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

a shamanskii like self adaptive levenberg marquardt method for nonlinear equations abstract in this paper we are concerned with the shamanskii like self adaptive levenberg marquardt methods for nonlinear equations we consider two choices of levenberg marquardt parameter one of them is the standard self adaptive levenberg marquardt parameter the other is nonmonotone self adaptive levenberg marquardt parameter by using the nonmonotone technique of grippo lampariello and lucidi under the error bound condition which is weaker than nonsingularity we show that the shamanskii like self adaptive levenberg marquardt methods converge with q order m 1 numerical experiments show the new algorithms are efficient and promising

a simplification of the zoeppritz equations the compressional wave reflection coefficient r given by the zoeppritz equations is simplified to the following r r0 a0r0 1 2 sin2 1 2 vpvp tan2 sin2 the first term gives the amplitude at normal incidence 0 the second term characterizes r at intermediate angles and the third term describes the approach to critical angle the coefficient of the second term is that combination of elastic properties which can be determined by analyzing the offset dependence of event amplitude in conventional multichannel reflection data if the event amplitude is normalized to its value for normal incidence then the quantity determined is a a0 1 1 2 r0 a0 specifies the normal gradual decrease of amplitude with offset its value is constrained well enough that the main information conveyed is r0 where is the contrast in poisson s ratio at the reflecting interface and r0 is the amplitude at normal incidence this simplified formula for r accounts for all of the relations between r

a filter trust region method for unconstrained optimization a new filter trust region algorithm for solving unconstrained nonlinear optimization problems is introduced based on the filter technique introduced by fletcher and leyffer it extends an existing technique of gould leyffer and toint siam j optim 15 2004 pp 17 38 for nonlinear equations and nonlinear least squares to the fully general unconstrained optimization problem the new algorithm is shown to be globally convergent to at least one second order critical point and numerical experiments indicate that it is very competitive with more classical trust region algorithms

tensor methods for nonlinear equations a new class of methods for solving systems of nonlinear equations called tensor methods is introduced tensor methods are general purpose methods intended especially for problems where the jacobian matrix at the solution is singular or ill conditioned they base each iteration on a quadratic model of the nonlinear function the standard linear model augmented by a simple second order term the second order term is selected so that the model interpolates function values from several previous iterations as well as the current function value and jacobian the tensor method requires no more function and derivative information per iteration and hardly more storage or arithmetic per iteration than a standard method based on newton s method in extensive computational tests a tensor algorithm is significantly more efficient than a similar algorithm based on the standard linear model both on standard nonsingular test problems and on problems where the jacobian at the solution is singular

local convergence analysis of tensor methods for nonlinear equations tensor methods for nonlinear equations base each iteration upon a standard linear model augmented by a low rank quadratic term that is selected in such a way that the mode is efficient to form store and solve these methods have been shown to be very efficient and robust computationally especially on problems where the jacobian matrix at the root has a small rank deficiency this paper analyzes the local convergence properties of two versions of tensor methods on problems where the jacobian matrix at the root has a null space of rank one both methods augment the standard linear model by a rank one quadratic term we show under mild conditions that the sequence of iterates generated by the tensor method based upon an ideal tensor model converges locally and two step q superlinearly to the solution with q order 3 2 and that the sequence of iterates generated by the tensor method based upon a practial tensor model converges locally and three step q superlinearly to the solution with q order 3 2 in the same situation it is known that standard methods converge linearly with constant converging to 1 2 hence tensor methods have theoretical advantages over standard methods our analysis also confirms that tensor methods converge at least quadratically on problems where the jacobian matrix at the root is nonsingular

comparison of fully implicit and impes formulations for simulation of water injection in fractured and unfractured media in this work we compare the fully implicit fi and implicit pressure explicit saturation impes formulations for the simulation of water injection in fractured media the system of partial differential equations is discretized within the discrete fracture framework using a control volume method a unique feature of the methodology is that there is no need for the computation of matrix fracture transfer terms the non linear system of equations resulting from the fi formulation is solved with state of the art newton and tensor methods direct and krylov iterative methods are employed to solve the system resulting from the newton linearization the performance of the fi and impes formulations is compared with numerical testing results show that the contrast between matrix and fracture properties affects the performance of both impes and fi formulations and that the tensor method outperforms all the newton solvers for the near singular jacobian matrix copyright 2006 john wiley sons ltd

