the convergence of subspace trust region methods the trust region method is an effective approach for solving optimization problems due to its robustness and strong convergence however the subproblem in the trust region method is difficult or time consuming to solve in practical computation especially in large scale problems in this paper we consider a new class of trust region methods specifically subspace trust region methods the subproblem in these methods has an adequate initial trust region radius and can be solved in a simple subspace it is easier to solve than the original subproblem because the dimension of the subproblem in the subspace is reduced substantially we investigate the global convergence and convergence rate of these methods

fast and robust fuzzy c means clustering algorithms incorporating local information for image segmentation fuzzy c means fcm algorithms with spatial constraints fcm s have been proven effective for image segmentation however they still have the following disadvantages 1 although the introduction of local spatial information to the corresponding objective functions enhances their insensitiveness to noise to some extent they still lack enough robustness to noise and outliers especially in absence of prior knowledge of the noise 2 in their objective functions there exists a crucial parameter a used to balance between robustness to noise and effectiveness of preserving the details of the image it is selected generally through experience and 3 the time of segmenting an image is dependent on the image size and hence the larger the size of the image the more the segmentation time in this paper by incorporating local spatial and gray information together a novel fast and robust fcm framework for image segmentation i e fast generalized fuzzy c means fgfcm clustering algorithms is proposed fgfcm can mitigate the disadvantages of fcm s and at the same time enhances the clustering performance furthermore fgfcm not only includes many existing algorithms such as fast fcm and enhanced fcm as its special cases but also can derive other new algorithms such as fgfcm s1 and fgfcm s2 proposed in the rest of this paper the major characteristics of fgfcm are 1 to use a new factor s i j as a local both spatial and gray similarity measure aiming to guarantee both noise immunity and detail preserving for image and meanwhile remove the empirically adjusted parameter a 2 fast clustering or segmenting image the segmenting time is only dependent on the number of the gray levels q rather than the size n q of the image and consequently its computational complexity is reduced from o nci 1 to o qci 2 where c is the number of the clusters i 1 and i 2

dogleg paths and trust region methods with back tracking technique for unconstrained optimization in this paper we improve approximate trust region methods via a class of dogleg paths for unconstrained optimization the dogleg paths include both definite and indefinite ones a hybrid strategy using both trust region and line search techniques is adopted which switches to back tracking steps when a trial step produced by the trust region subproblem is unacceptable we show that the algorithm preserves the strong convergence properties of trust region methods numerical results are presented and discussed

necessary and sufficient conditions for oscillations of higher order delay differential equations we obtain necessary and sufficient conditions under which ail solutions of certain n th order delay differential equations oscillate when n is odd and necessary and sufficient conditions under which all bounded solutions oscillate when n is even the conditions derived involve the characteristic equation of the delay differential equation

properties of kneser solutions for third order differential equations abstract we study asymptotic behavior of nonoscillatory solutions for the third order differential equations of the form r t y t p t y t 0 for property a of studied equations we extend the information about the asymptotic properties of positive decreasing solutions

newton type minimization via the lanczos method this paper discusses the use of the linear conjugate gradient method developed via the lanczos method in the solution of large scale unconstrained minimization problems at each iteration of a newton type method the direction of search is defined as the solution of a quadratic subproblem when the number of variables is very large this subproblem may be solved using the linear conjugate gradient method of hestenes and stiefel we show how the equivalent lanczos characterization of the linear conjugate gradient method may be exploited to define a modified newton method which can be applied to problems that do not necessarily have positive definite hessian matrices at all points of the region of interest this derivation also makes it possible to compute a negative curvature direction at a stationary point the idea of a truncated newton method is to perform only a limited number of iterations of the quadratic subproblem this effectively gives a search direction that interpolates between the steepest de

fuzzy c means clustering with local information and kernel metric for image segmentation in this paper we present an improved fuzzy c means fcm algorithm for image segmentation by introducing a tradeoff weighted fuzzy factor and a kernel metric the tradeoff weighted fuzzy factor depends on the space distance of all neighboring pixels and their gray level difference simultaneously by using this factor the new algorithm can accurately estimate the damping extent of neighboring pixels in order to further enhance its robustness to noise and outliers we introduce a kernel distance measure to its objective function the new algorithm adaptively determines the kernel parameter by using a fast bandwidth selection rule based on the distance variance of all data points in the collection furthermore the tradeoff weighted fuzzy factor and the kernel distance measure are both parameter free experimental results on synthetic and real images show that the new algorithm is effective and efficient and is relatively independent of this type of noise

oscillation of functional trinomial differential equations with positive and negative term abstract in the paper we establish new technique for investigation of properties of trinomial differential equations with positive and negative term b t a t x t p t f x t q t h x t 0 we offer new criteria for asymptotic properties of nonoscillatory solutions for the studied equations we support our results with illustrative examples

