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This paper studies the convergence of the proximal point method for quasiconvex functions in finite dimensional complete Riemannian manifolds. We prove initially that, in the general case, when the objective function is proper and lower semicontinuous, each accumulation point of the sequence generated by the method, if it exists, is a limiting critical point of the function. Then, under the assumptions that the sectional curvature of the manifold is bounded above by some non negative constant and the objective function is quasiconvex we analyze two cases. When the constant is zero, the global convergence of the algorithm to a limiting critical point is assured and if it is positive, we prove the local convergence for a class of quasiconvex functions, which includes Lipschitz functions. [ABSTRACT FROM AUTHOR]

In this article, we consider two parametric classes of derivative free iterative methods of high order of convergence in Banach spaces. We study local and semilocal convergence results in Banach spaces for both families of iterative processes. We carry out a numerical application to solve a nonlinear and nondifferentiable Fredholm integral equation in the Banach space of continuous functions with the maximum norm. [ABSTRACT FROM AUTHOR]

In this study, we introduce an iterative approach exhibiting sixth-order convergence for the solution of nonlinear equations. The method attains sixth-order convergence by using three evaluations of the function and two evaluations of the first-order derivative per iteration. We examined the theoretical convergence of our method through the convergence theorem, which substantiates the convergence order. Furthermore, we analyzed the local convergence of our proposed technique by employing a hypothesis that involves the first-order derivative of the function Θ alongside the Lipschitz conditions. To evaluate the performance and efficacy of our iterative method, we provide a comparative analysis against existing methods based on various standard numerical problems. Finally, graphical comparisons employing basins of attraction are presented to illustrate the dynamic behavior of the iterative method in the complex plane. [ABSTRACT FROM AUTHOR]

The recent advance of algorithms for nonlinear semidefinite optimization problems (NSDPs) is remarkable. Yamashita et al. first proposed a primal–dual interior point method (PDIPM) for solving NSDPs using the family of Monteiro–Zhang (MZ) search directions. Since then, various kinds of PDIPMs have been proposed for NSDPs, but, as far as we know, all of them are based on the MZ family. In this paper, we present a PDIPM equipped with the family of Monteiro–Tsuchiya (MT) directions, which were originally devised for solving linear semidefinite optimization problems as were the MZ family. We further prove local superlinear convergence to a Karush–Kuhn–Tucker point of the NSDP in the presence of certain general assumptions on scaling matrices, which are used in producing the MT search directions. Finally, we conduct numerical experiments to compare the efficiency among members of the MT family. [ABSTRACT FROM AUTHOR]

The aim of this paper is to extend the applicability of a two-step Gauss-Newton-Werner-type method (TGNWTM) for solving nonlinear least squares problems. The radius of convergence, error bounds and the information on the location of the solution are improved under the same information as in earlier studies. Numerical examples further validate the theoretical results.

Many applications from computational mathematics can be identified for a system of non-linear equations in more generalized Banach spaces. Analytical methods do not exist for solving these type of equations, and so we solve these equations using iterative methods. We introduced a new numerical technique for finding the roots of non-linear equations in Banach space. The method is tenth-order and it is an extension of the fifth-order method which is developed by Arroyo et.al. [1]. We provided a convergence analysis to demonstrate that the method exhibits tenth-order convergence. Also, we discussed the local convergence properties of the suggested method which depends on the fundamental supposition that the first-order Fréchet derivative of the involved function $ \Upsilon $ satisfies the Lipschitz conditions. This new approach is not only an extension of prior research, but also establishes a theoretical concept of the radius of convergence. We validated the efficacy of our method through various numerical examples. Our method is comparable with the methods of Tao Y et al. [2]. We also compared it with higher-order iterative methods, and we observed that it either performs similarly or better for the numerical examples. We also gave the basin of attraction to demonstrate the behaviour in the complex plane.

In this paper, we have constructed a family of three-step methods with sixth-order convergence and a novel approach to enhance the convergence order $ p $ of iterative methods for systems of nonlinear equations. Additionally, we propose a three-step scheme with convergence order $ p+3 $ (for $ p\geq3 $) and have extended it to a generalized $ (m+2) $-step scheme by merely incorporating one additional function evaluation, thus achieving convergence orders up to $ p+3m $, $ m\in\mathbb{N} $. We also provide a thorough local convergence analysis in Banach spaces, including the convergence radius and uniqueness results, under the assumption of a Lipschitz-continuous Fréchet derivative. Theoretical findings have been validated through numerical experiments. Lastly, the performance of these methods is showcased through the analysis of their basins of attraction and their application to systems of nonlinear equations.

