on sizing and shifting the bfgs update within the sized broyden family of secant updates in a numerical study contreras and tapia demonstrated the value of selectively sizing the bfgs secant update a conclusion of the current study is that there is value in following a sizing of the bfgs update with a shifting of the bfgs update our motivation for such a strategy comes from the known fact that the bfgs update is prone to producing updates with large eigenvalues hence sizing may contribute to near singularity and shifting can help compensate for this deficiency we follow contreras and tapia in our form of selective sizing our form of shifting can be viewed as switching from the bfgs update to an alternative member of the broyden family of updates our shift is obtained by minimizing a weighted form of the byrd nocedal measure over a sized broyden family of secant updates numerically our most effective shift is the one that gives the member of the class which is closest to steepest descent in the byrd nocedal measure

brief paper stabilization of some fractional delay systems of neutral type in this note we give new stability tests which enable one to fully characterize the h stability of systems with transfer function g s r s p s q s e s h where h 0 and p q r are real polynomials in the variable s m for 0 p s q s 1

a symmetric rank one method based on extra updating techniques for unconstrained optimization abstract in this paper we present a new symmetric rank one sr1 method for the solution of unconstrained optimization problems the proposed method involves an algorithm in which the usual sr1 hessian is updated a number of times in a way to be specified in some iterations to improve the performance of the hessian approximation in particular we discuss how to consider a criterion for indicating at each iteration whether it is necessary to employ extra updates however it is well known that there are some theoretical difficulties when applying the sr1 update even for a current positive definite hessian approximation it is possible that the sr1 update may not be defined or the sr1 update may not preserve positive definiteness at some iterations we then employ a restarting procedure that guarantees that updated matrices will be well defined while preserving positive definiteness of updates numerical results support these theoretical considerations they show that the implementation of the sr1 method using extra updating techniques improves the performance of the sr1 method substantially for a number of test problems from the literature

on stabilization and the existence of coprime factorizations shows that any transfer function matrix whose elements belong to the quotient field of h sub infinity and which is stabilizable has a matrix fraction representation over h sub infinity which is coprime in the sense that a matrix bezout identity can be satisfied

pid controller design for fractional order systems with time delays abstract classical proper pid controllers are designed for linear time invariant plants whose transfer functions are rational functions of s where 0 1 and s is the laplace transform variable effect of input output time delay on the range of allowable controller parameters is investigated the allowable pid controller parameters are determined from a small gain type of argument used earlier for finite dimensional plants

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

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

variable metric methods for unconstrainted optimization and nonlinear least squares variable metric or quasi newton methods are well known and commonly used in connection with unconstrained optimization since they have good theoretical and practical convergence properties although these methods were originally developed for small and moderate size dense problems their modifications based either on sparse partitioned or limited memory updates are very efficient on large scale sparse problems very significant applications of these methods also appear in nonlinear least squares approximation and nonsmooth optimization in this contribution we give an extensive review of variable metric methods and their use in various optimization fields

an improved multi step gradient type method for large scale optimization in this paper we propose an improved multi step diagonal updating method for large scale unconstrained optimization our approach is based on constructing a new gradient type method by means of interpolating curves we measure the distances required to parameterize the interpolating polynomials via a norm defined by a positive definite matrix by developing on implicit updating approach we can obtain an improved version of hessian approximation in diagonal matrix form while avoiding the computational expenses of actually calculating the improved version of the approximation matrix the effectiveness of our proposed method is evaluated by means of computational comparison with the bb method and its variants we show that our method is globally convergent and only requires o n memory allocations

on the performance of a new symmetric rank one method with restart for solving unconstrained optimization problems quasi newton qn methods are generally held to be the most efficient minimization methods for solving unconstrained optimization problems among the qn methods symmetric rank one sr1 is one of the very competitive formulas in the present paper we propose a new sr1 method the new technique attempts to improve the quality of the sr1 hessian by employing the scaling of the identity in a certain sense however since at some iterations these updates might be singular indefinite or undefined this paper proposes an updates criterion based on the eigenvalues of the sr1 update to measure this quality hence the new method is employed only to improve the approximation of the sr1 hessian it is shown that the numerical results support the theoretical considerations for the usefulness of this criterion and show that the proposed method improves the performance of the sr1 update substantially

