t s x 0 t s for all t s m n c we also obtain a similar description by supposing that is surjective and t s x 0 t s for all t s m n c

superlinear convergence of symmetric huang s class of methods in this paper the problem of minimizing the functionalf d r n r is considered typical assumptions onf are assumed a class of quasi newton methods namely huang s class of methods is used for finding an optimal solution of this problem a new theorem connected with this class is presented by means of this theorem some convergence results known up till now only for the methods which satisfy quasi newton condition are extended that is the results of superlinear convergence of variable metric methods in the cases of exact and asymptotically exact minimization and the so called direct prediction case this theorem allows to interpretate one of the parameters as the scaling parameter

nonlinear conjugate gradient methods with structured secant condition for nonlinear least squares problems in this paper we deal with conjugate gradient methods for solving nonlinear least squares problems several newton like methods have been studied for solving nonlinear least squares problems which include the gauss newton method the levenberg marquardt method and the structured quasi newton methods on the other hand conjugate gradient methods are appealing for general large scale nonlinear optimization problems by combining the structured secant condition and the idea of dai and liao 2001 20 the present paper proposes conjugate gradient methods that make use of the structure of the hessian of the objective function of nonlinear least squares problems the proposed methods are shown to be globally convergent under some assumptions finally some numerical results are given

a modified hestenes stiefel conjugate gradient method with sufficient descent condition and conjugacy condition in this paper by taking a little modification to the hestenes stiefel method we propose a new way to construct descent directions satisfying the sufficient descent condition also an adaptive conjugacy condition and a intrinsic self restarting mechanism are revealed a dynamical adjustment can be regarded as the inheritance and development of properties of standard hestenes stiefel method furthermore we establish global convergence for general nonconvex objective function under mild condition numerical results show that our presented methods can be efficient for solving large scale test problems and therefore is promising

local spectrum linear preservers at non fixed vectors abstract for a complex n n matrix t and a vector x c n we denote by t x respectively by r t x the local spectrum respectively the local spectral radius of t at x we prove that m n m n linear has the property that for each t m n there exists a nonzero x t c n such that t x t t x t if and only if there exists a m n invertible such that either t a t a 1 for each t m n or t a t t a 1 for each t m n modulo a multiplication by a unimodular complex number we arrive at the same conclusion by supposing that for each t m n there exists a nonzero x t c n such that r t x t r t x t

two new conjugate gradient methods based on modified secant equations following the approach proposed by dai and liao we introduce two nonlinear conjugate gradient methods for unconstrained optimization problems one of our proposed methods is based on a modified version of the secant equation proposed by zhang deng and chen and zhang and xu and the other is based on the modified bfgs update proposed by yuan an interesting feature of our methods is their account of both the gradient and function values under proper conditions we show that one of the proposed methods is globally convergent for general functions and that the other is globally convergent for uniformly convex functions to enhance the performance of the line search procedure we also propose a new approach for computing the initial steplength to be used for initiating the procedure we provide a comparison of implementations of our methods with the efficient conjugate gradient methods proposed by dai and liao and hestenes and stiefel numerical test results show the efficiency of our proposed methods

superlinear convergence of broyden s bounded θ class of methods in this paper the so called broyden s bounded class of methods is considered it contains as a subclass broyden s restricted class of methods in which the updating matrices retain symmetry and positive definiteness these iteration methods are used for solving unconstrained minimization problems of the following form f hat x min x in r n f x it is assumed that the step size coefficient k 1 in each iteration and the functionalf rn r1 satisfies the standard assumptions viz f is twice continuously differentiable and the hessian matrix is uniformly positive definite and bounded there exist constantsm m 0 such that m y 2 y m parallel y parallel 2 leqslant left langle y f hat x left y right rangle right leqslant m parallel y parallel 2 for ally rn and satisfies a lipschitz like condition at the optimal point hat x the gradient vanishes at hat x under these assumptions the local convergence of broyden s methods is proved furthermore the q superlinear convergence is shown