We develop the local convergence of the six order Contraharmonic-mean Newton’s method (CHN) to solve Banach space valued equations. Our analysis approach is two fold: The first way uses Taylor’s series and derivatives of higher orders. The second one uses only the first derivatives. We examine the theoretical results by solving a boundary value problem also using the examples relating the proposed method with other’s methods such as Newton’s, Kou’s and Jarratt’s to show that the proposed method performs better. The conjugate maps for second-degree polynomial are verified. We also calculate the fixed points (extraneous). The article is completed with the study of basins of attraction, which support and further validate the theoretical and numerical results.

To ensure the privacy of power system users while maximizing the efficiency of decentralized computational resources, this paper presents a novel decentralized power flow algorithm. Its core focus lies in preserving bus privacy through the integration of neighboring partial derivative information. Initially, the algorithm utilizes partial derivative data from adjacent buses to formulate a bus-level voltage iterative function, rooted in the bus power balance equation. Each bus in the power network is treated as an individual computational unit managed by an agent. This process involves scanning the network’s buses, computing estimated bus voltage values using a designated iteration formula, and iterating until convergence is achieved. Subsequent utilization of the latest neighbor information accelerates this iterative process. To fortify the algorithm’s reliability, a strategy assessing local convergence, utilizing the Jacobian matrix at the solution’s convergence, is proposed. The simulation results across various IEEE test systems unequivocally validate the proposed algorithm’s efficacy in power flow computation, showcasing accelerated speed and robust anti-interference capabilities. Compared to the traditional decentralized power flow method at the bus level, this algorithm significantly optimizes the iterative process, slashing the number of iterations by a minimum of 1.36 times. Notably, this efficiency boost amplifies in larger system scales, boasting an impressive reduction of up to 9.21 times fewer iterations.

In this article, we construct an efficient family of simultaneous methods for finding all the distinct as well as multiple roots of polynomial equations. Convergence analysis proves that the order of convergence of newly constructed family of simultaneous methods is seventeen. Fractal-based simultaneous iterative algorithms are thoroughly examined. Using self-similar features, fractal-based simultaneous schemes can converge to solutions faster, saving computational time and resources necessary for solving nonlinear equations. Fractals analysis illustrates the newly developed method's global convergence behavior when compared to single root-finding procedures for solving fractional order polynomials that arise in complex engineering applications. Some real problems from various branches of engineering along with some higher degree polynomials are considered as test examples to show the global convergence property of simultaneous methods, performance and efficiency of the proposed family of methods. Further computational efficiencies, CPU time and residual graphs are also drawn to validate the efficiency, robustness of the newly introduced family of methods as compared to the existing methods in the literature. [ABSTRACT FROM AUTHOR]

In this paper, a Newton-type iterative scheme for solving nonlinear systems is designed. In the process of proving the convergence order, we use the higher derivatives of the function and show that the convergence order of this iterative method is six. In order to avoid the influence of the existence of higher derivatives on the proof of convergence, we mainly discuss the convergence of this iterative method under weak conditions. In Banach space, the local convergence of the iterative scheme is established by using the ω -continuity condition of the first-order Fréchet derivative, and the application range of the iterative method is extended. In addition, we also give the radius of a convergence sphere and the uniqueness of its solution. Finally, the superiority of the new iterative method is illustrated by drawing attractive basins and comparing them with the average iterative times of other same-order iterative methods. Additionally, we utilize this iterative method to solve both nonlinear systems and nonlinear matrix sign functions. The applicability of this study is demonstrated by solving practical chemical problems. [ABSTRACT FROM AUTHOR]

Many applications from computational mathematics can be identified for a system of non-linear equations in more generalized Banach spaces. Analytical methods do not exist for solving these type of equations, and so we solve these equations using iterative methods. We introduced a new numerical technique for finding the roots of non-linear equations in Banach space. The method is tenth-order and it is an extension of the fifth-order method which is developed by Arroyo et.al. [1]. We provided a convergence analysis to demonstrate that the method exhibits tenth-order convergence. Also, we discussed the local convergence properties of the suggested method which depends on the fundamental supposition that the first-order Fréchet derivative of the involved function Υ satisfies the Lipschitz conditions. This new approach is not only an extension of prior research, but also establishes a theoretical concept of the radius of convergence. We validated the efficacy of our method through various numerical examples. Our method is comparable with the methods of Tao Y et al. [2]. We also compared it with higher-order iterative methods, and we observed that it either performs similarly or better for the numerical examples. We also gave the basin of attraction to demonstrate the behaviour in the complex plane. [ABSTRACT FROM AUTHOR]

