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In this paper, we study the simplest quadratic matrix equation: Q(X)=X2+BX+C=0$$ \mathcal{Q}(X)={X}^2+ BX+C=0 $$. We transform this equation into an equivalent fixed‐point equation, and based on it, we construct the Krasnoselskij method. From this transformation, we can obtain iterative schemes more accurate than the successive approximation method. Moreover, under suitable conditions, we establish different results for the existence and localization of a solution for this equation with the Krasnoselskij method. Finally, we see numerically that the predictor–corrector iterative scheme, with the Krasnoselskij method as a predictor and the Newton method as corrector method, can improve the numerical application of the Newton method when approximating a solution of the quadratic matrix equation. [ABSTRACT FROM AUTHOR]

The local convergence analysis of multi-step, high-order Jarratt-like schemes is extended for solving Banach space valued systems of equations using the derivative instead of up to the ninth derivative as in previous works. Our idea expands the usage of the scheme in cases not considered earlier and can also be utilized in other schemes, too. Experiments test the theoretical results. [ABSTRACT FROM AUTHOR]

In this work, we present probabilistic local convergence results for a stochastic semismooth Newton method for a class of stochastic composite optimization problems involving the sum of smooth nonconvex and nonsmooth convex terms in the objective function. We assume that the gradient and Hessian information of the smooth part of the objective function can only be approximated and accessed via calling stochastic first- and second-order oracles. The approach combines stochastic semismooth Newton steps, stochastic proximal gradient steps and a globalization strategy based on growth conditions. We present tail bounds and matrix concentration inequalities for the stochastic oracles that can be utilized to control the approximation errors via appropriately adjusting or increasing the sampling rates. Under standard local assumptions, we prove that the proposed algorithm locally turns into a pure stochastic semismooth Newton method and converges r-linearly or r-superlinearly with high probability. [ABSTRACT FROM AUTHOR]

The applicability of an efficient sixth convergence order scheme is extended for solving Banach space valued equations. In previous works, the seventh derivative has been used not appearing on the scheme. But we use only the first derivative that appears on the scheme. Moreover, bounds on the error distances and results on the uniqueness of the solution are provided (not given in earlier works) based on ω–continuity conditions. Numerical examples complete this article.

• A novel asymptotic optimization strategy is proposed to adjust the operational conditions iteratively based on the Bayesian optimization method. • The specified quality target is incorporated into the GP framework and EI acquisition function to improve the convergence property. • The improved EI is further combined with BOBYQA to accelerate the convergence, and an efficient warm start rule is developed when switching from BO to BOBYQA. • The comparison results in case studies show that the proposed quality control method can efficiently achieve the specified quality target. How to take as few experiments as possible to achieve the desired quality target is a challenging and essential topic of process quality control, especially for complex and expensive manufacturing processes with high-dimensional quality outputs. This paper proposes a Bayesian optimization (BO)-based strategy utilizing quality information with adaptive local convergence. First, a series of pretreatment setting rules of the Gaussian process model is proposed to reduce the uncertainty of BO. Then, the quality target is adopted into the Gaussian process model, and the acquisition function is improved. Finally, to balance global exploration and local exploitation and reduce the experimental cost, an adaptive criterion is designed to switch global BO search to local search. The operating conditions can be efficiently updated until the quality target is reached. Two applications with high-dimensional quality outputs are presented to demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

We study the local convergence of classical quasi-Newton methods for nonlinear optimization. Although it was well established a long time ago that asymptotically these methods converge superlinearly, the corresponding rates of convergence still remain unknown. In this paper, we address this problem. We obtain first explicit non-asymptotic rates of superlinear convergence for the standard quasi-Newton methods, which are based on the updating formulas from the convex Broyden class. In particular, for the well-known DFP and BFGS methods, we obtain the rates of the form (n L 2 μ 2 k) k / 2 and (nL μ k) k / 2 respectively, where k is the iteration counter, n is the dimension of the problem, μ is the strong convexity parameter, and L is the Lipschitz constant of the gradient. [ABSTRACT FROM AUTHOR]

