a limited memory bfgs type method for large scale unconstrained optimization in this paper a new numerical method for solving large scale unconstrained optimization problems is presented it is derived from a modified bfgs type update formula by wei li and qi it is observed that the update formula can be extended to the framework of limited memory scheme with hardly more storage or arithmetic operations under some suitable conditions the global convergence property is established the implementations of the method on a set of cute problems indicate that this extension is beneficial to the performance of the algorithm

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

the divergence of the bfgs and gauss newton methods we present examples of divergence for the bfgs and gauss newton methods these examples have objective functions with bounded level sets and other properties concerning the examples published recently in this journal like unit steps and convexity along the search lines as these other examples the iterates function values and gradients in the new examples fit into the general formulation in our previous work mascarenhas comput appl math 26 1 2007 which also presents an example of divergence for newton s method

on the convergence of the dfp algorithm for unconstrained optimization when there are only two variables let the dfp algorithm for unconstrained optimization be applied to an objective function that has continuous second derivatives and bounded level sets where each line search finds the first local minimum it is proved that the calculated gradients are not bounded away from zero if there are only two variables the new feature of this work is that there is no need for the objective function to be convex

on nesterov s nonsmooth chebyshev rosenbrock functions abstract we discuss two nonsmooth functions on r n introduced by nesterov we show that the first variant is partly smooth in the sense of lewis and that its only stationary point is the global minimizer in contrast we show that the second variant has 2 n 1 clarke stationary points none of them local minimizers except the global minimizer but also that its only mordukhovich stationary point is the global minimizer nonsmooth optimization algorithms from multiple starting points generate iterates that approximate all 2 n 1 clarke stationary points not only the global minimizer but it remains an open question as to whether the nonminimizing clarke stationary points are actually points of attraction for optimization algorithms

convergence properties of the bfgs algoritm the bfgs method is one of the most famous quasi newton algorithms for unconstrained optimization in 1984 powell presented an example of a function of two variables that shows that the polak ribiere polyak prp conjugate gradient method and the bfgs quasi newton method may cycle around eight nonstationary points if each line search picks a local minimum that provides a reduction in the objective function in this paper a new technique of choosing parameters is introduced and an example with only six cyclic points is provided it is also noted through the examples that the bfgs method with wolfe line searches need not converge for nonconvex objective functions

global convergence of the nonmonotone mbfgs method for nonconvex unconstrained minimization in this paper we propose a new nonmonotone armijo type line search and prove that the mbfgs method proposed by li and fukushima with this new line search converges globally for nonconvex minimization some numerical experiments show that this nonmonotone mbfgs method is efficient for the given test problems

the global convergence of the polak ribiere polyak conjugate gradient algorithm under inexact line search for nonconvex functions abstract powell 1984 and dai 2003 constructed respectively a counterexample to show that the polak ribiere polyak prp conjugate gradient algorithm fails to globally converge for nonconvex functions even when the exact line search technique is used which implies similar failure of the weak wolfe powell wwp inexact line search technique does another inexact line search technique exist that can ensure global convergence for nonconvex functions this paper gives a positive answer to this question and proposes a modified wwp mwwp inexact line search technique an algorithm is presented that uses the mwwp inexact line search technique the next point x k 1 generated by the prp formula is accepted if a positive condition holds and otherwise x k 1 is defined by a technique for projection onto a parabolic surface the proposed prp conjugate gradient algorithm is shown to possess global convergence under inexact line search for nonconvex functions numerical performance of the proposed algorithm is shown to be competitive with those of similar algorithms

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

a modified bfgs type quasi newton method with line search for symmetric nonlinear equations problems abstract in this paper using approximate gradient of the norm square metric function we present an inexact mbfgs method with line search for solving symmetric nonlinear equations which is a generalization of the mbfgs method proposed by li and fukushima 2001 for solving smooth unconstrained optimization the used approximate gradient of the proposed method only requires computing two residuals at every iteration and is easy to obtain by utilizing symmetric structure of the system we establish global convergence of the proposed method and report some numerical results

global convergence of a new hybrid gauss newton structured bfgs method for nonlinear least squares problems in this paper we propose a hybrid gauss newton structured bfgs method with a new update formula and a new switch criterion for the iterative matrix to solve nonlinear least squares problems we approximate the second term in the hessian by a positive definite bfgs matrix under suitable conditions global convergence of the proposed method with a backtracking line search is established moreover the proposed method automatically reduces to the gauss newton method for zero residual problems and the structured bfgs method for nonzero residual problems in a neighborhood of an accumulation point a locally quadratic convergence rate for zero residual problems and a locally superlinear convergence rate for nonzero residual problems are obtained for the proposed method some numerical results are given to compare the proposed method with some existing methods