In this paper, we have constructed a family of three-step methods with sixth-order convergence and a novel approach to enhance the convergence order p of iterative methods for systems of nonlinear equations. Additionally, we propose a three-step scheme with convergence order p + 3 (for p ≥ 3 ) and have extended it to a generalized (m + 2) -step scheme by merely incorporating one additional function evaluation, thus achieving convergence orders up to p + 3 m , m ∈ N . We also provide a thorough local convergence analysis in Banach spaces, including the convergence radius and uniqueness results, under the assumption of a Lipschitz-continuous Fréchet derivative. Theoretical findings have been validated through numerical experiments. Lastly, the performance of these methods is showcased through the analysis of their basins of attraction and their application to systems of nonlinear equations. In this paper, we have constructed a family of three-step methods with sixth-order convergence and a novel approach to enhance the convergence order of iterative methods for systems of nonlinear equations. Additionally, we propose a three-step scheme with convergence order (for ) and have extended it to a generalized -step scheme by merely incorporating one additional function evaluation, thus achieving convergence orders up to , . We also provide a thorough local convergence analysis in Banach spaces, including the convergence radius and uniqueness results, under the assumption of a Lipschitz-continuous Fréchet derivative. Theoretical findings have been validated through numerical experiments. Lastly, the performance of these methods is showcased through the analysis of their basins of attraction and their application to systems of nonlinear equations. [ABSTRACT FROM AUTHOR]

This research paper is determining the impact of integrated energy management and optimization control for Novel power systems. Research objectives have been made based on integrating power system for reducing power consumption based on optimization parameters. Optimal control is mainly dealing with control law by finding out a given system and delivering certain criteria for achieving goals. There are mainly three parts for problem optimization purposes those are included decision, constraints and objective functions. Four different objectives has been made those are mainly discussed about the importance of the power systems regarding integrated management system. There are various factors that are affecting optimizations including the role of the multigrain recursion, global convergence, properties of the optimization model and local convergence. Along with this, the optimization control system has been played a crucial role for the power systems development purposes. Research has been based on the effective design of energy management for power systems. The primary application for the optimization techniques is based on the storage system, electromagnetic-based design and mapping design for the microwave structure. The rate of unsustainable energy management is increasing and sustainable energy management system is decreasing. The conclusion has been based on the significance of the optimization control for the power systems. [ABSTRACT FROM AUTHOR]

We develop the local convergence of the six order Contraharmonic-mean Newton's method (CHN) to solve Banach space valued equations. Our analysis approach is two fold: The first way uses Taylor's series and derivatives of higher orders. The second one uses only the first derivatives. We examine the theoretical results by solving a boundary value problem also using the examples relating the proposed method with other's methods such as Newton's, Kou's and Jarratt's to show that the proposed method performs better. The conjugate maps for second-degree polynomial are verified. We also calculate the fixed points (extraneous). The article is completed with the study of basins of attraction, which support and further validate the theoretical and numerical results. [ABSTRACT FROM AUTHOR]

In this paper, with the aim of approximate the ball convergence of nonlinear equation, we study the local properties of a class of sixth-order Jarratt-type iterative methods in Banach spaces. By utilizing the first-order Fréchet derivative, we establish the local convergence. At last, the nonlinear equations related to the gas-state equation, the continuous stirred tank reactor (CSTR) and the problem of azeotropic point of a binary solutions are solved using this iterative method, and the applicability of the theoretical results is proved. [ABSTRACT FROM AUTHOR]

In this study, we introduce an iterative approach exhibiting sixth-order convergence for the solution of nonlinear equations. The method attains sixth-order convergence by using three evaluations of the function and two evaluations of the first-order derivative per iteration. We examined the theoretical convergence of our method through the convergence theorem, which substantiates the convergence order. Furthermore, we analyzed the local convergence of our proposed technique by employing a hypothesis that involves the first-order derivative of the function Θ alongside the Lipschitz conditions. To evaluate the performance and efficacy of our iterative method, we provide a comparative analysis against existing methods based on various standard numerical problems. Finally, graphical comparisons employing basins of attraction are presented to illustrate the dynamic behavior of the iterative method in the complex plane.

