Local convergence
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On the Convergence of Newton's Method in Power Flow Studies for DC Microgrids Open
The power flow is a non-linear problem that requires a Newton's method to be solved in dc microgrids with constant power terminals. This paper presents sufficient conditions for the quadratic convergence of the Newton's method in this type…
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Global optimality of local search for low rank matrix recovery Open
We show that there are no spurious local minima in the non-convex factorized parametrization of low-rank matrix recovery from incoherent linear measurements. With noisy measurements we show all local minima are very close to a global optim…
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Quantum gradient descent and Newton’s method for constrained polynomial optimization Open
Optimization problems in disciplines such as machine learning are commonly solved with iterative methods. Gradient descent algorithms find local minima by moving along the direction of steepest descent while Newton's method takes into acco…
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Nonlinear Preconditioning: How to Use a Nonlinear Schwarz Method to Precondition Newton's Method Open
For linear problems, domain decomposition methods can be used directly as iterative solvers, but also as preconditioners for Krylov methods. In practice, Krylov acceleration is almost always used, since the Krylov method finds a much bette…
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Sub-Sampled Newton Methods II: Local Convergence Rates Open
Many data-fitting applications require the solution of an optimization problem involving a sum of large number of functions of high dimensional parameter. Here, we consider the problem of minimizing a sum of $n$ functions over a convex con…
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Stability analysis of fourth-order iterative method for finding multiple roots of non-linear equations Open
The use of complex dynamics tools in order to deepen the knowledge of qualitative behaviour of iterative methods for solving non-linear equations is a growing area of research in the last few years with fruitful results. Most of the studie…
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On Convergence Rates of Linearized Proximal Algorithms for Convex Composite Optimization with Applications Open
In the present paper, we investigate a linearized proximal algorithm (LPA) for solving a convex composite optimization problem. Each iteration of the LPA is a proximal minimization of the convex composite function with the inner function b…
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Demystifying Why Local Aggregation Helps: Convergence Analysis of Hierarchical SGD Open
Hierarchical SGD (H-SGD) has emerged as a new distributed SGD algorithm for multi-level communication networks. In H-SGD, before each global aggregation, workers send their updated local models to local servers for aggregations. Despite re…
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A Primal-Dual Quasi-Newton Method for Exact Consensus Optimization Open
We introduce the primal-dual quasi-Newton (PD-QN) method as an approximated\nsecond order method for solving decentralized optimization problems. The PD-QN\nmethod performs quasi-Newton updates on both the primal and dual variables of\nthe…
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Local Averaging Helps: Hierarchical Federated Learning and Convergence Analysis. Open
Federated learning is an effective approach to realize collaborative learning among edge devices without exchanging raw data. In practice, these devices may connect to local hubs instead of connecting to the global server (aggregator) dire…
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Efficient High-Order Iterative Methods for Solving Nonlinear Systems and Their Application on Heat Conduction Problems Open
For solving nonlinear systems of big size, such as those obtained by applying finite differences for approximating the solution of diffusion problem and heat conduction equations, three-step iterative methods with eighth-order local conver…
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Survey of sequential convex programming and generalized Gauss-Newton methods Open
We provide an overview of a class of iterative convex approximation methods for nonlinear optimization problems with convex-over-nonlinear substructure. These problems are characterized by outer convexities on the one hand, and nonlinear, …
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Trusses Nonlinear Problems Solution with Numerical Methods of Cubic Convergence Order Open
A large part of the numerical procedures for obtaining the equilibrium path or load-displacement curve of structural problems with nonlinear behavior is based on the Newton-Raphson iterative scheme, to which is coupled the path-following m…
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A Hybrid Approach for Solving Systems of Nonlinear Equations Using Harris Hawks Optimization and Newton’s Method Open
Systems of nonlinear equations are known as the basis for many models of engineering and data science, and their accurate solutions are very critical in achieving progress in these fields. However, solving a system with multiple nonlinear …
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On iterative techniques for estimating all roots of nonlinear equation and its system with application in differential equation Open