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

elastic impedance parameterization and inversion with young s modulus and poisson s ratio young s modulus and poisson s ratio are related to quantitative reservoir properties such as porosity rock strength mineral and total organic carbon content and they can be used to infer preferential drilling locations or sweet spots conventionally they are computed and estimated with a rock physics law in terms of p wave s wave impedances velocities and density which may be directly inverted with prestack seismic data however the density term imbedded in young s modulus is difficult to estimate because it is less sensitive to seismicamplitude variations and the indirect way can create more uncertainty for the estimation of young s modulus and poisson s ratio this study combines the elastic impedance equation in terms of young s modulus and poisson s ratio and elastic impedance variation with incident angle inversion to produce as table and direct way to estimate the young s modulus and poisson s ratio with no need for density information from prestack seismic data we initially derive a novel elastic impedance equation in terms of young s modulus and poisson s ratio and then to enhance the estimation stability we develop the elastic impedance varying with incident angle inversion with damping singular value decomposition eva dsvd method to estimate the young s modulus and poisson sr atio this method is implemented in a two step inversion elastic impedance inversion and parameter estimation the introduction of a model constraint and dsvd algorithm in parameter estimation renders the eva dsvd inversion more stable tests on synthetic data show that the young s modulus and poisson s ratio are still estimated reasonable with moderate noise a test on a real data set shows that the estimated results are in good agreement with the results of well interpretation

a levenberg marquardt algorithm with correction for singular system of nonlinear equations in this paper we propose a new lm algorithm for the singular system of nonlinear equations f x 0 where f x r n r n is continuously differentiable and f x is lipschitz continuous at every iteration not only a lm step but also a correction step are computed the lm parameter is taken as m email protected f email protected d where email protected 0 2 and m k is updated by the trust region technique under the local error bound condition which is weaker than nonsingularity we prove the convergence rate of the algorithm is min 2 email protected numerical results show that the new lm algorithm performs very well

tests for a change point summary the problem considered is that of testing a sequence of independent normal random variables with constant known or unknown variance for no change in mean versus alternatives with a single change point various tests such as those based on the likelihood ratio and recursive residuals are studied power approximations are developed by integrating approximations for conditional boundary crossing probabilities a comparison of several tests is made

avo inversion and poroelasticity with p and s wave moduli the fluid term in the biot gassmann equation plays an important role in reservoir fluid discrimination the density term imbedded in the fluid term however is difficult to estimate because it is less sensitive to seismic amplitude variations we combined poroelasticity theory amplitude variation with offset avo inversion and identification of p and s wave moduli to present a stable and physically meaningful method to estimate the fluid term with no need for density information from prestack seismic data we used poroelasticity theory to express the fluid term as a function of p and s wave moduli the use of pand s wave moduli made the derivation physically meaningful and natural then we derived an avo approximation in terms of these moduli which can then be directly inverted from seismic data furthermore this practical and robust avo inversion technique was developed in a bayesian framework the objective was to obtain the maximum a posteriori solution for the p wave modulus s wave modulus and density gaussian and cauchy distributions were used for the likelihood and a priori probability distributions respectively the introduction of a low frequency constraint and statistical probability information to the objective function rendered the inversion more stable and less sensitive to the initial model tests on synthetic data showed that all the parameters can be estimated well when no noise is present and the estimated p and s wave moduli were still reasonable with moderate noise and rather smooth initial model parameters a test on a real data set showed that the estimated fluid term was in good agreement with the results of drilling

a revised modified cholesky factorization algorithm a modified cholesky factorization algorithm introduced originally by gill and murray and refined by gill murray and wright is used extensively in optimization algorithms since its introduction in 1990 a different modified cholesky factorization of schnabel and eskow has also gained widespread usage compared with the gill murray wright algorithm the schnabel eskow algorithm has a smaller a priori bound on the perturbation added to ensure positive definiteness and some computational advantages especially for large problems users of the schnabel eskow algorithm however have reported cases from two different contexts where it makes a far larger modification to the original matrix than is necessary and than is made by the gill murray wright method this paper reports on a simple modification to the schnabel eskow algorithm that appears to correct all the known computational difficulties with the method without harming its theoretical properties or its computational behavior in any other cases in new computational tests the modifications to the original matrix made by the new algorithm appear virtually always to be smaller than those made by the gill murray wright algorithm sometimes by significant amounts the perturbed matrix is allowed to be more ill conditioned with the new algorithm but this seems to be appropriate in the known contexts where the underlying problem is ill conditioned