In this article, we construct an efficient family of simultaneous methods for finding all the distinct as well as multiple roots of polynomial equations. Convergence analysis proves that the order of convergence of newly constructed family of simultaneous methods is seventeen. Fractal-based simultaneous iterative algorithms are thoroughly examined. Using self-similar features, fractal-based simultaneous schemes can converge to solutions faster, saving computational time and resources necessary for solving nonlinear equations. Fractals analysis illustrates the newly developed method’s global convergence behavior when compared to single root-finding procedures for solving fractional order polynomials that arise in complex engineering applications. Some real problems from various branches of engineering along with some higher degree polynomials are considered as test examples to show the global convergence property of simultaneous methods, performance and efficiency of the proposed family of methods. Further computational efficiencies, CPU time and residual graphs are also drawn to validate the efficiency, robustness of the newly introduced family of methods as compared to the existing methods in the literature.

In this paper, a Newton-type iterative scheme for solving nonlinear systems is designed. In the process of proving the convergence order, we use the higher derivatives of the function and show that the convergence order of this iterative method is six. In order to avoid the influence of the existence of higher derivatives on the proof of convergence, we mainly discuss the convergence of this iterative method under weak conditions. In Banach space, the local convergence of the iterative scheme is established by using the ω-continuity condition of the first-order Fréchet derivative, and the application range of the iterative method is extended. In addition, we also give the radius of a convergence sphere and the uniqueness of its solution. Finally, the superiority of the new iterative method is illustrated by drawing attractive basins and comparing them with the average iterative times of other same-order iterative methods. Additionally, we utilize this iterative method to solve both nonlinear systems and nonlinear matrix sign functions. The applicability of this study is demonstrated by solving practical chemical problems.

• 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]

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]

Firstly, we define an (r , r , ⋯ , r) -slice tensor, and then solve an (r , r , ⋯ , r) -slice tensor equation by direct method. Secondly, we propose a new splitting of the coefficient tensor to develop a slice tensor splitting method, which is different from the tensor splitting methods in [1,2]. At each iteration of this method, an (r , r , ⋯ , r) -slice tensor equation is solved. Finally, we prove the local convergence of the proposed method under appropriate conditions, and give numerical experiments to illustrate the effectiveness of this method. [ABSTRACT FROM AUTHOR]

Levenberg–Marquardt methods (LMMs) are the most typical algorithms for solving nonlinear equations F (x) = 0 , where F : R n → R m is a continuously differentiable function. They sequentially solve subproblems represented as squared residual of the Newton equations with the L 2 regularization to determine the search direction. However, since the subproblems of the LMMs are usually reduced to linear equations with n variables, it takes much time to solve them when m ≪ n. In this paper, we propose a new LMM which generalizes the L 2 regularization of the subproblems of the ordinary LMMs. By virtue of the generalization, we can choose a suitable regularization term for each given problem. Moreover, we show that a sequence generated by the proposed method converges globally and quadratically under some reasonable assumptions. Finally, we conduct numerical experiments to confirm that the proposed method performs better than the existing LMMs for some problems that satisfy m ≪ n. [ABSTRACT FROM AUTHOR]

Attitude motion parameters estimation of the space non-cooperative tumbling target is the premise of the on-orbit target capture task. However, existing methods cannot balance the estimation accuracy and efficiency. In this paper, an adaptive hybrid Kalman filter is presented to estimate the attitude, angular velocity and inertia motion parameters of the space non-cooperative tumbling target. First, the dynamic model of the non-cooperative tumbling target is derived, and the nonlinear discrete state-space equation is obtained. Then, a hybrid Kalman filter is designed by combining the extended Kalman filter and the unscented Kalman filter. The adaptive technique is introduced to track time-varying uncertain measurement noise caused by complex space disturbances. After that, based on the first-order linearization technique, the sufficient condition for the local convergence of the proposed method is proved. Finally, the numerical simulation experiment was conducted to compare the proposed method with the adaptive extended Kalman filter and the adaptive unscented Kalman filter. The experimental results show that the estimation accuracy of the proposed method is much higher than that of the adaptive extended Kalman filter and slightly lower than that of the adaptive unscented Kalman filter, and the iteration-average Central-Processing-Unit time is 64.2% smaller than that of the adaptive unscented Kalman filter and slightly longer than that of the adaptive extended Kalman filter. The proposed method achieves a much higher accuracy with a slight loss of iteration-average Central-Processing-Unit time. [ABSTRACT FROM AUTHOR]