Global Convergence Theorem Research Articles

New spectral PRP conjugate gradient method for unconstrained optimization

In this paper, a new spectral PRP conjugate gradient algorithm has been developed for solving unconstrained optimization problems, where the search direction was a kind of combination of the gradient and the obtained direction, and the steplength was obtained by the Wolfe-type inexact line search. It was proved that the search direction at each iteration is a descent direction of objective function. Under mild conditions, we have established the global convergence theorem of the proposed method. Numerical results showed that the algorithm is promising, particularly, compared with the existing several main methods.

On a third‐order Newton‐type method free of bilinear operators

AbstractThis paper is devoted to the study of a third‐order Newton‐type method. The method is free of bilinear operators, which constitutes the main limitation of the classical third‐order iterative schemes. First, a global convergence theorem in the real case is presented. Second, a semilocal convergence theorem and some examples are analyzed, including quadratic equations and integral equations. Finally, an approximation using divided differences is proposed and used for the approximation of boundary‐value problems. Copyright © 2009 John Wiley & Sons, Ltd.

Monocular vision-based iterative pose estimation algorithm from corresponding feature points

To determine the relative pose between an object and a single camera using 2D-to-3D point correspondences, a kind of iterative methods based on inverse projection ray approach is proposed. An iterative algorithm which is divided into depth recovery stage and absolute orientation stage is also proposed. In the first stage, the optimal translation vector is first computed in terms of rotation matrix via least square method, then the depths of the observed points are estimated by projecting the estimated point orthogonally to the inverse projection ray defined by the image point, and finally 3D points are reconstructed using the estimated depths from previous step. In the second stage, the optimal rotation matrix is estimated by applying Umeyama algorithm to fitting of the 3D model points and 3D estimated points. The above two stages are repeated until the result converges. The global convergence of the two-stage iterative algorithm is proven based on the global convergence theorem. Finally, a spacecraft docking application is implemented to test the effectiveness and convergence of the proposed algorithm by mathematical simulation and physical simulation.

A GLOBAL CONVERGENCE THEOREM IN BOOLEAN ALGEBRA

Robert has established a global convergence theorem in $\{0,1\}^{n}$: If a map $\hat{F}$ from $\{0,1\}^{n}$ to itself is contracting relative to the boolean vector distance $d$, then there exists a positive integer $p \leq n$ such that $\hat{F}^{p}$ is constant. In other words, $\hat{F}$ has a unique fixed point $\xi$ such that for any $x$ in $\{0,1\}^{n}$, we have $\hat{F}^{p}(x) = \xi$. The structure ($\{0,1\},+,\cdot,-,0,1$) may be regarded as the two-element boolean algebra. In this paper, this result is extended to any map $F$ from the product $X$ of $n$ finite boolean algebras to itself.

Remarks on feedforward circuits, adaptation, and pulse memory

This note studies feedforward circuits as models for perfect adaptation to step signals in biological systems. A global convergence theorem is proved in a general framework, which includes examples from the literature as particular cases. A notable aspect of these circuits is that they do not adapt to pulse signals, because they display a memory phenomenon. Estimates are given of the magnitude of this effect.

Global convergence for a 1-D inverse problem with application to imaging of land mines

A new globally convergent numerical method is developed for a 1-D coefficient inverse problem for a hyperbolic partial differential equation (PDE). The back reflected data are used. A version of the quasi-reversibility method is proposed. A global convergence theorem is proven via a Carleman estimate. The results of numerical experiments are presented.

Convergence analysis of quantum-behaved particle swarm optimization algorithm and study on its control parameter

Based on the analysis of particle swarm optimization algorithm, the particle is described in the quantum space and the potential energy field model is created. And then according to the swarms gregariousness, the quantum-behaved particle swarm optimization (QPSO) algorithm is derived. Within the framework of random algorithms global convergence theorem, the convergence of QPSO algorithm is discussed and is proved to be a kind of global convergence algorithm. Three kinds of control strategy are proposed for the unique parameter of QPSO algorithm and they are tested on five benchmark functions. According to the test results, some conclusions concerning the selection of the parameter are drawn.

A note on Newton’s method for stochastic differential equations and its error estimate

Kawabata and Yamada [3] proposed an implicit Newton method for stochastic differential equations and proved its convergence. They proved an error estimate in a sufficiently small time interval and extended it to a global convergence theorem by using open-closed method. In this note, the author gives an explicit Newton scheme which is equivalent to Kawabata-Yamada’s implicit formulation (Remark 1) and prove its direct error estimate (Theorem 2.1). His result could provide a key to solve the open problem of second order convergence (see Remark of Theorem 2.1 and [2]).

Convergence of a generalized Newton and an inexact generalized Newton algorithms for solving nonlinear equations with nondifferentiable terms

In this paper, we consider two versions of the Newton-type method for solving a nonlinear equations with nondifferentiable terms, which uses as iteration matrices, any matrix from B-differential of semismooth terms. Local and global convergence theorems for the generalized Newton and inexact generalized Newton method are proved. Linear convergence of the algorithms is obtained under very mild assumptions. The superlinear convergence holds under some conditions imposed on both terms of equation. Some numerical results indicate that both algorithms works quite well in practice.

A two-steps algorithm for approximating real roots of a polynomial in Bernstein basis

The surface/curve intersection problem, through the resultants process results in a high degree ( n ≥ 100 ) polynomial equation on [ 0 , 1 ] in the Bernstein basis. The knowledge of multiplicities of the roots is critical for the topological coherence of the results. In this aim, we propose an original two-steps algorithm based on successive differentiations which separates any root (even multiple) and guarantees that the assumptions of Newton global convergence theorem are satisfied. The complexity is ϑ ( n 4 ) but the algorithm can easily be parallelized. Experimental results show its efficiency when facing ill-conditioned polynomials.