The aim of this study is to analyze the local convergence of the multi-step Homeier's-like method for solving nonlinear equations in Banach space. Furthermore, we avoid hypotheses on high order derivatives which limit the applicability of the method. Instead, we only use hypotheses on the first derivative. Thus the applicability of the method has been extended by preserving the order of convergence. The convergence of the solution is proved under the weak hypotheses i.e. ω-continuity condition. Some numerical instances where earlier results cannot be applied to solve equations but our results can be applied are provided to validate the theoretical contribution. [ABSTRACT FROM AUTHOR]

We investigate the local convergence of the Gauss-Newton-Kurchatov method for solving nonlinear least squares problems. This method is a combination of Gauss-Newton and Kurchatov methods and it is used for problems with the decomposition of the operator. The convergence analysis of the method is performed under the generalized Lipshitz conditions. The conditions of convergence, radius and the convergence order of the considered method are established. Given numerical examples confirm the theoretical results.

A novel local and semi-local convergence theorem for the four-step nonlinear scheme is presented. Earlier studies on local convergence were conducted without particular assumption on Lipschitz constant. In first part, the main local convergence theorems with a weak ϰ-average (assuming it as a positively integrable function and dropping the essential property of ND) are obtained. In comparison to previous research, in another part, we employ majorizing sequences that are more accurate in their precision along with the certain form of ϰ average Lipschitz criteria. A finer local and semi-local convergence criteria, boosting its utility, by relaxing the assumptions is derived. Applications in engineering to a variety of specific cases, such as object motion governed by a system of differential equations, are illustrated.

We suggest the local analysis of a class of iterative schemes based on decomposition technique for solving Banach space valued nonlinear models. Earlier results used hypotheses up to the fourth derivative to establish convergence. But we only apply the first derivative in our convergence theorem. We also provide computable radius of convergence ball, error estimates and uniqueness of the solution results not studied in earlier works. Hence, we enhance the applicability of these schemes. Furthermore, we explore, using basin of attraction tool, the dynamics of the schemes when they are applied on various complex polynomials. This article is concluded with numerical experiments. [ABSTRACT FROM AUTHOR]

Stephenson (2018) established annealed local convergence of Boltzmann planar maps conditioned to be large. The present work uses results on rerooted multi-type branching trees to prove a quenched version of this limit. [ABSTRACT FROM AUTHOR]

We prove local convergence results for rerooted conditioned multi-type Galton–Watson trees. The limit objects are multitype variants of the random sin-tree constructed by Aldous (1991), and differ according to which types recur infinitely often along the backwards growing spine. [ABSTRACT FROM AUTHOR]

In this paper, we study local convergence of high-order Tensor Methods for solving convex optimization problems with composite objective. We justify local superlinear convergence under the assumption of uniform convexity of the smooth component, having Lipschitz-continuous high-order derivative. The convergence both in function value and in the norm of minimal subgradient is established. Global complexity bounds for the Composite Tensor Method in convex and uniformly convex cases are also discussed. Lastly, we show how local convergence of the methods can be globalized using the inexact proximal iterations. [ABSTRACT FROM AUTHOR]

In this paper, we consider N, a simply connected two-step nilpotent Lie group with N, its corresponding (two-step nilpotent) Lie algebra, and we study Newton's method for solving the equation f (x) = 0, where f: N !N is a mapping. Under certain generalized Lipschitz condition, we obtain the convergence radius of Newton's method and the estimation of the uniqueness ball of the zero point of f. Some applications to special cases including Kantorovich's condition and γ-condition are provided. The determination of an approximate zero point of an analytic mapping is also presented. [ABSTRACT FROM AUTHOR]

The main contribution of this study is that the applicability and convergence domain of a fifth-order convergent equation solver is extended. We use ω condition on the first Fréchet derivative to study the local analysis, and this expands the applicability of the formula for such problems where the earlier study based on Lipschitz constants cannot be used. Also, we avoid the use of the extra assumption on boundedness of the first derivative of the nonlinear operator. Our idea can be used on other iterative methods. Numerical tests confirmed that the proposed analysis produces a larger convergence domain, in comparison to the earlier study, without using additional conditions. [ABSTRACT FROM AUTHOR]

From the secant method and using symmetric divided differences, we construct a uniparametric family of secant-type iterative methods with quadratic convergence and better computational efficiency than the secant method. In addition, we experimentally analyze the accessibility of the family from a dynamic point of view and study the local convergence. [ABSTRACT FROM AUTHOR]