In this article, we construct a family of iterative methods for finding a single root of nonlinear equation and then generalize this family of iterative methods for determining all roots of nonlinear equations simultaneously. Further we ex…
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Convergence analysis of Sakurai–Torii–Sugiura iterative method for simultaneous approximation of polynomial zeros Open
In 1991, T. Sakurai, T. Torii and H. Sugiura presented a fourth-order iterative algorithm for finding all zeros of a polynomial simultaneously. In this paper, we provide a detailed convergence analysis (local and semilocal) of this method.…
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A hybrid semismooth quasi-Newton method for nonsmooth optimal control with PDEs Open
We propose a semismooth Newton-type method for nonsmooth optimal control problems. Its particular feature is the combination of a quasi-Newton method with a semismooth Newton method. This reduces the computational costs in comparison to se…
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Two new efficient sixth order iterative methods for solving nonlinear equations Open
In this paper, we present two new iterative methods, one of them is second derivative free, for solving nonlinear equations. We derive these methods based on the Taylor series expansion and Halley's method. The convergence analysis of the …
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Three iterative methods for solving second order nonlinear ODEs arising in physics Open
In this work, three iterative methods have been implemented to solve several second order nonlinear ODEs that arising in physics. The proposed iterative methods are Tamimi-Ansari method (TAM), Daftardar-Jafari method (DJM) and Banach contr…
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Convergence of the EDIIS Algorithm for Nonlinear Equations Open
The Energy Direct Inversion on the Iterative Subspace (EDIIS) algorithm was designed to globalize Anderson acceleration, a method for improving the performance of fixed point iteration. The motivating application is electronic structure co…
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Two new Hager–Zhang iterative schemes with improved parameter choices for monotone nonlinear systems and their applications in compressed sensing Open
Notwithstanding its efficiency and nice attributes, most research on the Hager–Zhang (HZ) iterative scheme are focused on unconstrained minimization problems. Inspired by this and recent extensions of the one-parameter HZ scheme to system …
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Semismooth Newton-type method for bilevel optimization: global convergence and extensive numerical experiments Open
We consider the standard optimistic bilevel optimization problem, in particular upper- and lower-level constraints can be coupled. By means of the lower-level value function, the problem is transformed into a single-level optimization prob…
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On the Convergence of Mirror Descent beyond Stochastic Convex Programming Open
International audience
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Newton-MR: Newton's Method Without Smoothness or Convexity Open
Establishing global convergence of Newton-CG has long been limited to making strong convexity assumptions. Hence, many Newton-type variants have been proposed which aim at extending Newton-CG beyond strongly convex problems. However, the a…
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Multistep Iterative Algorithms for Solving Nonlinear Equation Open
We present two new iterative methods, called the Secant and Dekker's Algorithms for solving nonlinear equations in this paper. These equations based on different values of load resistance R of a single diode scheme. We examined the effecti…
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A Special Iterative Algorithm for Solving Nonlinear Equations Open
In this work, we exhibit some variants iterative techniques free from second derivatives of the function foe solving nonlinear equation of the form and observe that number of iterations of the proposed method is five. Many numerical experi…
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Numerical Solving of Nonlinear Equation Using Iterative Algorithms Open
In the given algorithm, we propose a development to the evaluations of Newton's numerical algorithm. Derivation of the standard method (Newton Raphson method) involves first derivative of the function. It is shown that the number of iterat…
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Study of Dynamical Behavior and Stability of Iterative Methods for Nonlinear Equation with Applications in Engineering Open
In this article, we first construct a family of optimal 2-step iterative methods for finding a single root of the nonlinear equation using the procedure of weight function. We then extend these methods for determining all roots simultaneou…
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Computer Methodologies for the Comparison of Some Efficient Derivative Free Simultaneous Iterative Methods for Finding Roots of Non-Linear Equations Open
In this article, we construct the most powerful family of simultaneous iterative method with global convergence behavior among all the existing methods in literature for finding all roots of non-linear equations. Convergence analysis prove…
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Various Iterative Methods for Solving Nonlinear Equation Open
In recent articles, many researchers settled some refinements in iterative methods have been published in many scientific journals for solving nonlinear equations of a single diode mode of a solar cell. Thus, in our article, we have improv…