tensor methods for large sparse nonlinear least squares problems this paper introduces tensor methods for solving large sparse nonlinear least squares problems where the jacobian either is analytically available or is computed by finite difference approximations tensor methods have been shown to have very good computational performance for small to medium sized dense nonlinear least squares problems in this paper we consider the application of tensor methods to large sparse nonlinear least squares problems where a sparse factorization of the jacobian can be stored this involves an entirely new way of solving the tensor model that is efficient for sparse problems a number of interesting linear algebraic implementation issues are addressed the test results of the tensor method applied to a set of sparse nonlinear least squares problems compared with those of the standard gauss newton method reveal that the tensor method is significantly more robust and efficient than the standard gauss newton method

robust partial quadratic eigenvalue assignment with time delay using the receptance and the system matrices abstract in this paper we consider the robust partial quadratic eigenvalue assignment problem in vibration by active feedback control based on the receptance measurements and the system matrices we propose an optimization method for the robust and minimum norm partial quadratic eigenvalue assignment problem we provide a new cost function and the closed loop eigenvalue sensitivity and the feedback norms can be minimized simultaneously our method is also extended to the case of time delay between measurements of state and actuation of control numerical tests demonstrate the effectiveness of our method

high resolution velocity gathers and offset space reconstruction we present a high resolution procedure to reconstruct common midpoint cmp gathers first we describe the forward and inverse transformations between offset and velocity space then we formulate an underdetermined linear inverse problem in which the target is the artifacts free aperture compensated velocity gather we show that a sparse inversion leads to a solution that resembles the infinite aperture velocity gather the latter is the velocity gather that should have been estimated with a simple conjugate operator designed from an infinite aperture seismic array this high resolution velocity gather is then used to reconstruct the offset space the algorithm is formally derived using two basic principles first we use the principle of maximum entropy to translate prior information about the unknown parameters into a probabilistic framework in other words to assign a probability density function to our model second we apply bayes s rule to relate the a priori probability density function pdf with the pdf corresponding to the experimental uncertainties likelihood function to construct the a posteriori distribution of the unknown parameters finally the model is evaluated by maximizing the a posteriori distribution when the problem is correctly regularized the algorithm converges to a solution characterized by different degrees of sparseness depending on the required resolution the solutions exhibit minimum entropy when the entropy is measured in terms of burg s definition we emphasize two crucial differences in our approach with the familiar burg method of maximum entropy spectral analysis first burg s entropy is minimized rather than maximized which is equivalent to inferring as much as possible about the model from the data second our approach uses the data as constraints in contrast with the classic maximum entropy spectral analysis approach where the autocorrelation function is the constraint this implies that we recover not only amplitude information but also phase information which serves to extrapolate the data outside the original aperture of the array the tradeoff is controlled by a single parameter that under asymptotic conditions reduces the method to a damped least squares solution finally the high resolution or aperture compensated velocity gather is used to extrapolate near and far offset traces

a trust region method for solving generalized complementarity problems based on a semismooth equation reformulation using fischer s function a trust region algorithm is proposed for solving the generalized complementarity problem gcp the algorithm uses a generalized jacobian of the function involved in the semismooth equation and adopts the squared natural residual of the semismooth equation as a merit function the proposed algorithm is applicable to the nonlinear complementarity problem because the latter problem is a special case of the gcp global convergence and under a nonsingularity assumption local q superlinear or quadratic convergence of the algorithm are established moreover calculation of a generalized jacobian is discussed and numerical results are presented

a shamanskii like acceleration scheme for nonlinear equations at singular roots on applique une variante de la methode de shamanskii pour realiser une methode a convergence superlineaire de resolution d une classe d equations non lineaires ayant une derivee de frechet singuliere a la racine