a compact limited memory method for large scale unconstrained optimization abstract a compact limited memory method for solving large scale unconstrained optimization problems is proposed the compact representation of the quasi newton updating matrix is derived to the use in the form of limited memory update in which the vector y k is replaced by a modified vector y k so that more available information about the function can be employed to increase the accuracy of hessian approximations the global convergence of the proposed method is proved numerical tests on commonly used large scale test problems indicate that the proposed compact limited memory method is competitive and efficient

optimal conditioning and convergence in rank one quasi newton updates we show that the optimally conditioned update proposed by spedicato and greenstadt for solving nonlinear equations is obtainable in closed form however unlike its rank two counterpart designed for minimization conditioning in the rank one case is not necessarily a good measure of closeness we propose updates that compromise the two properties local and superlinear convergence is proved for a class of updates

a family of hager zhang conjugate gradient methods for system of monotone nonlinear equations abstract this paper presents two modified hager zhang hz conjugate gradient methods for solving large scale system of monotone nonlinear equations the methods were developed by combining modified forms of the one parameter method by hager and zhang 2006 and the hyperplane projection technique global convergence and numerical results of the methods are established preliminary numerical results show that the proposed methods are promising and more efficient compared to the methods presented by mushtak and keyvan 2018 and sun et al 2017

necessary and sufficient condition for extinction and persistence of sirs system with random perturbation abstract this paper characterizes the qualitative dynamics of a stochastic sirs epidemic model the study shows that the dynamics of the model are determined by a certain threshold quantity r s expressed in terms of the model parameters and the intensity of the noise if r s 1 the disease will be eliminated from the community whereas an epidemic occurs if r s 1 our results recover the known results in the earlier literature as special cases the presented results are illustrated by numerical simulations

how bad are the bfgs and dfp methods when the objective function is quadratic we study the use of the bfgs and dfp algorithms with step lengths of one for minimizing quadratic functions of only two variables the updating formulae in this case imply nonlinear three term recurrence relations between the eigenvalues of consecutive second derivative approximations which are analysed in order to explain some gross inefficiencies that can occur specifically the bfgs algorithm may require more than 10 iterations to achieve the first decimal place of accuracy while the performance of the dfp method is far worse the results help to explain why the dfp method is often less suitable than the bfgs algorithm for general unconstrained optimization calculations and they show that quadratic functions provide much information about efficiency when the current vector of variables is too far from the solution for an asymptotic convergence analysis

local and superlinear convergence of quasi newton methods based on modified secant conditions for solving unconstrained minimization problems quasi newton methods are popular iterative methods the secant condition which employs only the gradient information is imposed on these methods several researchers paid attention to other secant conditions to get a better approximation of the hessian matrix of the objective function recently zhang et al new quasi newton equation and related methods for unconstrained optimization j optim theory appl 102 1999 147 167 and zhang and xu properties and numerical performance of quasi newton methods with modified quasi newton equations j comput appl math 137 2001 269 278 proposed the modified secant condition which uses both gradient and function value information in order to get a higher order accuracy in approximating the second curvature of the objective function they showed the local and q superlinear convergence property of the bfgs like and dfp like updates based on their proposed secant condition in this paper we incorporate one parameter into this secant condition to smoothly switch the standard secant condition and the secant condition of zhang et al we consider a modified broyden family which includes the bfgs like and the dfp like updates proposed by zhang et al we prove the local and q superlinear convergence of our method

a double parameter scaled bfgs method for unconstrained optimization abstract a double parameter scaled bfgs method for unconstrained optimization is presented in this method the first two terms of the known bfgs update formula are scaled with a positive parameter while the third one is scaled with another positive parameter these parameters are selected in such a way as to improve the eigenvalues structure of the bfgs update the parameter scaling the first two terms of the bfgs update is determined by clustering the eigenvalues of the scaled bfgs matrix on the other hand the parameter scaling the third term is determined as a preconditioner to the hessian of the minimizing function combined with the minimization of the conjugacy condition from conjugate gradient methods under the inexact wolfe line search the global convergence of the double parameter scaled bfgs method is proved in very general conditions without assuming the convexity of the minimizing function using 80 unconstrained optimization test functions with a medium number of variables the preliminary numerical experiments show that this double parameter scaled bfgs method is more efficient than the standard bfgs update or than some other scaled bfgs methods

