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Consider the Erdős-Renyi random graph on n vertices where each edge is present independently with probability λ/n, with λ >0 fixed. For large n, a typical random graph locally behaves like a Galton-Watson tree with Poisson offspring distribution with mean λ. Here, we study large deviations from this typical behavior within the framework of the local weak convergence of finite graph sequences. The associated rate function is expressed in terms of an entropy functional on unimodular measures and takes finite values only at measures supported on trees. We also establish large deviations for other commonly studied random graph ensembles such as the uniform random graph with given number of edges growing linearly with the number of vertices, or the uniform random graph with given degree sequence. To prove our results, we introduce a new configuration model which allows one to sample uniform random graphs with a given neighborhood distribution, provided the latter is supported on trees. We also introduce a new class of unimodular random trees, which generalizes the usual Galton Watson tree with given degree distribution to the case of neighborhoods of arbitrary finite depth. These generalized Galton Watson trees turn out to be useful in the analysis of unimodular random trees and may be considered to be of interest in their own right. [ABSTRACT FROM AUTHOR]

We study the local convergence of an eighth order Newton-like method to approximate a locally-unique solution of a nonlinear equation. Earlier studies, such as Chen et al. (2015) show convergence under hypotheses on the seventh derivative or even higher, although only the first derivative and the divided difference appear in these methods. The convergence in this study is shown under hypotheses only on the first derivative. Hence, the applicability of the method is expanded. Finally, numerical examples are also provided to show that our results apply to solve equations in cases where earlier studies cannot apply. [ABSTRACT FROM AUTHOR]

We present a new semilocal convergence analysis for Secant method in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Our analysis includes the computation of the bounds on the limit points of the majorizing sequences involved. Under the same computational cost on the parameters involved our convergence criteria are weaker and the error bounds more precise than in earlier studies such as (Amat and Busquier, 2003; Amat et al., in press; Argyros and Hilout, 2012; Argyros et al., 2014; Argyros and Magreñán, 2014, 2015; Dennis, 1971; Ezquerro et al., 2000; Ortega and Rheinboldt, 1970; Potra and Pták, 1984; Schmidt, 1978). Numerical examples are also presented to illustrate the theoretical results obtained in this study. [ABSTRACT FROM AUTHOR]

We present a convergence ball comparison between three iterative methods for approximating a locally unique solution of a nonlinear equation in a Banach space setting. The convergence ball and error estimates are given for these methods under hypotheses only on the first Fréchet derivative in contrast to earlier studies such as Adomian (1994) [1], Babajee et al. (2008) [13], Cordero and Torregrosa (2007) [17], Cordero et al. [18], Darvishi and Barati (2007) [19], using hypotheses reaching up to the fourth Fréchet derivative although only the first derivative appears in these methods. This way we expand the applicability of these methods. Numerical examples are also presented in this study. [ABSTRACT FROM AUTHOR]

We present a unified convergence analysis of Inexact Newton-like methods in order to approximate a locally unique solution of a nonlinear operator equation containing a nondifferentiable term in a Banach space setting. The convergence conditions are more general and the error analysis more precise than in earlier studies such as (Argyros, 2007; Cătinaş, 2005; Cătinaş, 1994; Chen and Yamamoto, 1989; Dennis, 1968; Hernández and Romero, 2005; Potra and Pták, 1984; Rheinboldt, 1977). Special cases of our results can be used to find zeros of derivatives. Numerical examples are also provided in this study. [ABSTRACT FROM AUTHOR]

We present a local convergence analysis for general multi-point-Chebyshev–Halley-type methods (MMCHTM) of high convergence order in order to approximate a solution of an equation in a Banach space setting. MMCHTM includes earlier methods given by others as special cases. The convergence ball for a class of MMCHTM methods is obtained under weaker hypotheses than before. Numerical examples are also presented in this study. [ABSTRACT FROM AUTHOR]