on the global convergence of a filter sqp algorithm a mechanism for proving global convergence in sqp filter methods for nonlinear programming nlp is described such methods are characterized by their use of the dominance concept of multiobjective optimization instead of a penalty parameter whose adjustment can be problematic the main point of interest is to demonstrate how convergence for nlp can be induced without forcing sufficient descent in a penalty type merit function the proof relates to a prototypical algorithm within which is allowed a range of specific algorithm choices associated with the hessian matrix representation updating the trust region radius and feasibility restoration

parallel quasi newton methods for unconstrained optimization we discuss methods for solving the unconstrained optimization problem on parallel computers when the number of variables is sufficiently small that quasi newton methods can be used we concentrate mainly but not exclusively on problems where function evaluation is expensive first we discuss ways to parallelize both the function evaluation costs and the linear algebra calculations in the standard sequential secant method the bfgs method then we discuss new methods that are appropriate when there are enough processors to evaluate the function gradient and part but not all of the hessian at each iteration we develop new algorithms that utilize this information and analyze their convergence properties we present computational experiments showing that they are superior to parallelization either the bfgs methods or newton s method under our assumptions on the number of processors and cost of function evaluation finally we discuss ways to effectively utilize the gradient values at unsuccessful trial points that are available in our parallel methods and also in some sequential software packages

convergence rates for newton s method at singular points if newton s method is employed to find a root of a map from a banach space into itself and the derivative is singular at that root the convergence of the newton iterates to the root is linear rather than quadratic in this paper we give a detailed analysis of the linear convergence rates for several types of singular problems for some of these problems we describe modifications of newton s method which will restore quadratic convergence

on solving nonlinear equations with simple singularities or nearly singular solutions the paper begins with a review of the convergence theory for newton s method near simple and other regular singularities followed by a brief discussion of the inherent difficulty of singular problems and the effect of rounding errors on the achievable solution accuracy then several ways to accelerate the slow convergence of newton s method are described in some detail these are richardson extrapolation overrelaxation bordering by a singularity condition and appending the usual linear model by a quadratic term the last two approaches require limited second derivative information which can be obtained analytically by differencing or through secant updating numerical comparisons reported in the final sections lead to the conclusion that the collection of the additional information pays off in terms of speed and accuracy

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

a new acceleration method for newton s method at singular points when newton s method is used to find a root of a map from a banach space into itself and the derivative is singular at that root convergence of the newton sequence is in general linear in this paper we give a modification of newton s method that under certain conditions converges superlinearly our method is applicable under more general conditions than other techniques in particular it may be used for certain quadratic problems and problems in which the dimension of the null space of the derivative is larger than one

a note on the levenberg marquardt parameter abstract in c ma l jiang some research on levenberg marquardt method for the nonlinear equations appl math comput 184 2007 1032 1040 the lm parameter at the kth iteration is chosen as k f k 2 1 j k t f k 2 where f is the residual function j is the jacobi of f and 0 1 is a constant in this note we point out that the lm parameter can be any combination of f k 2 and j k t f k 2 provided it is positive furthermore we give a more general choice of the lm parameter and show that the lm method still preserves the quadratic convergence under the local error condition which is weaker than nonsingularity

on the global convergence of an slp filter algorithm that takes eqp steps a global convergence proof is presented for a class of trust region filter type methods for nonlinear programming such methods are characterized by their use of the dominance concept of multiobjective optimization instead of a penalty parameter whose adjustment can be problematic the methods are based on successively solving linear programming subproblems for which effective software is readily available the methods also permit the use of steps calculated on the basis of an equality constrained quadratic programming model which enables rapid convergence to take place for problems in which second order information is important the proof technique is presented in a fairly general context allowing a range of specific algorithm choices associated with choosing the quadratic model updating the trust region radius and with feasibility restoration

dealing with singularities in nonlinear unconstrained optimization we propose a new trust region based optimization algorithm for solving unconstrained nonlinear problems whose second derivatives matrix is singular at a local solution we give a theoretical characterization of the singularity in this context and we propose an iterative procedure which allows to identify a singularity in the objective function during the course of the optimization algorithm and artificially adds curvature to the objective function numerical tests are performed on a set of unconstrained nonlinear problems both singular and non singular results illustrate the significant performance improvement compared to classical trust region and filter algorithms proposed in the literature the approach is also shown to be competitive with tensor methods in terms of efficiency while reaching a higher level of robustness

