This paper presents convergence analysis of some algorithms for solving systems nonlinear equations defined by locally Lipschitzian functions. For the directional derivative-based and generalized Jacobian-based Newton methods, both iterates corresponding function values are locally, superlinearly convergent. Globally, a limiting point iterate sequence generated damped, method is zero system if only converges to this stepsize eventually becomes one, provided that strongly BD-regular...

In this paper we propose an iterative method for calculating the largest eigenvalue of irreducible nonnegative tensor. This is extension a Collatz (1942) spectral radius matrix. Numerical results show that our proposed promising. We also apply to studying higher-order Markov chains.

We introduce $M$-tensors. This concept extends the of $M$-matrices. denote $Z$-tensors as tensors with nonpositive off-diagonal entries. show that $M$-tensors must be and maximal diagonal entry nonnegative. The elements a symmetric $M$-tensor A is copositive. Based on spectral theory nonnegative tensors, we minimal value real parts all eigenvalues an its smallest H$^+$-eigenvalue also H-eigenvalue. $Z$-tensor if only H$^+$-eigenvalues are Some further properties given. strong $M$-tensors,...

Abstract In this paper, we first study the projections onto set of unit dual quaternions, and quaternion vectors with norms. Then propose a power method for computing dominant eigenvalue Hermitian matrix. For strict eigenvalue, show sequence generated by converges to its corresponding eigenvector linearly. general establish linear convergence standard part eigenvalue. Based upon these, reformulate simultaneous localization mapping problem as rank-one completion problem. A two-block...

This paper reports on some recent developments in the area of solving nonsmooth equations by generalized Newton methods. The emphasis is three topics: motivation, characterization superlinear convergence, and a new Gauss–Newton method for certain class equations. convergence extends classical result Dennis Moré smooth that Ip Kyparisis B-differentiable different from proposed recently Han, Pang, Rangaraj; it uses convex quadratic programs to generate descent directions least-squares merit function.

The smoothing Newton method for solving a system of nonsmooth equations <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F left-parenthesis x right-parenthesis equals 0"> <mml:semantics> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">F(x)=0</mml:annotation>...

We consider superlinearly convergent analogues of Newton methods for nondifferentiable operator equations in function spaces. The superlinear convergence analysis semismooth described by a locally Lipschitzian Rn is based on Rademacher's theorem which does not hold introduce concept slant differentiability and use it to study smoothing unified framework. show that slantly differentiable at point if only Lipschitz continuous point. An application the Dirichlet problems simple class nonsmooth...

In this paper, we introduce a constant positive linear dependence condition (CPLD), which is weaker than the Mangasarian--Fromovitz constraint qualification (MFCQ) and rank (CRCQ). We show that limit point of sequence approximating Karush--Kuhn--Tucker (KKT) points KKT if CPLD holds there. satisfying strong second-order sufficiency conditions (SSOSC) an isolated point. then establish convergence general sequential quadratical programming (SQP) method under SSOSC. Finally, apply these results...

We propose a simple and natural definition for the Laplacian signless tensors of uniform hypergraph.We study their H + -eigenvalues, i.e., H-eigenvalues with nonnegative H-eigenvectors, ++ positive H-eigenvectors.We show that each tensor, adjacency tensor has at most one -eigenvalue, but several other -eigenvalues.We identify largest smallest establish some maximum minimum properties these then define analytic connectivity hypergraph discuss its application in edge connectivity.