In the analysis of the fluid queues, it is necessary to obtain the nonnegative solution of a nonsymmetric algebraic Riccati matrix equation. Under suitable conditions, this solution can be obtained transforming algebraic Riccati equations into unilateral quadratic matrix equations. In this paper, we use an efficient iterative scheme to approximate a solution of this quadratic matrix equation. We improve the efficiency and the accuracy of Newton's method, widely used in the literature. Moreover, a local convergence result is proved. Finally, we apply this efficient method to approximate the solution of a particular noisy Wiener–Hopf problem and we compare it with Newton's method. Moreover, a predictor–corrector iterative scheme is constructed that improve the accessibility of the aforesaid method. [ABSTRACT FROM AUTHOR]

In this work we are going to use the Kurchatov–Schmidt–Schwetlick-like solver (KSSLS) and the Kurchatov-like solver (KLS) to locate a zero, denoted by x ∗ of operator F. We define F as F : D ⊆ B 1 ⟶ B 2 where B 1 and B 2 stand for Banach spaces, D ⊆ B 1 be a convex set and F be a differentiable mapping according to Fréchet. Under these conditions, for all n = 0 , 1 , 2 , ... and 0 ≤ i ≤ m − 1 using Taylor expansion, KSSLS and KLS, when B 1 = B 2 and high order derivatives and divided differences not appearing in these solvers, the results obtained are the restart of the utilization of these iterative solvers. Moreover, we show under the same set of conditions that the local convergence radii are the same, the uniqueness balls coincide but the error estimates on ‖ x n − x ∗ ‖ differ. It is worth noticing our results improve the corresponding ones (Grau-Sánchez et al., 2011; Kurchatov, 1971 and Shakno, 2009). Finally, we apply our theoretical results to some numerical examples in order to prove the improvement. [ABSTRACT FROM AUTHOR]

Sufficient conditions for a weak local minimizer in the classical calculus of variations can be expressed without reference to conjugate points. The local quadratic convergence of Newton’s method follows from these sufficient conditions. Newton’s method is applied in the minimization form; that is, the step is generated by minimizing the local quadratic approximation. This allows the extension to a globally convergent line search based algorithm (which will be presented in a future paper). [ABSTRACT FROM PUBLISHER]

This paper studies adaptive regularized methods for nonlinear least-squares problems where the model of the objective function used at each iteration is either the Euclidean residual regularized by a quadratic term or the Gauss-Newton model regularized by a cubic term. For suitable choices of the regularization parameter the role of the regularization term is to provide global convergence. In this paper we investigate the impact of the regularization term on the local convergence rate of the methods and establish that, under the well-known error-bound condition, quadratic convergence to zero-residual solutions is enforced. This result extends the existing analysis on the local convergence properties of adaptive regularized methods. In fact, the known results were derived under the standard full-rank condition on the Jacobian at a zero-residual solution while the error-bound condition is weaker than the full-rank condition and allows the solution set to be locally nonunique. [ABSTRACT FROM AUTHOR]

We present a local convergence analysis for eighth-order variants of Newton's method in order to approximate a solution of a nonlinear equation. We use hypotheses up to the first derivative in contrast to earlier studies such as [7]-[11], [20] using hypotheses up to the seventh derivative. This way the applicability of these methods is extended under weaker hypotheses. Moreover the radius of convergence and computable error bounds on the distances involved are also given in this study. Numerical examples are also presented in this study. [ABSTRACT FROM AUTHOR]

In this paper we study local convergence of inexact Newton methods of the form (Æ'(xk) + Ak(xk+1 - xk) + F(xk+1) ∩ Rk(xk) ≠ à with Ak ∈ H(xk) for solving the generalized equation Æ'(x) + F(x) ∋ 0 in Banach spaces, where the function Æ' is continuous but not necessarily smooth and Æ' is a set-valued mapping with closed graph. The mapping H plays the role of a generalized set-valued derivative of f which in finite dimensions may be represented by Clarke's generalized Jacobian, while in Banach spaces it may be identified with Ioffe's strict prederivative. The set-valued mappings Rk represent inexactness. We utilize conditions divided into three groups: the first concerns the kind of nonsmoothness of the function f, the second involves metric regularity properties of an approximation of the mapping Æ'+F, and the third is about the sequence of mappings Rk. Under various combinations of these conditions we show linear, superlinear, or quadratic convergence of the method. In the second part of the paper we give two generalizations of the Dennis-Moré theorem. As corollaries, we obtain results regarding convergence of inexact semismooth quasi-Newton-type methods and Dennis-Moré theorems for semismooth equations. [ABSTRACT FROM AUTHOR]