a high order modified levenberg marquardt method for systems of nonlinear equations with fourth order convergence fan 2014 presented an accelerated modified levenberg marquardt method for nonlinear equations at every iteration the accelerated modified lm method computed not only a lm trial step but also an additional approximate lm step which employed a line search in this paper based on the accelerated modified lm method we compute the approximate lm step one more time at every iteration and obtain a high order accelerating modified levenberg marquardt method under the local error bound condition which is weaker than nonsingularity the convergence order of this new method is shown to be fourth a globally convergence is also given by the trust region technique numerical results show that the new method is efficient and could save many calculations of the jacobian

tests for change points with epidemic alternatives summary the purpose of this paper is to discuss the tests to detect an epidemic alternative in the mean value of a sequence of independent normal variables various test statistics such as the likelihood ratio the score like statistic levin kline s statistic the semilikelihood ratio and the recursive residual are studied the large deviation approximations to the significance levels and powers are developed by integrating approximations to conditional boundary crossing probabilities some results of monte carlo experiments confirm the accuracy of these approximations a numerical comparison of the different tests is made

estimation and comparison of multiple change point models this paper provides a new bayesian approach for models with multiple change points the centerpiece of the approach is a formulation of the change point model in terms of a latent discrete state variable that indicates the regime from which a particular observation has been drawn this state variable is specified to evolve according to a discrete time discrete state markov process with the transition probabilities constrained so that the state variable can either stay at the current value or jump to the next higher value this parameterization exactly reproduces the change point model the model is estimated by markov chain monte carlo methods using an approach that is based on chib 1996 this methodology is quite valuable since it allows for the fitting of more complex change point models than was possible before methods for the computation of bayes factors are also developed all the techniques are illustrated using simulated and real data sets

boundary crossing probabilities and statistical applications abstract this paper surveys recent results involving boundary crossing probabilities and related statistical applications the first part is concerned with problems of sequential analysis especially repeated significance tests and their application to sequential clinical trials involving survival data the second part develops the probability theory motivated by the problems of part 1 a method for computing first passage distributions of brownian motion to linear boundaries is introduced and then modified to handle problems in discrete time and those involving nonlinear boundaries the third part is concerned with fixed sample statistical problems especially change point problems which involve boundary crossing probabilities examples are given of problems for which methods of part 2 appear adequate and of problems which require new methods

a higher order levenberg marquardt method for nonlinear equations in this paper we present a high order levenberg marquardt method for nonlinear equations at every iteration not only a lm step is computed but also two approximate lm steps are computed which use the previous calculated jacobian under the local error bound condition which is weaker than nonsingularity the new method has biquadratic convergence a globally convergent lm algorithm is also given by the trust region technique numerical results show that the new fourth order lm algorithm performs very well and could save many jacobian calculations especially for large scale problems

approximate tail probabilities for the maxima of some random fields abstract hogan and siegmund 1986 adapt the method developed by pickands 1969 qualls and watanabe 1973 and bickel and rosenblatt 1973 to obtain explicit large deviation approximations for the maxima of several gaussian random fields arising in statistics using a special argument for one particular case they suggest a heuristic second order approximation for that case and they show by a monte carlo experiment that the second order approximation frequently gives considerably better numerical results the purpose of this paper is to show that the method developed by woodroofe 1976 1982 for problems in one dimensional time can be adapted to study maxima of random fields overall it involves simpler computations than the previous method and consequently seems potentially capable of delivering a genuine second order approximation should one seem desirable

a conservative test for multiple comparison based on highly correlated test statistics in genetics we often encounter a large number of highly correlated test statistics the most famous conservative bound for multiple comparison is bonferroni s bound which is suitable when the test statistics are independent but not when the test statistics are highly correlated this article proposes a new conservative bound that is easily calculated without multiple integration and is a good approximation when the test statistics are highly correlated the performance of the proposed method is evaluated by simulation and real data analysis