an adaptive three term conjugate gradient method based on self scaling memoryless bfgs matrix abstract due to its simplicity and low memory requirement conjugate gradient methods are widely used for solving large scale unconstrained optimization problems in this paper we propose a three term conjugate gradient method the search direction is given by a symmetrical perry matrix which contains a positive parameter the value of this parameter is determined by minimizing the distance of this matrix and the self scaling memoryless bfgs matrix in the frobenius norm the sufficient descent property of the generated directions holds independent of line searches the global convergence of the given method is established under wolfe line search for general non convex functions numerical experiments show that the proposed method is promising

properties and numerical performance of quasi newton methods with modified quasi newton equations quasi newton qn equation plays a core role in contemporary nonlinear optimization the usual qn equation employs only the gradients but ignores the available function value information in this paper we derive a class of modified qn equations with a vector parameter which use both available gradient and function value information the modified quasi newton methods maintain most properties of the usual quasi newton methods meanwhile they achieve a higher order accuracy in approximating the second order curvature of the problem functions than the usual ones do numerical experiments are reported which support the theoretical analyses and show the advantages of the modified qn methods over the usual ones

conjugate gradient methods based on secant conditions that generate descent search directions for unconstrained optimization conjugate gradient methods have been paid attention to because they can be directly applied to large scale unconstrained optimization problems in order to incorporate second order information of the objective function into conjugate gradient methods dai and liao 2001 proposed a conjugate gradient method based on the secant condition however their method does not necessarily generate a descent search direction on the other hand hager and zhang 2005 proposed another conjugate gradient method which always generates a descent search direction in this paper combining dai liao s idea and hager zhang s idea we propose conjugate gradient methods based on secant conditions that generate descent search directions in addition we prove global convergence properties of the proposed methods finally preliminary numerical results are given

the convergence of matrices generated by rank 2 methods from the restricted β class of broyden it is shown that the matricesb k generated by any method from the restricted class of broyden converge if the method is applied to the unconstrained minimization of a functionf c 2 r n with lipschitz continuous 2 f x and if the method is such that it generates vectorsx k converging sufficiently fast sum limits k x k x infty to a local minimumx off with positive definite 2 f x this result not only holds for constant step sizes k 1 in each iterationx k x k 1 x k k b k 1 f x k of these methods but also for step sizes determined by asymptotically exact line searches the paper generalizes corresponding results of ge ren pu and powell 6 for the dfp and bfgs methods used in conjunction with step sizes k 1

a stochastic hepatitis b epidemic model driven by levy noise abstract in this paper we investigate a stochastic hepatitis b epidemic model driven by levy noise we prove the existence of a unique global positive solution and we present sufficient conditions to derive extinction and persistence in mean of the disease

spectral scaling quasi newton methods with updates from the one parameter of the broyden family in this paper based on the spectral scaling secant condition w y cheng d h li spectral scaling bfgs method journal of optimization theory and applications 146 2010 305 319 we propose spectral scaling one parameter broyden family methods which allow for negative values of the parameter we show that the proposed methods possess some good properties such as quadratic termination property and single step convergence rate not inferior to that of the steepest descent method when minimizing an n dimensional quadratic function under appropriate conditions we establish the global convergence of the proposed methods for uniformly convex functions numerical results from problems in the cute test set show that the proposed methods are promising

on the use of function values in unconstrained optimisation abstract by the use of a nonlinear model for the gradient of the objective function along a chosen direction we show how information available via values of the objective function may be efficiently utilised in an optimisation method of quasi newton type numerical experiments indicate that computational gains are possible by such means