A new fundamental solution semigroup for operator differential Riccati equations is developed. This fundamental solution semigroup is constructed via an auxiliary finite horizon optimal control problem whose value functional growth with respect to time horizon is determined by a particular solution of the operator differential Riccati equation of interest. By exploiting semiconvexity of this value functional, and the attendant max-plus linearity and semigroup properties of the associated dynamic programming evolution operator, a semigroup of max-plus integral operators is constructed in a dual space defined via the Legendre-Fenchel transform. It is demonstrated that this semigroup of max-plus integral operators can be used to propagate all solutions of the operator differential Riccati equation that are initialized from a specified class of initial conditions. As this semigroup of max-plus integral operators can be identified with a semigroup of quadratic kernels, an explicit recipe for the aforementioned solution propagation is also rendered possible. [ABSTRACT FROM AUTHOR]

Anderson(m) is a method for acceleration of fixed point iteration which stores m+1 prior evaluations of the fixed point map and computes the new iteration as a linear combination of those evaluations. Anderson(0) is fixed point iteration. In this paper we show that Anderson(m) is locally r-linearly convergent if the fixed point map is a contraction and the coefficients in the linear combination remain bounded. Without assumptions on the coefficients, we prove q-linear convergence of Anderson(1) and, in the case of linear problems, Anderson(m). We observe that the optimization problem for the coefficients can be formulated and solved in nonstandard ways and report on numerical experiments which illustrate the ideas. [ABSTRACT FROM AUTHOR]

A new predictor–corrector iterative procedure, that combines Newton’s method as predictor scheme and a fifth-order iterative method as a corrector, is designed for solving nonlinear equations in Banach spaces. We analyze the local order of convergence and the regions of accessibility of the new method comparing it with Newton’s method, both theoretical and numerically. [ABSTRACT FROM AUTHOR]

We present a local convergence analysis of a class of multi-point Super-Halley methods in order to approximate a solution of nonlinear equations in a Banach space setting. The semilocal convergence analysis of this method was studied by Wang et. al. [24], This work provides computable convergence ball and computable error bounds under weaker conditions than in [24], Numerical examples are also provided in this study. [ABSTRACT FROM AUTHOR]

In this work, we introduce a family of Least Change Secant Update Methods for solving Nonlinear Complementarity Problems based on its reformulation as a nonsmooth system using the one-parametric class of nonlinear complementarity functions introduced by Kanzow and Kleinmichel. We prove local and superlinear convergence for the algorithms. Some numerical experiments show a good performance of this algorithm. [ABSTRACT FROM AUTHOR]

We present a local convergence analysis for deformed Chebyshev methods in order to approximate a solution of a nonlinear equation in a Banach space setting. Our methods include the Chebyshev and other high-order methods under hypotheses up to the first Fréchet derivative in contrast to earlier studies using hypotheses up to the second or third Fréchet derivative. The convergence ball and error estimates are given for these methods. Numerical examples are also provided in this study. [ABSTRACT FROM AUTHOR]