approximate solution of the trust region problem by minimization over two dimensional subspaces the trust region problem minimization of a quadratic function subject to a spherical trust region constraint occurs in many optimization algorithms in a previous paper the authors introduced an inexpensive approximate solution technique for this problem that involves the solution of a two dimensional trust region problem they showed that using this approximation in an unconstrained optimization algorithm leads to the same theoretical global and local convergence properties as are obtained using the exact solution to the trust region problem this paper reports computational results showing that the two dimensional minimization approach gives nearly optimal reductions in then dimension quadratic model over a wide range of test cases we also show that there is very little difference in efficiency and reliability between using the approximate or exact trust region step in solving standard test problems for unconstrained optimization these results may encourage the application of similar approximate trust region techniques in other contexts

high resolution three term avo inversion by means of a trivariate cauchy probability distribution three term avo inversion can be used to estimate p wave velocity s wave velocity and density perturbations from reflection seismic data the density term however exhibits little sensitivity to amplitudes and therefore its inversion is unstable one way to stabilize the density term is by including a scale matrix that provides correlation information between the three unknown avo parameters we investigate a bayesian procedure to include sparsity and a scale matrix in the three term avo inversion problem to this end we model the prior distribution of the avo parameters via a trivariate cauchy distribution we found an iterative algorithm to solve the bayesian inversion and in addition comparisons are provided with the classical inversion approach that uses a multivariate gaussian prior it is important to point out that the multivariate gaussian prior allows us to include the correlation of the avo parameters in the solution of the inverse problem the trivariate cauchy prior not only permits us

a modified brown algorithm for solving singular nonlinear systems with rank defects a modified brown algorithm for solving a class of singular nonlinear systems f x 0 where x f rn is presented this method is constructed by combining the discreted brown algorithm with the space transforming method the second order information of f x at a point is not required calculating which is different from the tensor method and the hoy s method the q quadratic convergence of this algorithm and some numerical examples are given as well

the cusum test of homogeneity with an application in spontaneous abortion epidemiology we test the hypothesis of homogeneity over time in the rate of an event with a modification of page s cumulative sum cusum procedure the modification draws on an indifference zone approach to testing that is well suited to the epidemiologic study of retrospective data we provide procedures for evaluating the significance level of the largest observed cusum value occurring in a sequence of fixed length we draw inferences unconditionally with a known baseline rate of the event and conditionally on observed margins with unknown baseline rate data collected over a five year period from one new york city hospital serve to illustrate the modified cusum test in a study of fluctuations in the proportions of chromosomally normal trisomic x monosomic and triploid spontaneous abortions

tensor methods for large sparse systems of nonlinear equations this paper introduces tensor methods for solving large sparse systems of nonlinear equations tensor methods for nonlinear equations were developed in the context of solving small to medium sized dense problems they base each iteration on a quadratic model of the nonlinear equations where the second order term is selected so that the model requires no more derivative or function information per iteration than standard linear model based methods and hardly more storage or arithmetic operations per iteration computational experiments on small to medium sized problems have shown tensor methods to be considerably more efficient than standard newton based methods with a particularly large advantage on singular problems this paper considers the extension of this approach to solve large sparse problems the key issue considered is how to make efficient use of sparsity in forming and solving the tensor model problem at each iteration accomplishing this turns out to require an entirely new way of solving the tensor model that successfully exploits the sparsity of the jacobian whether the jacobian is nonsingular or singular we develop such an approach and based upon it an efficient tensor method for solving large sparse systems of nonlinear equations test results indicate that this tensor method is significantly more efficient and robust than an efficient sparse newton based method in terms of iterations function evaluations and execution time 1998 the mathematical programming society inc published by elsevier science b v

a modified two steps levenberg marquardt method for nonlinear equations the modified levenberg marquardt mlm method to solve nonlinear equations was introduced by fan 2012 this method uses an addition of the levenberg marquardt step and an approximate lm step as the trial step at every iteration using a trust region technique the global and cubic convergence of the mlm method under the local error bound condition is proved fan 2012 recently fan proposed an accelerated mlm algorithm by using a line search strategy to generate a modified lm step and showed that the convergence rate of the algorithm is min 1 2 3 which results the cubic convergence for 1 fan 2014 in this paper by introducing an adaptive lm parameter for amlm algorithm we propose an efficient amlm algorithm the cubic convergence of the new algorithm is presented while numerical experiments show the new algorithm is promising