In this paper, a new version of a secant-type method for solving constrained mixed generalized equations is addressed. The method is a combination of the secant method applied to generalized equations with the conditional gradient method. We use the contraction mapping principle to establish the convergence results. Moreover, by assuming the Lipschitz condition on the gradient and the metric regularity property, we show that the sequence generated by the proposed algorithm is well-defined and locally convergent for a solution with linear or superlinear rate. [ABSTRACT FROM AUTHOR]

In this paper, we study the nonlinear matrix equation (NME) X+∑i=1mAi*X−1Ai=Q. We transform this equation into an equivalent zero-point equation, then we use Newton’s iteration method to solve the equivalent equation. Under some mild conditions, we obtain the domain of approximation solutions and prove that the sequence of approximation solutions generated by Newton’s iteration method converges to the unique solution of this equation. In addition, the error estimation of the approximation solution is given. Finally, the comparison of two well-known approaches with Newton’s iteration method by some numerical examples demonstrates the superiority of Newton’s iteration method in the convergence speed.

A local and semi-local convergence is developed of a class of iterative methods without derivatives for solving nonlinear Banach space valued operator equations under the classical Lipschitz conditions for first-order divided differences. Special cases of this method are well-known iterative algorithms, in particular, the Secant, Kurchatov, and Steffensen methods as well as the Newton method. For the semi-local convergence analysis, we use a technique of recurrent functions and majorizing scalar sequences. First, the convergence of the scalar sequence is proved and its limit is determined. It is then shown that the sequence obtained by the proposed method is bounded by this scalar sequence. In the local convergence analysis, a computable radius of convergence is determined. Finally, the results of the numerical experiments are given that confirm obtained theoretical estimates.

In this paper, we provide a detailed local convergence analysis of a one-parameter family of iteration methods for the simultaneous approximation of polynomial zeros due to Ivanov (Numer. Algor. 75(4): 1193–1204, 2017). Thus, we obtain two local convergence theorems that provide sufficient conditions to guarantee the Q-cubic convergence of all members of the family. Among the other contributions, our results unify the latest such kind of results of the well known Dochev–Byrnev and Ehrlich methods. Several practical applications are further given to emphasize the advantages of the studied family of methods and to show the applicability of the theoretical results.

In this work, a time-fractional biharmonic equation with a Caputo derivative of fractional order α ∈ (0 , 1) is considered, whose solutions exhibit a weak singularity at initial time t = 0. For this problem, a system of two second-order differential equations is derived by introducing a intermediate variable p = − Δ u , then discretised the system using the standard finite element method in space together with the L1 discretisation of Caputo derivative on graded mesh in time. The H 1 -norm stability result of the method is established, and then a sharp H 1 -norm local convergent result is presented. Finally, numerical experiments are provided to further verify our theoretical analysis for each fixed value of α. [ABSTRACT FROM AUTHOR]

In this paper, a class of efficient iterative methods with increasing order of convergence for solving systems of nonlinear equations is developed and analyzed. The methodology uses well-known third-order Potra–Pták iteration in the first step and Newton-like iterations in the subsequent steps. Novelty of the methods is the increase in convergence order by an amount three per step at the cost of only one additional function evaluation. In addition, the algorithm uses a single inverse operator in each iteration, which makes it computationally more efficient and attractive. Local convergence is studied in the more general setting of a Banach space under suitable assumptions. Theoretical results of convergence and computational efficiency are verified through numerical experimentation. Comparison of numerical results indicates that the developed algorithms outperform the other similar algorithms available in the literature, particularly when applied to solve the large systems of equations. The basins of attraction of some of the existing methods along with the proposed method are given to exhibit their performance. [ABSTRACT FROM AUTHOR]

Optimization is important in machine learning problems, and quasi-Newton methods have a reputation as the most efficient numerical methods for smooth unconstrained optimization. In this paper, we study the explicit superlinear convergence rates of quasi-Newton methods and address two open problems mentioned by Rodomanov and Nesterov (2021b). First, we extend Rodomanov and Nesterov (2021b)'s results to random quasi-Newton methods, which include common DFP, BFGS, SR1 methods. Such random methods employ a random direction for updating the approximate Hessian matrix in each iteration. Second, we focus on the specific quasi-Newton methods: SR1 and BFGS methods. We provide improved versions of greedy and random methods with provable better explicit (local) superlinear convergence rates. Our analysis is closely related to the approximation of a given Hessian matrix, unconstrained quadratic objective, as well as the general strongly convex, smooth, and strongly self-concordant functions. [ABSTRACT FROM AUTHOR]