scaled memoryless symmetric rank one method for large scale optimization this paper concerns the memoryless quasi newton method that is precisely the quasi newton method for which the approximation to the inverse of hessian at each step is updated from the identity matrix hence its search direction can be computed without the storage of matrices in this paper a scaled memoryless symmetric rank one sr1 method for solving large scale unconstrained optimization problems is developed the basic idea is to incorporate the sr1 update within the framework of the memoryless quasi newton method however it is well known that the sr1 update may not preserve positive definiteness even when updated from a positive definite matrix therefore we propose the memoryless sr1 method which is updated from a positive scaled of the identity where the scaling factor is derived in such a way that positive definiteness of the updating matrices are preserved and at the same time improves the condition of the scaled memoryless sr1 update under very mild conditions it is shown that for strictly convex objective functions the method is globally convergent with a linear rate of convergence numerical results show that the optimally scaled memoryless sr1 method is very encouraging

fuzzy wavelet neural network models for prediction and identification of dynamical systems this paper presents fuzzy wavelet neural network fwnn models for prediction and identification of nonlinear dynamical systems the proposed fwnn models are obtained from the traditional takagi sugeno kang fuzzy system by replacing the then part of fuzzy rules with wavelet basis functions that have the ability to localize both in time and frequency domains the first and last model use summation and multiplication of dilated and translated versions of single dimensional wavelet basis functions respectively and in the second model then parts of the rules consist of radial function of wavelets gaussian type of activation functions are used in if part of the fuzzy rules a fast gradient based training algorithm i e the broyden fletcher goldfarb shanno method is used to find the optimal values for unknown parameters of the fwnn models simulation examples are also given to compare the effectiveness of the models with the other known methods in the literature according to simulation results we see that the proposed fwnn models have impressive generalization ability

on the behavior of broyden s class of quasi newton methods this paper analyzes algorithms from the broyden class of quasi newton methods for nonlinear unconstrained optimization this class depends on a parameter phi k for which the choices phi k 0 and phi k 1 give the well known bfgs and dfp methods this paper examines algorithms that allow for negative values of the parameter phi k it shows that severe restrictions have to be imposed on the selection of phi k to guarantee q superlinear convergence it is argued that negative values of phi k are desirable and conditions on phi k that guarantee superlinear convergence are given however practical algorithms that preserve the excellent properties of the bfgs method are not easy to design

a broyden class of quasi newton methods for riemannian optimization this paper develops and analyzes a generalization of the broyden class of quasi newton methods to the problem of minimizing a smooth objective function f on a riemannian manifold a condition on vector transport and retraction that guarantees convergence and facilitates efficient computation is derived experimental evidence is presented demonstrating the value of the extension to the riemannian broyden class through superior performance for some problems compared to existing riemannian bfgs methods in particular those that depend on differentiated retraction

global analysis of the dennis wolkowicz least change secant algorithm this paper explores the global convergence of the least change secant method proposed by dennis and wolkowicz siam j numer anal 30 1993 pp 1291 1314 this method is a member of the broyden family but it doesn t necessarily belong to the broyden convex family furthermore it can be very close to the dfp method we will prove that this method with the wolfe line search on uniformly convex objective functions is globally convergent

connectedness of users items networks and recommender systems recommender systems have become an important issue in network science collaborative filtering and its variants are the most widely used approaches for building recommender systems which have received great attention in both academia and industry in this paper we studied the relationship between recommender systems and connectivity of users items bipartite network this results in a novel recommendation algorithm in our method recommended items are selected based on the eigenvector corresponding to the algebraic connectivity of the graph the second smallest eigenvalue of the laplacian matrix since recommending an item to a user equals to adding a new link to the users items bipartite graph the intuition behind the proposed approach is that the items should be recommended to the users such that the least increase in the connectedness of the network i e the algebraic connectivity is obtained through experiments on a number of benchmark datasets we showed that the proposed connectivity based recommendation method has comparable results to a number of commonly used recommendation methods these results shed light on the relation between the evolution of users behavior and network topology

