benchmarking optimization software with performance profiles we propose performance profiles distribution functions for a performance metric as a tool for benchmarking and comparing optimization software we show that performance profiles combine the best features of other tools for performance evaluation

moment lyapunov exponent and stability index for linear conservative system with small random perturbation asymptotic expansion series for the moment lyapunov exponent and stability index are constructed and justified for the two dimensional linear stochastic system close to a harmonic oscillator as an example a one degree of freedom mechanical system parametrically excited in stiffness and damping is considered and several terms of the expansion are obtained

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 the global convergence of the bfgs method for nonconvex unconstrained optimization problems this paper is concerned with the open problem of whether the bfgs method with inexact line search converges globally when applied to nonconvex unconstrained optimization problems we propose a cautious bfgs update and prove that the method with either a wolfe type or an armijo type line search converges globally if the function to be minimized has lipschitz continuous gradients

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

asymptotic analysis of the lyapunov exponent and rotation number of the random oscillator and application we construct asymptotic expansions for the exponential growth rate lyapunov exponent and rotation number of the random oscillator when the noise is large small rapidly varying or slowly varying we then apply our results to problems in the stability of the random oscillator the spectrum of the one dimensional random schr6dinger operator and wave propagation in a one dimensional random medium

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

moment lyapunov exponents of the stochastic parametrical hill s equation abstract the lyapunov exponent and moment lyapunov exponents of hill s equation with frequency and damping coefficient fluctuated by white noise stochastic process are investigated a perturbation approach is used to obtain explicit expressions for these exponents in the presence of small intensity noises the results are applied to the study of the almost sure and the moment stability of the stationary solutions of the thin simply supported beam subjected to axial compressions and time varying damping which are small intensity stochastic excitations

a formula connecting sample and moment stability of linear stochastic systems given the linear stochastic system dot x a xi t x x in mathbb r d xi t some nice diffusion sample and moment stability of x equiv 0 are studied via the sample lyapunov exponent lambda and the lyapunov exponent of the pth mean g p of a solution generalizing a result of molchanov for the undamped linear oscillator it is shown that g p is a smooth convex function of p in mathbb r with g 0 0 and g 0 lambda for trace a 0 a second zero of g at p d is detected this allows one to draw simple but quite complete conclusions about the relation between sample and moment stability

moment lyapunov exponents of a two dimensional system under bounded noise parametric excitation the moment lyapunov exponents of a two dimensional system under bounded noise parametric excitation are studied in this paper the method of regular perturbation is applied to obtain weak noise expansions of the moment lyapunov exponent lyapunov exponent and stability index in terms of the small fluctuation parameter

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

new quasi newton methods for unconstrained optimization problems abstract many methods for solving minimization problems are variants of newton method which requires the specification of the hessian matrix of second derivatives quasi newton methods are intended for the situation where the hessian is expensive or difficult to calculate quasi newton methods use only first derivatives to build an approximate hessian over a number of iterations this approximation is updated each iteration by a matrix of low rank in unconstrained minimization the original quasi newton equation is bk 1sk yk where yk is the difference of the gradients at the last two iterates in this paper we first propose a new quasi newton equation b k 1 s k y k in which y k is decided by the sum of yk and aksk where ak is some matrix then we give two choices of ak which carry some second order information from the hessian of the objective function the three corresponding bfgs type algorithms are proved to possess global convergence property the superlinear convergence of the one algorithm is proved extensive numerical experiments have been conducted which show that the proposed algorithms are very encouraging

moment lyapunov exponents of a two dimensional system under real noise excitation abstract the moment lyapunov exponents of a two dimensional system under real noise excitation an ornstein uhlenbeck process are studied in this paper the method of regular perturbation is applied to obtain the weak noise expansions of the moment lyapunov exponent lyapunov exponent and stability index in terms of the small fluctuation parameter

a theoretical and experimental study of the symmetric rank one update this paper first discusses computational experience using the sri update in conventional line search and trust region algorithms for unconstrained optimization the experiments show that the sri is very competitive with the widely used bfgs method they also indicate two interesting features the final hessian approximations produced by the sri method are not generally appreciably better than those produced by the bfgs and the sequences of steps produced by the sri do not usually seem to have the uniform linear independence property that is assumed in recent convergence analysis this paper presents a new analysis that shows that the sri method with a line search is n 1 step q superlinearly convergent without the assumption of linearly independent iterates this analysis assumes that the hessian approximations are positive definite and bounded asymptotically which from computational experience are reasonable assumptions

structural modification of systems using discretization and generalized sampled data hold functions this paper investigates the decentralized control of lti continuous time plants using generalized sampled data hold functions gshf gshfs can be used to modify the structure of the digraph of the resultant discrete plant by removing certain interconnections in the discrete time equivalent model to form a hierarchical system model of the plant this is a new application of discretization and has as its motivation the design of decentralized controllers using centralized methods

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

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

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

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

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 modified bfgs method and its global convergence in nonconvex minimization in this paper we propose a modification of the bfgs method for unconstrained optimization a remarkable feature of the proposed method is that it possesses a global convergence property even without convexity assumption on the objective function under certain conditions we also establish superlinear convergence of the method

mean square stability of a second order parametric linear system excited by a colored gaussian noise abstract the stochastic stability of a second order linear parametric oscillator whose stiffness is perturbed by a stationary zero mean colored gaussian process is investigated in this paper referring to the stability in the second response moments mean square stability the stochastic perturbation is the output of a linear filter excited by a stationary gaussian white noise stochastic process and the hypothesis of weak excitation is not formulated the first assumption allows using ito s stochastic differential calculus the moment equations of the response are written till the second order by means of ito s differential rule they form an infinite hierarchy that is the equations for the moments of order r contain moments of order larger than r in order to close the hierarchy the cumulant neglect closure method is applied herein the moment equation set is closed by neglecting the cumulants of third order nonlinear moment equations result they are linearized and the limit of stability is searched by studying the eigenvalues of the matrix of the coefficients of the linearized moment equations several numerical analyses are performed to evaluate the critical mean square value of the excitation when a system parameter is varied for the cases of both first and second order parametric excitation a first order excitation an ornstein uhlenbeck process does not cause stochastic resonance while a second order excitation does the first result was not known previously