preservers of local spectrum of matrix jordan triple products abstract let m n be the space of n n complex matrices and for a nonzero vector e c n and t m n let t e denote the local spectrum of t at e maps on m n which preserve the local spectrum of jordan triple product of matrices at e in a sense that t s t e t s t e t s m n are characterized with no surjectivity assumption on them

tensor methods for unconstrained optimization using second derivatives a new type of method for unconstrained optimization called a tensor method is introduced it is related in its basic philosophy to the tensor methods for nonlinear equations for schnabel and frank siam j numer anal 21 1984 pp 815 843 but beyond that the methods have significant differences the tensor method for unconstrained optimization bases each iteration upon a fourth order model of the objective function this model consists of the quadratic portion of the taylor series plus low rank third and fourth order terms that cause the model to interpolate already calculated function and gradient values from one or more previous iterates this paper also shows that the costs of forming storing and solving the tensor model are not significantly more than these costs for a standard method based upon a quadratic taylor series model test results are presented for sets of problems where the hessian at the minimizer is nonsingular and where it is singular on all the test sets the tensor method

using function values in multi step quasi newton methods in previous work the authors 1993 1994 developed the concept of multi step quasi newton methods based on the use of interpolating polynomials determined by data from the m most recent steps different methods for parametrizing these polynomials were studied by the authors 1993 and several methods were shown empirically to yield substantial gains over the standard one step bfgs method for unconstrained optimization in this paper we will consider the issue of how to incorporate function value information within the framework of such multi step methods this is achieved in the case of two step methods through the use of a carefully chosen rational form to interpolate the three most recent iterates the results of numerical experiments on the new methods are reported

the revised dfp algorithm without exact line search in this paper we discuss the convergence of the dfp algorithm with revised search direction under some inexact line searches we prove that the algorithm is globally convergent for continuously differentiable functions and the rate of convergence of the algorithm is one step superlinear and n step second order for uniformly convex objective functions from the proof of this paper we obtain the superlinear and n step second order convergence of the dfp algorithm for uniformly convex objective functions

globally convergent modified perry s conjugate gradient method abstract conjugate gradient methods are probably the most famous iterative methods for solving large scale optimization problems in scientific and engineering computation characterized by the simplicity of their iteration and their low memory requirements in this paper we propose a new conjugate gradient method which is based on the mbfgs secant condition by modifying perry s method our proposed method ensures sufficient descent independent of the accuracy of the line search and it is globally convergent under some assumptions numerical experiments are also presented

on the use of curvature estimates in quasi newtonian methods abstract in previous work ford and ghandhari 1989 considered the use of nonlinear gradient models for quasi newton methods by means of such models values of the objective function in addition to values of the gradient were utilised in updating formulae numerical experiments provided evidence that such an approach could yield gains in terms of the evaluations required but at the cost of the solution of a nonlinear equation at each iteration in a subsequent paper ford and ghandhari to appear have developed alternative models which enable the parameter to be determined explicitly in this paper we shall consider further models of the general type introduced in the latter paper which determine the parameter explicitly by employing estimates of the curvature of the objective function thus following a technique first introduced by biggs 1971 1973 we conclude by presenting the results of numerical tests on the new methods

multi step quasi newton methods for optimization abstract quasi newton methods update at each iteration the existing hessian approximation or its inverse by means of data deriving from the step just completed we show how multi step methods employing in addition data from previous iterations may be constructed by means of interpolating polynomials leading to a generalization of the secant or quasi newton equation the issue of positive definiteness in the hessian approximation is addressed and shown to depend on a generalized version of the condition which is required to hold in the original single step methods the results of extensive numerical experimentation indicate strongly that computational advantages can accrue from such an approach by comparison with single step methods particularly as the dimension of the problem increases