descent directions of quasi newton methods for symmetric nonlinear equations in general when a quasi newton method is applied to solve a system of nonlinear equations the quasi newton direction is not necessarily a descent direction for the norm function in this paper we show that when applied to solve symmetric nonlinear equations a quasi newton method with positive definite iterative matrices may generate descent directions for the norm function on the basis of a gauss newton based bfgs method d h li and m fukushima siam j numer anal 37 1999 pp 152 172 we develop a norm descent bfgs method for solving symmetric nonlinear equations under mild conditions we establish the global and superlinear convergence of the method the proposed method shares some favorable properties of the bfgs method for solving unconstrained optimization problems a the generated sequence of the quasi newton matrices is positive definite b the generated sequence of iterates is norm descent c a global convergence theorem is established without nonsingularity assumption on the jacobian preliminary numerical results are reported which positively support the method

the bfgs method with exact line searches fails for non convex objective functions this work shows that the bfgs method and other methods in the broyden class with exact line searches may fail for non convex objective functions

a globally convergent bfgs method with nonmonotone line search for non convex minimization in this paper we propose a modified bfgs broyden fletcher goldfarb shanno method with nonmonotone line search for unconstrained optimization under some mild conditions we show that the method is globally convergent without a convexity assumption on the objective function we also report some preliminary numerical results to show the efficiency of the proposed method

nonsmooth optimization via quasi newton methods we investigate the behavior of quasi newton algorithms applied to minimize a nonsmooth function f not necessarily convex we introduce an inex act line search that generates a sequence of nested intervals containing a set of points of nonzero measure that satisfy the armijo and wolfe conditions if f is absolutely continuous along the line furthermore the line search is guaranteed to terminate if f is semi algebraic it seems quite difficult to establish a convergence theorem for quasi newton methods applied to such general classes of functions so we give a care ful analysis of a special but illuminating case the euclidean norm in one variable using the inexact line search and in two variables assuming that the line search is exact in practice we find that when f is locally lipschitz and semi algebraic with bounded sublevel sets the bfgs broyden fletcher goldfarb shanno method with the inexact line search almost always generates sequences whose cluster points are clarke stationary and with function values converging r linearly to a clarke station ary value we give references documenting the successful use of bfgs in a variety of nonsmooth applications particularly the design of low order controllers for linear dynamical systems we conclude with a challenging open question

global dynamics of a hyperbolic parabolic model arising from chemotaxis we prove global existence and qualitative behavior of classical solutions for a hyperbolic parabolic system describing chemotaxis on bounded domains it is shown that classical solutions to the initial boundary value problem of the one dimensional model exist globally in time for large initial data and the solutions converge to constant equilibrium states exponentially in time which rigorously demonstrates the collapsing of cell populations in chemotaxis moreover similar results are established for the multidimensional model when the initial data are small

a globally convergent bfgs method for nonlinear monotone equations without any merit functions r since 1965 there has been significant progress in the theoretical study on quasi newton methods for solving nonlinear equations especially in the local convergence analysis however the study on global convergence of quasi newton methods is relatively fewer especially for the bfgs method to ensure global convergence some merit function such as the squared norm merit function is typically used in this paper we propose an algorithm for solving nonlinear monotone equations which combines the bfgs method and the hyperplane projection method we also prove that the proposed bfgs method converges globally if the equation is monotone and lipschitz continuous without differentiability requirement on the equation which makes it possible to solve some nonsmooth equations an attractive property of the proposed method is that its global convergence is independent of any merit function we also report some numerical results to show efficiency of the proposed method

galerkin method for unsplit 3 d dirac equation using atomically kinetically balanced b spline basis a galerkin method is developed to solve the time dependent dirac equation in prolate spheroidal coordinates for an electron molecular two center system the initial state is evaluated from a variational principle using a kinetic atomic balanced basis which allows for an efficient and accurate determination of the dirac spectrum and eigenfunctions b spline basis functions are used to obtain high accuracy this numerical method is used to compute the energy spectrum of the two center problem and then the evolution of eigenstate wavefunctions in an external electromagnetic field

a perfect example for the bfgs method consider the bfgs quasi newton method applied to a general non convex function that has continuous second derivatives this paper aims to construct a four dimensional example such that the bfgs method need not converge the example is perfect in the following sense a all the stepsizes are exactly equal to one the unit stepsize can also be accepted by various line searches including the wolfe line search and the arjimo line search b the objective function is strongly convex along each search direction although it is not in itself the unit stepsize is the unique minimizer of each line search function hence the example also applies to the global line search and the line search that always picks the first local minimizer c the objective function is polynomial and hence is infinitely continuously differentiable if relaxing the convexity requirement of the line search function namely b we are able to construct a relatively simple polynomial example

shape sensitivity analysis for an interface problem via minimax differentiability this paper deals with the shape sensitivity analysis for the conductivity problem with discontinuous coefficients we consider the geometric inverse problem of locating the interfaces where the jump of the coefficients occurs we propose a reconstruction method based on the minimization of the kohn vogelius functional the reformulation of the functional in a suitable saddle point problem form allows us to obtain the optimality conditions by using the differentiability of the minimax combined with the function space parametrization technique an important advantage of this approach is that it avoids the differentiability of the state variable with respect to the domain the reconstruction is then performed with an iterative algorithm using an appropriate quasi newton method we give some numerical examples showing the efficiency of the method