The complex geometry of cancellous bone tissue makes it difficult to generate finite element (FE) models. Only a few studies investigated the convergence behavior at the tissue scale using Cartesian meshes. However, these studies were not conducted according to an ideal patch test and the postelastic convergence behavior was not reported. In this study, the third principal strain and stress, and the displacement obtained from human micro finite element (microFE) models of lower resolutions were compared against the model of 19.5 pm as a reference, representing the original spatial resolution of microCT data. Uniaxial compression simulations using both linear-elastic and nonlinear constitutive equations were performed. The results showed a decrease in percentage difference in all three values as the element size decreased, with the displacement converging the fastest among the three. Simulations carried out using a nonlinear constitutive equation however, failed to show convergence for the third principal strains and stresses. It was concluded that at the tissue level, Cartesian meshes of human cancellous bone tissue were able to reach a converged solution in all three parameters investigated for linear simulation and only in displacement for nonlinear simulation. These parameters can be used as references in the future for studies in local biomechanical behavior of human cancellous bone tissues with linear simulation. The convergence behavior for human cancellous bone tissue using nonlinear constitutive equations needs further investigation. [ABSTRACT FROM AUTHOR]

Abstract: We present sufficient convergence conditions for two-step Newton methods in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. The advantages of our approach over other studies such as Argyros et al. (2010) [5], Chen et al. (2010) [11], Ezquerro et al. (2000) [16], Ezquerro et al. (2009) [15], Hernández and Romero (2005) [18], Kantorovich and Akilov (1982) [19], Parida and Gupta (2007) [21], Potra (1982) [23], Proinov (2010) [25], Traub (1964) [26] for the semilocal convergence case are: weaker sufficient convergence conditions, more precise error bounds on the distances involved and at least as precise information on the location of the solution. In the local convergence case more precise error estimates are presented. These advantages are obtained under the same computational cost as in the earlier stated studies. Numerical examples involving Hammerstein nonlinear integral equations where the older convergence conditions are not satisfied but the new conditions are satisfied are also presented in this study for the semilocal convergence case. In the local case, numerical examples and a larger convergence ball are obtained. [Copyright &y& Elsevier]

Solving equations of the form H (x) = 0 is one of the most faced problem in mathematics and in other science fields such as chemistry or physics. This kind of equations cannot be solved without the use of iterative methods. The Steffensen-type methods, defined using divided differences are derivative free, are usually considered to solve these problems when H is a non-differentiable operator due to its accuracy and efficiency. However, in general, the accessibility of these iterative methods is small. The main interest of this paper is to improve the accessibility of Steffensen-type methods, this is the set of starting points that converge to the roots applying those methods. So, by means of using a predictor–corrector iterative process we can improve this accessibility. For this, we use a predictor iterative process, using symmetric divided differences, with good accessibility and then, as corrector method, we consider the Center-Steffensen method with quadratic convergence. In addition, the dynamical studies presented show, in an experimental way, that this iterative process also improves the region of accessibility of Steffensen-type methods. Moreover, we analyze the semilocal convergence of the predictor–corrector iterative process proposed in two cases: when H is differentiable and H is non-differentiable. Summing up, we present an effective alternative for Newton's method to non-differentiable operators, where this method cannot be applied. The theoretical results are illustrated with numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, we establish two local convergence theorems that provide initial conditions and error estimates to guarantee the Q-convergence of an extended version of Chebyshev–Halley family of iterative methods for multiple polynomial zeros due to Osada (J. Comput. Appl. Math. 2008, 216, 585–599). Our results unify and complement earlier local convergence results about Halley, Chebyshev and Super–Halley methods for multiple polynomial zeros. To the best of our knowledge, the results about the Osada's method for multiple polynomial zeros are the first of their kind in the literature. Moreover, our unified approach allows us to compare the convergence domains and error estimates of the mentioned famous methods and several new randomly generated methods. [ABSTRACT FROM AUTHOR]

We present a unified local and semilocal convergence analysis for secant-type methods in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Our analysis includes the computation of the bounds on the limit points of the majorizing sequences involved. Under the same computational cost, using both Lipschtiz and center Lipschitz conditions, our convergence criteria can be: weaker; the error bounds more precise and the convergence balls larger than in earlier studies. Special cases such us Newton's method or Secant method are also presented. Numerical examples, including a Chandrasekhar equation and a boundary value problem, are also presented to illustrate the theoretical results obtained in this study. [ABSTRACT FROM AUTHOR]