convergence of quasi newton matrices generated by the symmetric rank one update quasi newton algorithms for unconstrained nonlinear minimization generate a sequence of matrices that can be considered as approximations of the objective function second derivatives this paper gives conditions under which these approximations can be proved to converge globally to the true hessian matrix in the case where the symmetric rank one update formula is used the rate of convergence is also examined and proven to be improving with the rate of convergence of the underlying iterates the theory is confirmed by some numerical experiments that also show the convergence of the hessian approximations to be substantially slower for other known quasi newton formulae

a new conjugate gradient algorithm for training neural networks based on a modified secant equation conjugate gradient methods have been established as excellent neural network training methods due to the simplicity of their iteration numerical efficiency and their low memory requirements in this work we propose a conjugate gradient neural network training algorithm which guarantees sufficient descent using any line search avoiding thereby the usually inefficient restarts moreover it approximates the second order curvature information of the error surface with a high order accuracy by utilizing a new modified secant condition under mild conditions we establish that the global convergence of our proposed method experimental results provide evidence that our proposed method is in general superior to the classical conjugate gradient training methods and has a potential to significantly enhance the computational efficiency and robustness of the training process

on hager and zhang s conjugate gradient method with guaranteed descent abstract in this paper we point out that hager zhang family of conjugate gradient methods are based on the spectral scaling secant equation are really members of dai liao type of gradient methods with common constant parameters we propose a possible efficient parameter for hager zhang s conjugate gradient method with guaranteed descent cg descent and present a conjugate gradient method cg descent1 the cg descent1 method preserves the same global convergence properties as cg descent method of hager and zhang and is numerically tested compared with cg descent method and other conjugate gradient methods considering the difference of the search directions approximation to the search direction of the memoryless bfgs method we give the reason why the proposed method is more efficient than cg descent method under some conditions and further find an optimal conjugate gradient method and an approximation optimal conjugate method in hager zhang family of conjugate gradient methods

a nonlinear model for function value multistep methods abstract we develop a framework employing scaling functions for the construction of multistep quasi newton methods for unconstrained optimization these methods utilize values of the objective function they are constructed via interpolants of the m 1 most recent iterates gradient evaluations and possess a free parameter which introduces an additional degree of flexibility this permits the interpolating functions to assimilate information in the form of function values which is readily available at each iteration motivated by previous experience 1 with the use of function values in multistep methods we investigate the incorporation of this information in the construction of the hessian approximation at each iteration in an attempt to accelerate convergence we concentrate on a specific example from the general family of methods corresponding to a particular choice of the scaling function and from it derive three new algorithms the relative numerical performance of these methods is assessed and the most successful of them is then compared with the standard bfgs method and with an earlier algorithm utilizing function values also developed by the authors 1

nonmonotone bfgs trained recurrent neural networks for temporal sequence processing in this paper we propose a nonmonotone approach to recurrent neural networks training for temporal sequence processing applications this approach allows learning performance to deteriorate in some iterations nevertheless the network s performance is improved over time a self scaling bfgs is equipped with an adaptive nonmonotone technique that employs approximations of the lipschitz constant and is tested on a set of sequence processing problems simulation results show that the proposed algorithm outperforms the bfgs as well as other methods previously applied to these sequences providing an effective modification that is capable of training recurrent networks of various architectures

a new super memory gradient method with curve search rule in this paper we propose a new super memory gradient method with curve search rule for unconstrained optimization problems the method uses previous multi step iterative information and some curve search rules to generate new iterative points at each iteration this makes the new method converge stably and be more suitable for solving large scale optimization problems than other similar methods we analyze the global convergence and convergence rate under some mild conditions numerical experiments show that some new algorithms are available and effective in practical computation