On the global convergence of Chebyshev's iterative method

In [A. Melman, Geometry and convergence of Euler's and Halley's methods, SIAM Rev. 39(4) (1997) 728–735] the geometry and global convergence of Euler's and Halley's methods was studied. Now we complete Melman's paper by considering other classical third-order method: Chebyshev's method. By using the geometric interpretation of this method a global convergence theorem is performed. A comparison of the different hypothesis of convergence is also presented.

Some remarks on the inverse eigenvalue problem for real symmetric Toeplitz matrices

A theorem about the bounds of solutions of the Toeplitz Inverse Eigenvalue Problem is introduced and proved. It can be applied to make a better starting generator for iterative numerical methods. This application is tested through a short Mathematica program. Also an optimisation method for solving the Toeplitz Inverse Eigenvalue Problem with a global convergence property is presented. A global convergence theorem is proved.

Convergence of nonmonotone line search method

In this paper, we develop a new nonmonotone line search for general line search method and establish some global convergence theorems. The new nonmonotone line search is a novel form of the nonmonotone Armijo line search and allows one to choose a larger step size at each iteration, which is available in constructing new line search methods and possibly reduces the function evaluations at each iteration. Moreover, we analyze the convergence rate of some special line search methods with the new line search. Preliminary numerical results show that some line search methods with the new nonmonotone line search are available and efficient in practical computation.

Dynamical systems method for ill-posed equations with monotone operators

Consider an operator equation B( u) − f = 0 in a real Hilbert space. Let us call this equation ill-posed if the operator B′( u) is not boundedly invertible, and well-posed otherwise. The dynamical systems method (DSM) for solving this equation consists of a construction of a Cauchy problem, which has the following properties: (1) it has a global solution for an arbitrary initial data, (2) this solution tends to a limit as time tends to infinity, (3) the limit is the minimal-norm solution to the equation B( u) = f. A global convergence theorem is proved for DSM for equation B( u) − f = 0 with monotone C loc 2 operators B.

Mean Curvature Flows of Lagrangian Submanifolds with Convex Potentials

This article studies the mean curvature flow of Lagrangian submanifolds. In particular, we prove the following global existence and convergence theorem: if the potential function of a Lagrangian graph in T^{2n} is convex, then the flow exists for all time and converges smoothly to a flat Lagrangian submanifold.

Probabilistic version of the method of feasible directions

A probabilistic version of the method of feasible directions (MFD) for solving nonlinear programming (NLP) problems of the type min{ f( x): f j ( x)⩽0, j=1,2,…, m} is presented. Randomization is applied to modify the algorithm and a global convergence Theorem is used in the analysis of convergence. Some numerical experiments on problems with known solutions serve to compare this method with the traditional deterministic versions.

Descent Directions of Quasi-Newton Methods for Symmetric Nonlinear Equations

In general, when a quasi-Newton method is applied to solve a system of nonlinear equations, the quasi-Newton direction is not necessarily a descent direction for the norm function. In this paper, we show that when applied to solve symmetric nonlinear equations, a quasi-Newton method with positive definite iterative matrices may generate descent directions for the norm function. On the basis of a Gauss--Newton based BFGS method [D. H. Li and M. Fukushima, SIAM J. Numer. Anal., 37 (1999), pp. 152--172], we develop a norm descent BFGS method for solving symmetric nonlinear equations. Under mild conditions, we establish the global and superlinear convergence of the method. The proposed method shares some favorable properties of the BFGS method for solving unconstrained optimization problems: (a) the generated sequence of the quasi-Newton matrices is positive definite; (b) the generated sequence of iterates is norm descent; (c) a global convergence theorem is established without nonsingularity assumption on the Jacobian. Preliminary numerical results are reported, which positively support the method.

Inexact-Restoration Method with Lagrangian Tangent Decrease and New Merit Function for Nonlinear Programming

A new inexact-restoration method for nonlinear programming is introduced. The iteration of the main algorithm has two phases. In Phase 1, feasibility is improved explicitly; in Phase 2, optimality is improved on a tangent approximation of the constraints. Trust regions are used for reducing the step when the trial point is not good enough. The trust region is not centered in the current point, as in many nonlinear programming algorithms, but in the intermediate more feasible point. Therefore, in this semifeasible approach, the more feasible intermediate point is considered to be essentially better than the current point. This is the first method in which intermediate-point-centered trust regions are combined with the decrease of the Lagrangian in the tangent approximation to the constraints. The merit function used in this paper is also new: it consists of a convex combination of the Lagrangian and the nonsquared norm of the constraints. The Euclidean norm is used for simplicity, but other norms for measuring infeasibility are admissible. Global convergence theorems are proved, a theoretically justified algorithm for the first phase is introduced, and some numerical insight is given.

A new semilocal convergence theorem for Newton's method in banach space using hypotheses on the second Fréchet-derivative

A new global Kantorovich-type convergence theorem for Newton's method in Banach space is provided for approximating a solution of a nonlinear equation. It is assumed that a solution exists and the second Fréchet-derivative of the operator involved satisfies a Lipschitz condition. Our convergence condition differs from earlier ones, and therefore it has theoretical and practical value. Finally, a simple numerical example is provided to show that our results apply, where earlier ones fail.

Analysis of a SIS epidemic model with stage structure and a delay

A disease transmission model of SIS type with stage structure and a delay corresponding to the infectious period is formulated. By constructing the difference operator and using global convergence theorem, we show that the disease-free equilibrium is globally asymptotically stable. The stability of an endemic equilibrium are investigated for the differential difference equations.