a modified nearly exact method for solving low rank trust region subproblem in this paper we first discuss how the nearly exact ne method proposed by more and sorensen 14 for solving trust region tr subproblems can be modified to solve large scale low rank tr subproblems efficiently our modified algorithm completely avoids computation of cholesky factorizations by instead relying primarily on the sherman morrison woodbury formula for computing inverses of diagonal plus low rank type matrices we also implement a specific version of the modified log barrier mlb algorithm proposed by polyak 17 where the generated log barrier subproblems are solved by a trust region method the corresponding direction finding tr subproblems are of the low rank type and are then solved by our modified ne method we finally discuss the computational results of our implementation of the mlb method and its comparison with a version of lancelot 5 based on a collection extracted from cuter 12 of nonlinear programming problems with simple bound constraints

on kneser solutions of nonlinear third order differential equations sufficient conditions are given under which the third order differential equation with the middle term y q t y r t f y 0 q positive has a kneser solution y the derivatives of which change their signs

oscillation of third order trinomial delay differential equations the purpose of this paper is to study oscillation and asymptotic behavior of solutions to a third order linear delay differential equation y t p t y t q t y tau t 0 new comparison th

newton s method with a model trust region modification a modified newton method for unconstrained minimization is presented and analyzed the modification is based upon the model trust region approach this report contains a thorough analysis of the locally constrained quadratic minimizations that arise as subproblems in the modified newton iteration several promising alternatives are presented for solving these subproblems in ways that overcome certain theoretical difficulties exposed by this analysis very strong convergence results are presented concerning the minimization algorithm in particular the explicit use of second order information is justified by demonstrating that the iterates converge to a point which satisfies the second order necessary conditions for minimization with the exception of very pathological cases this occurs whenever the algorithm is applied to problems with continuous second partial derivatives

properties of the third order trinomial differential equations with delay argument abstract in this paper we study properties of the third order trinomial delay differential equation y t p t y t g t y t 0 by transforming this equation to binomial second third order differential equation employing suitable comparison theorems we establish new results on asymptotic behavior of solutions of the studied equation

a class of indefinite dogleg path methods for unconstrained minimization in this paper we propose a convenient curvilinear search method to solve trust region problems arising from unconstrained optimization problems the curvilinear paths we set forth are dogleg paths generated mainly by employing bunch parlett factorization for general symmetric matrices that may be indefinite this method is easy to implement and globally convergent it is proved that the method satisfies the first and second order stationary point convergence properties and that the convergence rate is quadratic under commonly used conditions on functions numerical experiments are conducted to compare this method with some existing methods

a unified approach to the implementation of several restart fixed point algorithms and a new variable dimension algorithm we present a unified description of a class of restart fixed point algorithms including merrill s method and several variable dimension algorithms for their implementation on computers based on this description we show that some pivot saving techniques originally developed for the homotopy methods can be applied to the class we also propose a new variable dimension algorithm having 3 n 1 rays along which we can move toward a solution some numerical comparisons of the simplicial restart algorithms merrill s method the 2n method the octahedral method and the new one support that the latter two methods are more efficient than the others

oscillation criteria of a certain class of third order nonlinear delay differential equations with damping abstract in this paper we are concerned with the oscillation of third order nonlinear delay differential equations of the form r 2 t r 1 t y p t y q t f y g t 0 by using a generalized riccati transformation and integral averaging technique we establish some new sufficient conditions which insure that any solution of this equation oscillates or converges to zero in particular several examples are given to illustrate the importance of our results

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

an efficient implementation of merrill s method for sparse or partially separable systems of nonlinear equations when merrill s method without an extra dimension is applied to solving sparse or partially separable systems of nonlinear equations the computational efficiency can be further improved i e a larger piece of linearity can be traversed in one step by using a suitable linear system one of the linear systems is updated and the corresponding technique is shown to update information about the linear system and the large piece in the implementation of the method some numerical results of the method support the claim that it is efficient also some mistakes from a previous paper in which the main technique exploiting sparsity was proposed are corrected

on third order differential equations with property a and b we present some new results concerning relationships between oscillation and property a oscillation and property b the obtained results have necessary and sufficient character they are applied to the nonlinear third order differential equations too

on functional inequalities and their applications in the oscillation theory in the paper we present several functional inequalities and offer their application for higher order advanced differential equations of the formx n t q t x s t 0 to be oscillatory the conditions obtained essentially improve complements and extends many other known results and involves unimprovable constants

partial update newton methods for unary factorable and partially separable optimization a modified newton method for solving unary optimization problems that is based upon only partially updating an approximation to the hessian matrix at each iteration is developed using rank one updates two partial updating criteria are presented the first enables the method to retain the quadratic convergence property of the classical newton method while the second enables it to achieve the superlinear convergence property of quasi ewton methods globally convergent modifications of the partial update newton method are also given finally the methods and proofs of their convergence are extended to partially separable and factorable optimization problems

integral criteria for oscillation of third order nonlinear differential equations abstract in this paper we are concerned with the oscillation of third order nonlinear differential equations of the form r 2 t r 1 t y p t y q t f y 0 t t 0 where t 0 0 is a fixed real number r 1 r 2 p q i t 0 r are nonnegative continuous functions with r 1 0 r 2 0 f r r is a continuous function by making use of a generalized riccati transformation we establish some new sufficient integral conditions under which the equation has at least one oscillatory solution examples are given to illustrate the importance of our results