a bregman extension of quasi newton updates ii analysis of robustness properties in part i of this series of articles we introduced the information geometric framework of quasi newton methods and gave an extension of hessian update formulas based on the bregman divergence the purpose of this article is to investigate the convergence and robustness properties of extended hessian update formulas fletcher has studied a variational problem which derives the approximate hessian update formula of the quasi newton methods we point out that the variational problem is identical to optimization of the kullback leibler divergence which is a discrepancy measure between two probability distributions then we introduce the bregman divergence as an extension of the kullback leibler divergence and derive extended quasi newton update formulas based on the variational problem with the bregman divergence the proposed update formulas belong to a class of self scaling quasi newton methods we study the convergence property of the proposed quasi newton method moreover we apply tools in the robust statistics to analyze the robustness properties of hessian update formulas against numerical rounding errors or a shift of tuning parameters included in line search methods of the step length as the main contribution of this paper we present that the influence of perturbations in the line search is bounded only for the standard bfgs formula for the hessian approximation numerical studies are conducted to verify the usefulness of the tools borrowed from the robust statistics

fresh look into the design and computation of optimal output feedback controls for linear multivariable systems this paper presents as its main result a new and efficient method for the design of optimal dynamic compensators for linear multivariable control systems the discussion starts with a fresh look into the design of optimal constant gain output feedback in this context three new ideas are introduced that prove their usefulness particularly in the dynamic compensator case 1 deterministic measurement disturbances are included in the formulation of the underlying optimization problem 2 the relationship between a deterministic and a stochastic optimal control problem is shown 3 by proper modifications of the quadratic cost function the effort involved in the numerical solution of the optimal output feedback problem is considerably reduced furthermore a canonical form is introduced for the dynamic compensator and an algorithm is developed for the numerical computation of its free parameters it is based on quasi newtonian gradient methods and an efficient line search procedure some features of the

preservers of local spectra of matrix sums abstract let m n c be the set of n n complex matrices and for a nonzero vector e c n and t m n c let t e denote the local spectrum of t at e we characterize surjective maps on m n c which satisfy t s e t s e for all matrices t and s we arrive at the same conclusion by supposing that t s e t s e for all matrices t and s without the surjectivity assumption on continuous maps from m n c onto itself that preserve the local spectral radius of the sum of matrices at a nonzero fixed vector are also characterized

new quasi newton methods via higher order tensor models many researches attempt to improve the efficiency of the usual quasi newton qn methods by accelerating the performance of the algorithm without causing more storage demand they aim to employ more available information from the function values and gradient to approximate the curvature of the objective function in this paper we derive a new qn method of this type using a fourth order tensor model and show that it is superior with respect to the prior modification of wei et al 2006 4 convergence analysis gives the local convergence property of this method and numerical results show the advantage of the modified qn method

linear maps on mn c preserving the local spectrum let x0 cn be a nonzero vector we prove that if a linear map mn c mn c preserves the local spectrum at x0 i e t x0 t x0 for all t mn c then there exists an invertible matrix a such that a x0 x0 and t ata 1 for every t

matrix algebras in quasi newton methods for unconstrained minimization in this paper a new class of quasi newton methods named qn is introduced in order to solve unconstrained minimization problems the novel approach which generalizes classical bfgs methods is based on a hessian updating formula involving an algebra of matrices simultaneously diagonalized by a fast unitary transform the complexity per step of qn methods is o n log n thereby improving considerably bfgs computational efficiency moreover since qn s iterative scheme utilizes single indexed arrays only o n memory allocations are required global convergence properties are investigated in particular a global convergence result is obtained under suitable assumptions on f numerical experiences 7 confirm that qn methods are particularly recommended for large scale problems

a new descent algorithm with curve search rule in the paper we present a new descent algorithm with curve search rule for unconstrained minimization problems at each iteration the next iterative point is determined by means of a curve search rule it is particular that the search direction and the step size is determined simultaneously at each iteration of the new algorithm the algorithm similarly to conjugate gradient methods avoids the computation and storage of some matrices associated with the hessian of objective functions unlike ode methods it is not required to solve ordinary differential equations at each iteration although the convergence rate of our algorithm is not as fast as that of newton like methods it is suitable to solve large scale minimization problems