In this article, we propose a nonmonotone linesearch sequential quadratic programming method for general constrained optimization problems without a penalty function or a filter. The algorithm proposed here is a development of the algorithm in Xue et al. [17]. Compared with the former, the novelty of the method we propose is that the new algorithm will achieve the local convergence under weaker assumptions. In order to avoid the Maratos effect, we use the second-order correction in this method, which need not be computed at each iteration. In other words, after a certain number of iterations, there is no need to compute the second-order correction step any more. The global convergence and the locally superlinear convergence of our method are proved under some suitable conditions. [ABSTRACT FROM PUBLISHER]

We unify and extend some Newtonian iterative frameworks developed earlier in the literature, which results in a collection of convenient tools for local convergence analysis of various algorithms under various sets of assumptions including strong metric regularity, semistability, or upper-Lipschitz stability, the latter allowing for nonisolated solutions. These abstract schemes are further applied for deriving sharp local convergence results for some constrained optimization algorithms under reduced smoothness hypotheses. Specifically, we consider applications to the augmented Lagrangian method and to the linearly constrained Lagrangian method for problems with Lipschitzian derivatives but possibly without second derivatives, and our local convergence analysis for these methods improves all the existing theories of this kind. [ABSTRACT FROM AUTHOR]

Abstract: A local convergence analysis of the Gauss–Newton method for solving injective-overdetermined systems of nonlinear equations under a majorant condition is provided. The convergence as well as results on its rate are established without a convexity hypothesis on the derivative of the majorant function. The optimal convergence radius, the biggest range for uniqueness of the solution along with some other special cases are also obtained. [Copyright &y& Elsevier]

We study the local convergence analysis of a fifth order method and its multi-step version in Banach spaces. The hypotheses used are based on the first Fréchet-derivative only. The new approach provides a computable radius of convergence, error bounds on the distances involved, and estimates on the uniqueness of the solution. Such estimates are not provided in the approaches using Taylor expansions of higher order derivatives, which may not exist or may be very expensive or impossible to compute. Numerical examples are provided to validate the theoretical results. Convergence domains of the methods are also checked through complex geometry shown by drawing basins of attraction. The boundaries of the basins show fractal-like shapes through which the basins are symmetric. [ABSTRACT FROM AUTHOR]

Abstract: Following an idea similar to that given by Dennis and Schnabel (1996) in [2], we prove a local convergence result for Newton’s method under generalized conditions of Kantorovich type. [Copyright &y& Elsevier]

Abstract: We present a class of trust region algorithms that do not use any penalty function or a filter for nonlinear equality constrained optimization. In each iteration, the infeasibility is controlled by a progressively decreasing upper limit and trial steps are computed by a Byrd–Omojokun-type trust region strategy. Measures of optimality and infeasibility are computed, whose relationship serves as a criterion on which the algorithm decides which one to focus on improving. As a result, the algorithm keeps a balance between the improvements on optimality and feasibility even if no restoration phase which is required by filter methods is used. The framework of the algorithm ensures the global convergence without assuming regularity or boundedness on the iterative sequence. By using a second order correction strategy, Marato’s effect is avoided and fast local convergence is shown. The preliminary numerical results are reported. [Copyright &y& Elsevier]

We consider generalized Nash equilibrium problems (GNEP) from a structural and computational point of view. In GNEP the players' feasible sets may depend on the other players' strategies. Moreover, the players may share common constraints. In particular, the latter leads to the stable appearance of Nash equilibria which are Fritz-John (FJ) points, but not Karush-Kuhn-Tucker (KKT) points. Basic to our approach is the representation of FJ points as zeros of an appropriate underdetermined system of nonsmooth equations. Here, additional nonsmooth variables are taken into account. We prove that the set of FJ points (together with corresponding active Lagrange multipliers)--generically--constitutes a Lipschitz manifold. Its dimension is (N - 1) |J0|. where N is the number of players and |J0| is the number of active common constraints. In a structural analysis of Nash equilibria the number (AT -- 1) |J0| plays a crucial role. In fact, this number encodes both the possible degeneracies for the players' parametric subproblems and the dimension of the set of Nash equilibria. In particular, in the nondegenerate case, the dimension of the set of Nash equilibria locally equals (N - 1) |J0|. For the computation of FJ points we propose a nonsmooth projection method (NPM) which aims at finding solutions of an underdetermined system of nonsmooth equations. NPM is shown to be well defined for GNEP. Local convergence of NPM is conjectured for GNEP under generic assumptions and its proof is challenging. However, we indicate special cases (known from the literature) in which convergence holds. [ABSTRACT FROM AUTHOR]