We extend the applicability of a fourth-order convergent nonlinear system solver by providing its local convergence analysis under Lipschitz continuous Fréchet derivative in Banach spaces. Our analysis only uses the first-order Fréchet derivative to ensure the convergence and provides the uniqueness of the solution, the radius of convergence ball and the computable error bounds. This study is applicable in solving such problems for which earlier studies are not effective. Furthermore, the convergence region for the scheme to approximate the zeros of various polynomials is studied using basins of attraction tool. Various computational tests are conducted to validate that our analysis is beneficial when prior studies fail to solve problems. [ABSTRACT FROM AUTHOR]

Different set of criteria based on the seventh derivative are used for convergence of sixth order methods. Then, these methods are compared using numerical examples. But we do not know: if the results of those comparisons are true if the examples change; the largest radii of convergence; error estimates on distance between the iterate and solution, and uniqueness results that are computable. We address these concerns using only the first derivative and a common set of criteria. Numerical experiments are used to test the convergence criteria and further validate the theoretical results. Our technique can be used to make comparisons between other methods of the same order.

We present two theoretical results and two surprising conjectures concerning convergence properties of Broyden's method for smooth nonlinear systems of equations. First, we show that when Broyden's method is applied to a nonlinear mapping F : R n → R n with n - d affine component functions and the initial matrix B 0 is chosen suitably, then the generated sequence (u k , F (u k) , B k) k ≥ 1 can be identified with a lower-dimensional sequence that is also generated by Broyden's method. This property enables us to prove, second, that for such mixed linear–nonlinear systems of equations a proper choice of B 0 ensures 2d-step q-quadratic convergence, which improves upon the previously known 2n steps. Numerical experiments of high precision confirm the faster convergence and show that it is not available if B 0 deviates from the correct choice. In addition, the experiments suggest two surprising possibilities: It seems that Broyden's method is (2 d - 1) -step q-quadratically convergent for d > 1 and that it admits a q-order of convergence of 2 1 / (2 d) . These conjectures are new even for d = n . [ABSTRACT FROM AUTHOR]

We present the local convergence of a Newton-type solver for equations involving Banach space valued operators. The eighth order of convergence was shown earlier in the special case of the k--dimensional Euclidean space, using hypotheses up to the eighth derivative although these derivatives do not appear in the method. We show convergence using only the first derivative. This way we extend the applicability of the methods. Numerical examples are used to show the convergence conditions. Finally, the basins of attraction of the method, on some test problems are presented. [ABSTRACT FROM AUTHOR]

We present the semi-local convergence analysis of a Potra-type method to solve equations involving Banach space valued operators. The analysis is based on our ideas of recurrent functions and restricted convergence region. The study is completed using numerical examples.

Abstract We extend the applicability of a cubically convergent nonlinear system solver using Lipschitz continuous first-order Fréchet derivative in Banach spaces. This analysis avoids the usual application of Taylor expansion in convergence analysis and extends the applicability of the scheme by applying the technique based on the first-order derivative only. Also, our study provides the radius of convergence ball and computable error bounds along with the uniqueness of the solution. Furthermore, the generalization of this analysis using Hölder condition is provided. Various numerical tests confirm that our analysis produces better results and it is useful in solving such problems where previous methods can not be implemented.

In this paper, we have studied local convergence of Super-Halley method in Banach spaces under the assumption of second order majorant conditions. This approach allows us to obtain generalization of earlier convergence analysis under majorizing sequences. Two important special cases of the convergence analysis based on the premises of Kantorovich and Smale type conditions have also been concluded. To show efficacy of our approach we have given three numerical examples.

We present a local convergence analysis Potra-Ptak-type method with optimal fourth order of convergence in order to approximate a solution of a nonlinear equation. In earlier studies such as [1], [5]-[28] hypotheses up to the fourth derivative are used. In this paper we use hypotheses up to the first derivative only, so that 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.