modifying the bfgs update by a new column scaling technique letb be a positive definite symmetric approximation to the second derivative matrix of the objective function of a minimization calculation we study modifications of the bfgs method that apply a scaling technique to the columns of a conjugate direction matrixz satisfyingz t bz i for a simple scaling technique similar to the ones considered by powell 1987 and 1989 we show that due to a two iteration cycle linear convergence can occur when the method is applied to a quadratic function and wolfe s line search is employed although for exact line searches quadratic termination can be proved we then suggest a different scaling technique that prevents certain columns thought to contain important curvature information from being scaled for this algorithm we prove global and superlinear convergence and demonstrate the superiority of our method over the bfgs formula on a range of illconditioned optimization problems moreover we present an implementation of our algorithm that requires only 3n 2 o n multiplications per iteration

a novel energy based approach for merging finite elements a novel approach for merging two intersecting finite elements is presented and demonstrated the solution mimics concepts from biology and uses principles rooted in continuum mechanics the problem of attaching or merging two coincident finite elements is common when using the plastering technique as part of the advancing front method this problem is particularly challenging for 3 d meshes of non convex shapes some automatic meshing methods require portions of the partially formed mesh to coincide and merge this problem is generally solved with heuristic rules which lack generality and may have difficulties with unforeseen situations the problem of merging two overlapping polyhedra may also appear in other applications such as computer graphics and cad software a new approach to address the problem of merging is presented here this solution does not utilize heuristic rules but rather uses an approach based on minimization of strain energy a fully automatic merging routine has been created that can address in an optimum way any situation of two nearby or overlapping elements that are to be merged this approach with minor adjustments is suitable for most types of 3 d elements

a new class of spectral conjugate gradient methods based on a modified secant equation for unconstrained optimization conjugate gradient methods have played a special role for solving large scale optimization problems due to the simplicity of their iteration convergence properties and their low memory requirements in this work we propose a new class of spectral conjugate gradient methods which ensures sufficient descent independent of the accuracy of the line search moreover an attractive property of our proposed methods is that they achieve a high order accuracy in approximating the second order curvature information of the objective function by utilizing the modified secant condition proposed by babaie kafaki et al s babaie kafaki r ghanbari n mahdavi amiri two new conjugate gradient methods based on modified secant equations journal of computational and applied mathematics 234 2010 1374 1386 further a global convergence result for general functions is established provided that the line search satisfies the wolfe conditions our numerical experiments indicate that our proposed methods are preferable and in general superior to the classical conjugate gradient methods in terms of efficiency and robustness

dynamics of a stochastic sis model with double epidemic diseases driven by levy jumps abstract this paper is to investigate the dynamics of a stochastic sis epidemic model with saturated incidence rate and double epidemic diseases which make the research more complex the environment variability in this study is characterized by white noise and jump noise sufficient conditions for the extinction and persistence in the mean of two epidemic diseases are obtained it is shown that the two diseases can coexist under appropriate conditions finally numerical simulations are introduced to illustrate the results developed

new control allocation algorithms in fixed point framework for overactuated systems with actuator saturation abstractin this paper two control allocation algorithms have been proposed for overactuated systems the algorithms are developed by formulating constrained control allocation problem into an equivalent fixed point framework the first algorithm follows sequential solving method while the other one involves a zero finding technique by the newton method in order to consider limiting constraints the saturation function has been taken into account that leads to a nonsmooth zero finding problem and the proposed method guarantees superlinear convergence the second algorithm is easy to implement and faster than the algorithm developed based on the sequential solving technique to demonstrate the effectiveness of the proposed algorithms a detail simulation study has been carried out in which one example considers both the actuator rate and amplitude constraints

some general techniques on linear preserver problems abstract several general techniques on linear preserver problems are described the first one is based on a transfer principle in model theoretic algebra that allows one to extend linear preserver results on complex matrices to matrices over other algebraically closed fields of characteristic 0 the second one concerns the use of some simple geometric technique to reduce linear preserver problems to standard types so that known results can be applied the third one is about solving linear preserver problems on more general operator algebras by reducing the problems to idempotent preservers numerous examples will be given to demonstrate the proposed techniques