a new variational result for quasi newton formulae the recent measure function of byrd and nocedal siam j numer anal 26 1989 pp 727 739 is considered and simple proofs of some of its properties are given it is then shown that the bfgs and dfp formulae satisfy a least change property with respect to this new measure

the least prior deviation quasi newton update we propose a new choice for the parameter in the broyden class and derive and discuss properties of the resulting self complementary quasi newton update our derivation uses a variational principle that minimizes the extent to which the quasi newton relation is violated on a prior step we discuss the merits of the variational principle used here vis a vis the other principle in common use which minimizes deviation from the current hessian or hessian inverse approximation in an appropriate frobenius matrix norm one notable advantage of our principle is an inherent symmetry that results in the same update being obtained regardless of whether the hessian matrix or the inverse hessian matrix is updated

a partitioned psb method for partially separable unconstrained optimization problems in this paper we propose a partitioned psb method for solving partially separable unconstrained optimization problems by using a projection technique we construct a sufficient descent direction under appropriate conditions we show that the partitioned psb method with projected direction is globally and superlinearly convergent for uniformly convex problems in particular the unit step length is accepted after finitely many iterations finally some numerical results are presented which show that the partitioned psb method is effective and competitive

embedded in the shadow of the separator eigenvectors to the second smallest eigenvalue of the laplace matrix of a graph also known as fiedler vectors are the basic ingredient in spectral graph partitioning heuristics maximizing this second smallest eigenvalue over all nonnegative edge weightings with bounded total weight yields the absolute algebraic connectivity introduced by fiedler who proved tight connections of this value to the connectivity of the graph our objective is to gain a better understanding of the connections between separators and the eigenspace of this eigenvalue by studying the dual semidefinite optimization problem to the absolute algebraic connectivity by exploiting optimality conditions we show that this problem is equivalent to finding an embedding of the n nodes of the graph in n space so that their barycenter is the origin the distance between adjacent nodes is bounded by one and the nodes are spread as much as possible the sum of the squared norms is maximized for connected graphs we prove that for any separator in the graph at least one of the two separated node sets is embedded in the shadow with the origin being the light source of the convex hull of the separator furthermore we show that there always exists an optimal embedding whose dimension is bounded by the tree width of the graph plus one

brief analysis of fractional delay systems of retarded and neutral type this paper analyses several properties linked to the robust control of fractional differential systems with delays the bibo stability of both retarded and neutral fractional delay systems is related to the location of their poles in the particular case of retarded systems we give some properties of the poles of the system and the singular values of its hankel operator

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

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

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 solving large scale limited memory quasi newton equations abstract we consider the problem of solving linear systems of equations arising with limited memory members of the restricted broyden class of updates and the symmetric rank one sr1 update in this paper we propose a new approach based on a practical implementation of the compact representation for the inverse of these limited memory matrices numerical results suggest that the proposed method compares favorably in speed and accuracy to other algorithms and is competitive with several update specific methods available to only a few members of the broyden class of updates using the proposed approach has an additional benefit the condition number of the system matrix can be computed efficiently

coprime factorization for regular linear systems mild sufficient conditions are given for the existence of a doubly coprime factorization of the transfer function of a regular linear system as well as formulae for such a factorization the results are illustrated by two examples of delay systems one of which has infinitely many unstable poles

statistical quasi newton a new look at least change a new method for quasi newton minimization outperforms bfgs by combining least change updates of the hessian with step sizes estimated from a wishart model of uncertainty the hessian update is in the broyden family but uses a negative parameter outside the convex range that is usually regarded as the safe zone for broyden updates although full newton steps based on this update tend to be too long excellent performance is obtained with shorter steps estimated from the wishart model in numerical comparisons to bfgs the new statistical quasi newton sqn algorithm typically converges with about 25 fewer iterations functions and gradient evaluations on the top 1 3 hardest unconstrained problems in the cute library typical improvement on the 1 3 easiest problems is about 5 the framework used to derive sqn provides a simple way to understand differences among various broyden updates such as bfgs and dfp and shows that these methods do not preserve accuracy of the hessian in a certain sense while the new method does in fact bfgs dfp and all other updates with nonnegative broyden parameters tend to inflate hessian estimates and this accounts for their observed propensity to correct eigenvalues that are too small more readily than eigenvalues that are too large numerical results on three new test functions validate these conclusions