a multidimensional filter algorithm for nonlinear equations and nonlinear least squares we introduce a new algorithm for the solution of systems of nonlinear equations and nonlinear least squares problems that attempts to combine the efficiency of filter techniques and the robustness of trust region methods the algorithm is shown under reasonable assumptions to globally converge to zeros of the system or to first order stationary points of the euclidean norm of its residual preliminary numerical experience is presented that shows substantial gains in efficiency over the traditional monotone trust region approach

an efficient two step trust region algorithm for exactly determined consistent systems of nonlinear equations abstract in this paper we propose a modified two step trust region algorithm for solving nonlinear systems two step trust region algorithms at every iteration use a trust region step and an approximate step by saving the jacobian matrix to have fewer computations we introduce a convex combination to modify the trust region subproblems along with an additional criterion to verify whether the first step is accepted or not we establish global and quadratic convergence of the algorithm under some mild assumptions numerical results show that the modified algorithm is efficient and promising

a corrected levenberg marquardt algorithm with a nonmonotone line search for the system of nonlinear equations in this paper we propose a corrected levenberg marquardt method for the system of nonlinear equations in which not only an l m step and a corrected step are computed at every iteration but also a nonmonotone line search to find a new iteration point will be performed if a trial step is not accepted to ensure the global convergence of the new method a new nonmonotone line search technique is introduced for the merit function the cubic convergence of the new method is proved under the local error bound condition which is weaker than nonsingularity some numerical results are reported which shows that the algorithm is quite effective

superlinearly convergent algorithms for solving singular equations and smooth reformulations of complementarity problems we propose a new algorithm for solving smooth nonlinear equations in the case where their solutions can be singular compared to other techniques for computing singular solutions a distinctive feature of our approach is that we do not employ second derivatives of the equation mapping in the algorithm and we do not assume their existence in the convergence analysis important examples of once but not twice differentiable equations whose solutions are inherently singular are smooth equation based reformulations of the nonlinear complementarity problems reformulations of complementarity problems serve both as illustration of and motivation for our approach and one of them we consider in detail we show that the proposed method possesses local superlinear quadratic convergence under reasonable assumptions we further demonstrate that these assumptions are in general not weaker and not stronger than regularity conditions employed in the context of other superlinearly convergent newton type algorithms for solving complementarity problems which are typically based on nonsmooth reformulations therefore our approach appears to be an interesting complement to the existing ones

analysis of newton s method at irregular singularities recent work on newton s method at singularities of the jacobian has established linear convergence under certain regularity assumptions here newton s method is analyzed in the neighborhood of irregular singularities which include all minimizers at which the hessian has a one dimensional null space depending on a parameter specific to the underlying optimization problem or system of simultaneous equations newton s method is found either to converge with a limiting ratio of about frac 2 3 or to diverge from arbitrarily close starting points or to behave in a certain sense chaotically

fixed point quasi newton methods this article studies iterative methods defined by x k 1 phi x k e k where x k in mathbb r n and e k lie on a space of parameters sufficient conditions are established for local convergence and for convergence at an ideal linear or superlinear rate a theory of least change secant update methods for this class of processes is developed several examples are given showing a wide range of applications of the new theory key words quasi

three steps modified levenberg marquardt method with a new line search for systems of nonlinear equations three steps modified levenberg marquardt method for nonlinear equations was introduced by yang 2013 this method uses the addition of the levenberg marquardt lm step and two approximate lm steps as the trial step at every iteration using trust region technique the global and biquadratic convergence of the method is proved by yang the main aim of this paper is to introduce a new line search strategy while investigating the convergence properties of the method with this line search technique since the search direction of yang method may be not a descent direction standard line searches cannot be used directly in this paper we propose a new nonmonotone third order armijo type line search technique which guarantees the global convergence of this method while we use an adaptive lm parameter it is proved that the convergence order of the new method is biquadratic numerical results show the new algorithm is efficient and promising