Consider the following variational inclusion problem: where f is differentiable in a neighbourhood of a solution and g is differentiable at , and F is a set-valued mapping, and the method introduced in Jean-Alexis and Pietrus [C. Jean-Alexis and A. Pietrus, On the convergence of some methods for variational inclusions, Rev. R. Acad. Cien. serie A. Mat. 102(2) (2008), pp. 355–361] for solving this problem: where ∇f(x) denotes the Fréchet derivative of f at x and [x, y; g] the first-order divided difference of g on the points x and y. Local converge analysis are provided for the method under the weaker conditions than Jean-Alexis and Pietrus (2008). Moreover, if ∇f and the first-order divided difference of g are p-Hölder continuous at a solution, then we show that this method converges superlinearly. In particular, our results extend the corresponding ones Jean-Alexis and Pietrus (2008), and fix a gap in the proof in (Jean-Alexis and Pietrus (2008), Theorem 1). [ABSTRACT FROM AUTHOR]

A local convergence analysis of Newton's method for finding a singularity of a differentiable vector field defined on a complete Riemannian manifold, based on the majorant principle, is presented in this paper. This analysis provides a clear relationship between the majorant function, which relaxes the Lipschitz continuity of the derivative, and the vector field under consideration. It also allows us to obtain the optimal convergence radius and the biggest range for the uniqueness of the solution and to unify some previously unrelated results. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, the multistability of a class of Amari’s -divergence based nonnegative matrix factorization learning algorithms is analyzed. The analysis results show that invariant sets for the update algorithms can be constructed. In these invariant sets, the non-convergence of the discussed algorithms can be guaranteed. Based on Lyapunov’s stability theorem, the local convergence of this class of learning algorithms is proved in the domain of their update rules. In the simulation, the analysis results are applied to image representation. Experiment results demonstrate that selecting suitable initial data for different applications of these nonnegative matrix factorization algorithms is very important. [Copyright &y& Elsevier]

SUMMARY In this paper we consider the Pad'e family of iterations for computing the matrix sign function and the Padé family of iterations for computing the matrix p-sector function. We prove that all the iterations of the Padé family for the matrix sign function have a common convergence region. It completes a similar result of Kenney and Laub for half of the Padé family. We show that the iterations of the Padé family for the matrix p-sector function are well defined in an analogous common region, depending on p. For this purpose we proved that the Padé approximants to the function (1− z)−σ, 0<σ<1, are a quotient of hypergeometric functions whose poles we have localized. Furthermore we proved that the coefficients of the power expansion of a certain analytic function form a positive sequence and in a special case this sequence has the log-concavity property. Copyright © 2011 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We use Lipschitz and center-Lipschitz conditions to provide an improved local convergence analysis for a certain class of iterative methods with cubic order of convergence. It turns out that under the same computational cost as before, we obtain a larger radius of convergence and tighter error bounds. Numerical examples are also provided in this study. [ABSTRACT FROM AUTHOR]

Abstract: A local convergence analysis for an efficient Secant-type method (STM) for solving nonlinear equations is given using both the Lipschitz continuous and center-Lipshchitz continuous divided differences of order one. An estimate of the radius of the convergence ball of (STM) is provided, the error estimate matching its convergence order is established. Numerical examples validating the theoretical results are also provided in the concluding section of this study. [Copyright &y& Elsevier]