sizing and least change secant methods oren and luenberger introduced in 1974 a strategy for replacing hessian approximations by their scalar multiples and then performing quasi newton updates generally least change secant updates such as the bfgs or dfp updates oren and luenberger management sci 20 1974 pp 845 862 in this paper the function omega a left frac operatorname trace a n operatorname det a 1 n right is shown to be a measure of change with a direct connection to the oren luenberger strategy this measure is interesting because it is related to the ell 2 condition number but it takes all the eigenvalues of a into account rather than just the extremes if the class of possible updates is restricted to the broyden class i e scalar premultiples are not allowed then the optimal update depends on the dimension of the problem it may or may not be in the convex class but it becomes the bfgs update as the dimension increases this seems to be yet another explanation for why

the fastest mixing markov process on a graph and a connection to a maximum variance unfolding problem we consider a markov process on a connected graph with edges labeled with transition rates between the adjacent vertices the distribution of the markov process converges to the uniform distribution at a rate determined by the second smallest eigenvalue lambda 2 of the laplacian of the weighted graph in this paper we consider the problem of assigning transition rates to the edges so as to maximize lambda 2 subject to a linear constraint on the rates this is the problem of finding the fastest mixing markov process fmmp on the graph we show that the fmmp problem is a convex optimization problem which can in turn be expressed as a semidefinite program and therefore effectively solved numerically we formulate a dual of the fmmp problem and show that it has a natural geometric interpretation as a maximum variance unfolding mvu problem the problem of choosing a set of points to be as far apart as possible measured by their variance while respecting local distance constraints this mvu problem is closely related to a problem recently proposed by weinberger and saul as a method for unfolding high dimensional data that lies on a low dimensional manifold the duality between the fmmp and mvu problems sheds light on both problems and allows us to characterize and in some cases find optimal solutions

improved hessian approximation with modified secant equations for symmetric rank one method symmetric rank one sr1 is one of the competitive formulas among the quasi newton qn methods in this paper we propose some modified sr1 updates based on the modified secant equations which use both gradient and function information furthermore to avoid the loss of positive definiteness and zero denominators of the new sr1 updates we apply a restart procedure to this update three new algorithms are given to improve the hessian approximation with modified secant equations for the sr1 method numerical results show that the proposed algorithms are very encouraging and the advantage of the proposed algorithms over the standard sr1 and bfgs updates is clearly observed

solving hankel matrix approximation problem using semidefinite programming positive semidefinite hankel matrices arise in many important applications some of their properties may be lost due to rounding or truncation errors incurred during evaluation the problem is to find the nearest matrix to a given matrix to retrieve these properties the problem is converted into a semidefinite programming problem as well as a problem comprising a semidefined program and second order cone problem the duality and optimality conditions are obtained and the primal dual algorithm is outlined explicit expressions for a diagonal preconditioned and crossover criteria have been presented computational results are presented a possibility for further improvement is indicated

measures for symmetric rank one updates measures of deviation of a symmetric positive definite matrix from the identity are derived they give rise to symmetric rank one sr1 type updates the measures are motivated by considering the volume of the symmetric difference of the two ellipsoids which arise from the current and updated quadratic models in quasi newton methods the measure defined by the problem maximize the determinant subject to a bound of 1 on the largest eigenvalue yields the sr1 update the measure a 1 a det a 1 n yields the optimally conditioned sized symmetric rank one updates the volume considerations also suggest a correction for the initial stepsize for these sized updates it is then shown that the optimal updates as well as the oren luenberger self scaling updates are all optimal updates for the measure the l2 condition number moreover all four sized updates result in the same largest and smallest scaled eigenvalue and corresponding eigenvector in fact the inversesized bfgs is the mean of the