The local convergence analysis of the multi-step seventh order method to solve nonlinear equations is presented in this paper. The point of this paper is that our proposed study requires a weak hypothesis where the Fréchet derivative of the nonlinear operator satisfies the ψ-continuity condition, which thereby extends the applicability of the method when both Lipschitz and Hölder conditions fail. The convergence in this study is considered under the hypotheses on the first-order derivative without involving derivatives of the higher-order. To find a subset of the original convergence domain, a strategy is devised here. As a result, the new Lipschitz constants are at least as tight as the old ones, allowing for a more precise convergence analysis in the local convergence case. Some concrete numerical examples showing the performance of the method over some existing schemes are presented in this article.

The local convergence analysis of multi-step, high-order Jarratt-like schemes is extended for solving Banach space valued systems of equations using the derivative instead of up to the ninth derivative as in previous works. Our idea expands the usage of the scheme in cases not considered earlier and can also be utilized in other schemes, too. Experiments test the theoretical results.

We provide the local and semi-local convergence analysis of the most celebrated cubically convergent Traub's iterative method (TM) for obtaining solutions of Banach space valued nonlinear operator equations. The significance of our work is that the convergence study only needs the continuity condition on the first-order Fréchet derivative and avoids the use of higher order derivatives, which do not occur in this scheme. Also, the proposed local analysis extends the domain of convergence and applicability of this scheme. Using basins of attraction technique the complex dynamics of the scheme are also explored when it is applied on various complex polynomials. Finally, convergence radii for benchmark numerical problems are computed applying our analytical results. From these numerical tests, it is confirmed that the proposed analysis provides a larger convergence domain in comparison with the earlier work. [ABSTRACT FROM AUTHOR]

In earlier studies different methods of same convergence order are campared using numerical examples. The drawback of this approach is that, we do not know: if the results of those comparisons are true if the examples change; the largest radii of convergence; error estimates on distance between the iterate and solution, and uniqueness results that are computable. In this paper we campare the ball convergence of two-step iterative methods for solving the equation G (x) = 0 using only the first derivative and a common set of criteria. Numerical experiments are used to test the convergence criteria and further validate the theoretical results. Our technique can be used to make comparisons between other methods of the same order. [ABSTRACT FROM AUTHOR]

Nonconvex distributed constrained optimization over multi-agent systems plays an increasingly important role in control community. However, the duality gap for a nonconvex distributed constrained optimization problem is not always zero. This brings some difficulties for the algorithm design and convergence analysis about such problems. In this paper, to eliminate the duality gap, the original distributed optimization problem is converted to an equivalent distributed one by using a partial power reformulation. Next, for solving this equivalence problem, a continuous-time algorithm is proposed over the multi-agent system where each agent exchanges local information with other agents under an undirected communication graph. It is proved that the proposed continuous-time algorithm locally converges to a strict local optimal solution of the considered original nonconvex distributed optimization problem. Numerical examples show the validness of the presented algorithm. Finally, the algorithm is applied for optimal placement. [ABSTRACT FROM AUTHOR]

A numerical algorithm for the solution of an inverse coefficient problem for nonstationary, nonlinear production-destruction type model is proposed and tested on an example of the Lorenz'63 system. With an ensemble of adjoint problem solutions, the inverse problem is transformed into a quasi-linear matrix problem and solved with Newton-type algorithm. Two different ways of the adjoint ensemble construction are compared. In the first one, a trigonometric basis is used. In the second one in situ measurements are taken into account. Local convergence properties of the algorithm are studied numerically to find out when the use of more data can lead to the degradation of the reconstruction results. [ABSTRACT FROM AUTHOR]

We study the local convergence analysis of two higher-order methods using Hölder continuity condition on the first Frechet derivative to solve nonlinear equations in Banach spaces. Hölder continuous first derivative is used to extend the applicability of the method on such problems for which Lipschitz condition fails. Also, this convergence analysis generalizes the local convergence analysis based on Lipschitz continuity condition. Our analysis provides the radius of convergence ball and error bounds along with the uniqueness of the solution. Numerical examples like Hammerstein integral equation and a system of nonlinear equations are solved to verify our theoretical results. [ABSTRACT FROM AUTHOR]

We are motivated by a Kung-Traub-type method for solving equations on the real line. In particular, we extend this method for Banach space valued operators. The radius of convergence is also obtained as well as error bounds on the distances involved and a uniqueness result. Our convergence analysis avoids Taylor expansions and the computation of higher order than one derivatives. [ABSTRACT FROM AUTHOR]