Abstract: In this paper, we present a local convergence analysis of inexact Gauss–Newton like methods for solving nonlinear least squares problems. Under the hypothesis that the derivative of the function associated with the least squares problem satisfies a majorant condition, we obtain that the method is well-defined and converges. Our analysis provides a clear relationship between the majorant function and the function associated with the least squares problem. It also allows us to obtain an estimate of convergence ball for inexact Gauss–Newton like methods and some important, special cases. [Copyright &y& Elsevier]

Abstract: It is proved that among the rational iterations locally converging with order to the sign function, the Padé iterations and their reciprocals are the unique with the lowest sum of the degrees of numerator and denominator. [Copyright &y& Elsevier]

Abstract: We provide a local convergence analysis for Newton’s method under a weak majorant condition in a Banach space setting. Our results provide under the same information a larger radius of convergence and tighter error estimates on the distances involved than before . Special cases and numerical examples are also provided in this study. [Copyright &y& Elsevier]

Abstract: Iterative methods, such as Newton’s, behave poorly when solving ill-conditioned problems: they become slow (first order), and decrease their accuracy. In this paper we analyze deeply and widely the convergence of a modified Newton method, which we call perturbed Newton, in order to overcome the usual disadvantages Newton’s one presents. The basic point of this method is the dependence of a parameter affording a degree of freedom that introduces regularization. Choices for that parameter are proposed. The theoretical analysis will be illustrated through examples. [Copyright &y& Elsevier]

Dual interior point methods for solving linear semidefinite programming problems are proposed. These methods are an extension of dual barrier-projection methods for linear programs. It is shown that the proposed methods converge locally at a linear rate provided that the solutions to the primal and dual problems are nondegenerate. [ABSTRACT FROM AUTHOR]

In this paper, the estimate of the radius of the convergence ball of the modified Halley's method for finding multiple zeros of nonlinear equations is provided under the hypotheses that the derivative f of function f is Hölder continuous, and f is bounded. The uniqueness ball of solution is also established. Finally, some examples are provided to show applications of our theorem. [ABSTRACT FROM AUTHOR]

This paper considers an outer approximation projection method for variational inequalities, in which the projections are not performed on the original set that appears in the variational inequality, but on a polyhedral convex set defined by the linearized constraints. It shows that the method converges linearly, when the starting point is sufficiently close to the solution and the step lengths are sufficiently small. [ABSTRACT FROM AUTHOR]

n Adaptive Regularisation algorithm using Cubics (ARC) is proposed for unconstrained optimization, generalizing at the same time an unpublished method due to Griewank (Technical Report NA/12, 1981, DAMTP, University of Cambridge), an algorithm by Nesterov and Polyak (Math Program 108(1):177-205, 2006) and a proposal by Weiser et al. (Optim Methods Softw 22(3):413-431, 2007). At each iteration of our approach, an approximate global minimizer of a local cubic regularisation of the objective function is determined, and this ensures a significant improvement in the objective so long as the Hessian of the objective is locally Lipschitz continuous. The new method uses an adaptive estimation of the local Lipschitz constant and approximations to the global model-minimizer which remain computationally-viable even for large-scale problems. We show that the excellent global and local convergence properties obtained by Nesterov and Polyak are retained, and sometimes extended to a wider class of problems, by our ARC approach. Preliminary numerical experiments with small-scale test problems from the CUTEr set show encouraging performance of the ARC algorithm when compared to a basic trust-region implementation. [ABSTRACT FROM AUTHOR]

The Particle Swarm Optimiser (PSO) is a population based stochastic optimisation algorithm, empirically shown to be efficient and robust. This paper provides a proof to show that the original PSO does not have guaranteed convergence to a local optimum. A flaw in the original PSO is identified which causes stagnation of the swarm. Correction of this flaw results in a PSO algorithm with guaranteed convergence to a local minimum. Further extensions with provable global convergence are also described. Experimental results are provided to elucidate the behavior of the modified PSO as well as PSO variations with global convergence. [ABSTRACT FROM AUTHOR]