analysis of a self scaling quasi newton method we study the self scaling bfgs method of oren and luenberger 1974 for solving unconstrained optimization problems for general convex functions we prove that the method is globally convergent with inexact line searches we also show that the directions generated by the self scaling bfgs method approach newton s direction asymptotically this would ensure superlinear convergence if in addition the search directions were well scaled but we show that this is not always the case we find that the method has a major drawback to achieve superlinear convergence it may be necessary to evaluate the function twice per iteration even very near the solution an example is constructed to show that the step sizes required to achieve a superlinear rate converge to 2 and 0 5 alternately

generalizing inverse compositional and esm image alignment inverse compositional ic image alignment baker and matthews in int j comput vis 56 3 221 255 2004 uses the symmetry between the roles of the fixed and moving images for faster processing however it requires implementation of compositional optimizer update steps the ic approach can be viewed as an efficient way of computing the similarity measure derivative relative to the fixed image warp parameters since the mapping between the fixed and moving warp parameters is continuous and differentiable this derivative can be converted into the moving warp space using the chain rule this avoids the need for compositional update steps our generalization also allows the efficient second order method esm malis in proceedings of the 2004 ieee international conference on robotics and automation icra04 pp 1843 1848 2004 benhimane and malis in ieee rsj international conference on intelligent robots and systems 2004 malis and benhimane in robot auton syst 52 1 39 52 2005 to be applied to general parameterizations of the transformation experiments using multiple similarity measures and optimizers show that our generalized ic method equals or exceeds the performance of the original ic approach the generalized esm approach is more reliable than the classic approach as it increases the capture radius of the optimization

the quasi cauchy relation and diagonal updating the quasi cauchy qc relation is the weak quasi newton relation of dennis and wolkowicz siam j numer anal 30 1993 pp 1291 1314 with the added restriction that full matrices are replaced by diagonal matrices this relation is justified and explored and in particular two basic variational techniques for updating diagonal matrices that satisfy it are formulated for purposes of illustration a numerical experiment is described where a diagonal updated matrix with hereditary positive definiteness is used to precondition cauchy s steepest descent direction the resulting qc algorithm is shown to be significantly accelerated in the concluding section the following topics are briefly discussed additional variational principles use of diagonal updates within other optimization algorithms together with some further numerical experience summarized in an appendix and an interesting connection between qc diagonal updating and trust region techniques

stabilization of miso fractional systems with delays abstract we consider multi input single output miso fractional systems of commensurate fractional orders with different input or output delays we derive explicit expressions of left and right coprime factorizations over h and of the associated bezout factors of the transfer matrix of the systems these factors allow the construction of the youla kucera parametrization of the set of stabilizing controllers which guarantee the internal stability of the closed loop systems

robust pole assignment for synthesizing feedback control systems using recurrent neural networks this paper presents a neurodynamic optimization approach to robust pole assignment for synthesizing linear control systems via state and output feedback the problem is formulated as a pseudoconvex optimization problem with robustness measure i e the spectral condition number as the objective function and linear matrix equality constraints for exact pole assignment two coupled recurrent neural networks are applied for solving the formulated problem in real time in contrast to existing approaches the exponential convergence of the proposed neurodynamics to global optimal solutions can be guaranteed even with lower model complexity in terms of the number of variables simulation results of the proposed neurodynamic approach for 11 benchmark problems are reported to demonstrate its superiority

quasi newton methods based on ordinary differential equation approach for unconstrained nonlinear optimization in this paper we propose a hybrid ode based quasi newton qn method for unconstrained optimization problems which combines the idea of low order implicit runge kutta rk techniques for gradient systems with the qn type updates of the jacobian matrix such as the symmetric rank one sr1 update the main idea of this approach is to associate a qn matrix to approximate numerically the jacobian matrix in the gradient system fundamentally this is a gradient system based on the first order optimality conditions of the optimization problem to further extend the methods for solving large scale problems a feature incorporated to the proposed methods is that a limited memory setting for matrix vector multiplications is used thus avoiding the computational and storage issues when computing jacobian information under suitable assumptions global convergence of the proposed method is proved practical insights on the effectiveness of these approaches on a set of test functions are given by a numerical comparison with that of the limited memory bfgs algorithm l bfgs and conjugate gradient algorithm cg

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

an algorithm for stabilization of fractional order time delay systems using fractional order pid controllers this technical note presents a solution to the problem of stabilizing a given fractional order system with time delay using fractional order pllambdadmu controllers it is based on determining a set of global stability regions in the kp ki kd space corresponding to the fractional orders lambda and mu in the range of 0 2 and then choosing the biggest global stability region in this set this method can be also used to find the set of stabilizing controllers that guarantees prespecified gain and phase margin requirements the algorithm is simple and has reliable result which is illustrated by an example and hence is practically useful in the analysis and design of fractional order control systems

quasi newton bundle type methods for nondifferentiable convex optimization in this paper we provide implementable methods for solving nondifferentiable convex optimization problems a typical method minimizes an approximate moreau yosida regularization using a quasi newton technique with inexact function and gradient values which are generated by a finite inner bundle algorithm for a bfgs bundle type method global and superlinear convergence results for the outer iteration sequence are obtained

a matrix free quasi newton method for solving large scale nonlinear systems one of the widely used methods for solving a nonlinear system of equations is the quasi newton method the basic idea underlining this type of method is to approximate the solution of newton s equation by means of approximating the jacobian matrix via quasi newton update application of quasi newton methods for large scale problems requires in principle vast computational resource to form and store an approximation to the jacobian matrix of the underlying problem hence this paper proposes an approximation for newton step based on the update of approximation requiring a computational effort similar to that of matrix free settings it is made possible by approximating the jacobian into a diagonal matrix using the least change secant updating strategy commonly employed in the development of quasi newton methods under suitable assumptions local convergence of the proposed method is proved for nonsingular systems numerical experiments on popular test problems confirm the effectiveness of the approach in comparison with newton s chord newton s and broyden s methods

on the behaviour of a combined extra updating self scaling bfgs method this paper attempts to combine the best features of certain extra updating bfgs method and self scaling bfgs method it describes an algorithm similar to the bfgs method except that extra self scaling updates are employed at some iterations the bfgs hessian is scaled and updated a number of times depending on the information of the first order derivatives the results of numerical tests on certain sets of standard problems are reported it is shown that the new algorithm is very competitive with the bfgs method in several cases

approximate and exact completion problems for euclidean distance matrices using semidefinite programming abstract a partial pre distance matrix a is a matrix with zero diagonal and with certain elements fixed to given nonnegative values the other elements are considered free the euclidean distance matrix completion problem chooses nonnegative values for the free elements in order to obtain a euclidean distance matrix edm the nearest or approximate euclidean distance matrix problem is to find a euclidean distance matrix edm that is nearest in the frobenius norm to the matrix a when the free variables are discounted in this paper we introduce two algorithms one for the exact completion problem and one for the approximate completion problem both use a reformulation of edm into a semidefinite programming problem sdp the first algorithm is based on an implicit equation for the completion that for many instances provides an explicit solution the other algorithm is based on primal dual interior point methods that exploit the structure and sparsity included are results on maps that arise that keep the edm and sdp cones invariant we briefly discuss numerical tests

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

rate based flow controllers for communication networks in the presence of uncertain time varying multiple time delays an h based robust controller is designed for a rate feedback flow control problem in single bottleneck communication networks the controller guarantees stability robustness to uncertain time varying multiple time delays in different channels it also brings the queue length at the bottleneck node to the desired steady state value asymptotically and satisfies a weighted fairness condition lower bounds for stability margins for uncertainty in the time delays and for the rate of change of the time delays are derived a number of simulations are included to demonstrate the time domain performance of the controller trade offs between robustness and time domain performance are also discussed

a reduced hessian method for large scale constrained optimization we propose a quasi newton algorithm for solving large optimization problems with nonlinear equality constraints it is designed for problems with few degrees of freedom and is motivated by the need to use sparse matrix factorizations the algorithm incorporates a correction vector that approximates the cross term z t wy p y in order to estimate the curvature in both the range and null spaces of the constraints the algorithm can be considered to be in some sense a practical implementation of an algorithm of coleman and conn we give conditions under which local and superlinear convergence is obtained