Abstract: The Gauss–Newton method for solving nonlinear least squares problems is studied in this paper. Under the hypothesis that the derivative of the function associated with the least square problem satisfies a majorant condition, a local convergence analysis is presented. This analysis allows us to obtain the optimal convergence radius and the biggest range for the uniqueness of stationary point, and to unify two previous and unrelated results. [ABSTRACT FROM AUTHOR]

n interior point method (IPM) defines a search direction at each interior point of the feasible region. These search directions form a direction field, which in turn gives rise to a system of ordinary differential equations (ODEs). Thus, it is natural to define the underlying paths of the IPM as solutions of the system of ODEs. In Sim and Zhao (Math. Program. Ser. A 110:475-499, ), these off-central paths are shown to be well-defined analytic curves and any of their accumulation points is a solution to the given monotone semidefinite linear complementarity problem (SDLCP). In Sim and Zhao (Math. Program. Ser. A 110:475-499, ; J. Optim. Theory Appl. 137:11-25, ) and Sim (J. Optim. Theory Appl. 141:193-215, ), the asymptotic behavior of off-central paths corresponding to the HKM direction is studied. In particular, in Sim and Zhao (Math. Program. Ser. A 110:475-499, ), the authors study the asymptotic behavior of these paths for a simple example, while, in Sim and Zhao (J. Optim. Theory Appl. 137:11-25, ) and Sim (J. Optim. Theory Appl. 141:193-215, ), the asymptotic behavior of these paths for a general SDLCP is studied. In this paper, we study off-central paths corresponding to another well-known direction, the Nesterov-Todd (NT) direction. Again, we give necessary and sufficient conditions for these off-central paths to be analytic w.r.t. $\sqrt{\mu}$ and then w.r.t. μ, at solutions of a general SDLCP. Also, as in Sim and Zhao (Math. Program. Ser. A 110:475-499, ), we present off-central path examples using the same SDP, whose first derivatives are likely to be unbounded as they approach the solution of the SDP. We work under the assumption that the given SDLCP satisfies a strict complementarity condition. [ABSTRACT FROM AUTHOR]

Abstract: A local convergence analysis of Newton’s method for solving nonlinear equations, under a majorant condition, is presented in this paper. Without assuming convexity of the derivative of the majorant function, which relaxes the Lipschitz condition on the operator under consideration, convergence, the biggest range for uniqueness of the solution, the optimal convergence radius and results on the convergence rate are established. Besides, two special cases of the general theory are presented as applications. [ABSTRACT FROM AUTHOR]

Quasi-Newton methods for solving singular systems of nonlinear equations are considered in this paper. Singular roots cause a number of problems in implementation of iterative methods and in general deteriorate the rate of convergence. We propose two modifications of QN methods based on Newton's and Shamanski's method for singular problems. The proposed algorithms belong to the class of two-step iterations. Influence of iterative rule for matrix updates and the choice of parameters that keep iterative sequence within convergence region are empirically analyzed and some conclusions are obtained. [ABSTRACT FROM AUTHOR]

In Gonzaga et al. [A globally convergent filter method for nonlinear programming, SIAM J. Optimiz. 14 (2003), pp. 646-669] we discuss general conditions to ensure global convergence of inexact restoration filter algorithms for non-linear programming. In this article we show how to avoid the Maratos effect by means of a second-order correction. The algorithms are based on feasibility and optimality phases, which can be either independent or not. The optimality phase differs from the original one only when a full Newton step for the tangential minimization of the Lagrangian is efficient but not acceptable by the filter method. In this case a second-order corrector step tries to produce an acceptable point keeping the efficiency of the rejected step. The resulting point is tested by trust region criteria. Under the usual hypotheses, the algorithm inherits the quadratic convergence properties of the feasibility and optimality phases. This article includes a comparison between classical Sequential Quadratic Programming (SQP) and Inexact Restoration (IR) iterations, showing that both methods share the same asymptotic convergence properties. [ABSTRACT FROM AUTHOR]