Branin, F.H., Widely Convergent Methods for Finding Multiple Solutions of Simultaneous Nonlinear Equations, IBM Journal of Research Developments, 504 (1972).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....we make the following (4.1) Hypotheses. 1) f : R N # R N is a C # map; 2) # # R N is an open bounded set having a smooth connected boundary ##; 3) 0 is a regular value of f . The global Newton method calculates a zero point of f in #. This method has been promulgated by Branin [6] and has found frequent use in scientific applications. Smale [20] has studied this method from a theoretical standpoint and given an existence theorem which we will state below. The method consists of the following steps. 4.2) Global Newton Method. 1) Choose a starting point p # ##; 2) ....

F. H. Branin, Jr., Widely convergent methods for finding multiple solutions of simultaneous nonlinear equations, IBM J. Res. Develop. 16 (1972), 504--522.

....0 ; x(0) 0; where the starting point x 0 is choosen at random and a Bayesian stopping rule is used. In the next subsections we shall describe Griewank s method and the method of Snyman and Fatti more closely and then we will give a short description of a method initially introduced by Branin in [4] which is based on the first order differential equation d dt F (x) Sigma F (x) 0; x(0) x 0 Today this method is called continuous Newton method and in chapter 3 we shall take another look at this method from a different perspective. TRAJECTORY METHODS IN GLOBAL OPTIMIZATION 5 2.1. ....

....but one must keep in mind that this is actually a variant of multistart which a piori has an asymptotic convergence guarantee and that the efficiency of such procedures depends very heavily on the choosen stopping rule and the appropriate setting of the parameters. 2.3. Branin s method In [4] Branin considers the problem (P) of finding possibly all solutions to F (x) 0 on IR n . He introduces the first order differential equation d dt F (x) F (x) 0; x(0) x 0 ; which gives x = GammaDF (x) Gamma1 F (x) x(0) x 0 : 5) 8 IMMO DIENER as long as the Jacobian DF (x) is ....

F.H. Branin. A widely convergent method for finding multiple solutions of simultaneous nonlinear equations. I.B.M Journal of Research and Development, 16:504--522, 1972.

....or repellors for the trajectories. Now the right hand side of the equation dNDE is defined on the whole space and vanishes in the zeros of F . However, additional singularities might occur in points where F (x) 6= 0 but still g DF (x)F (x) 0. Such points were called extraneous singularities in [1]. The structure of such extraneous singularities has been studied in several papers by Jongen, Jonker and Twilt (see [11] and references therein) In [1] Branin suggested not to stop at a zero of F but to project the last step across the zero, reverse the sign on the right hand side of dNDE and to ....

....additional singularities might occur in points where F (x) 6= 0 but still g DF (x)F (x) 0. Such points were called extraneous singularities in [1] The structure of such extraneous singularities has been studied in several papers by Jongen, Jonker and Twilt (see [11] and references therein) In [1] Branin suggested not to stop at a zero of F but to project the last step across the zero, reverse the sign on the right hand side of dNDE and to continue integration, possibly finding further zeros of F . He conjectured that in the absence of extraneous singularities one could find all zeros of ....

F.H. Branin. A widely convergent method for finding multiple solutions of simultaneous nonlinear equations. I.B.M Journal of Research and Development, (1972) 504--522. 14 Newton leaves

....by Lester Ingber and other contributors is available at URL http: www.ingber.com or ftp: ftp.ingber.com. 3 We used the critical distance parameter i = 2 with 100 points generated per iteration. merits. RAST is a scaled Rastrigin function [7] HUMP is the six hump camelback function [11]. G P is the Goldstein Price function [11] GW1 and GW100 are 6 dimensional Griewank functions with bounds of each dimension [ Gamma1; 1] and [ Gamma100; 100] respectively [7] SWISS is a 4 D paraboloid with a lattice of many circular pits [5] CMMR is a 4 D paraboloid with a grid of deep troughs ....

....is available at URL http: www.ingber.com or ftp: ftp.ingber.com. 3 We used the critical distance parameter i = 2 with 100 points generated per iteration. merits. RAST is a scaled Rastrigin function [7] HUMP is the six hump camelback function [11] G P is the Goldstein Price function [11]. GW1 and GW100 are 6 dimensional Griewank functions with bounds of each dimension [ Gamma1; 1] and [ Gamma100; 100] respectively [7] SWISS is a 4 D paraboloid with a lattice of many circular pits [5] CMMR is a 4 D paraboloid with a grid of deep troughs [12] GW100, SWISS, and CMMR have many ....

F.H. Branin. Widely convergent method for finding multiple solutions of simultaneous nonlinear equations. I.B.M. J. R&D., Sept 1972.

....to the local behavior of the function to be minimized. The major advantages are that they are easy to implement and they lead to the global (rather than local) minimum. Convergence can be very slow, and the performance of the method is highly problem dependent. Trajectory description methods [3, 16, 30] can be used to deal with the minimization of multimodal functions by generating trajectories in the domain of interest which will pass through all the minima of the function, and thus ensure locating the global minumum. This method requires the evaluation of the Hessian and its inverse, which is ....

F. H. Branin. Widely convergent method for finding multiple solutions of simultaneous nonlinear equations. IBM J. Res. Develop, 16:504--522, 1972.

.... 0; 2.8) will be called the tangent vector induced by A. Making use of this definition, the above solution curve c(s) is characterized as the solution of the initial value problem u = t(H 0 (u) u(0) u 0 (2:9) which in this context is occasionally attributed to Davidenko (1953) see also Branin (1972). Note that the domain fu 2 R N 1 : u is a regular pointg is open. This differential equation is not used in efficient path following algorithms, but it serves as a useful device in analyzing the path. Two examples are: Lemma 2.3 Let (a; b) be the maximal interval of existence for (2.9) If a ....

....such questions. One may interpret Newton s method as the numerical integration of the 18 E. Allgower and K. Georg differential equation x = GammaG 0 (x) Gamma1 G(x) using Euler s method with unit step size. The idea of using the above flow to find zero points of G was exploited by Branin (1972). Smale (1976) gave conditions on Omega under which the flow leads to a zero point of G in Omega Gamma Such numerical methods have been referred to as global Newton methods. Keller (1978) observed that the above flow can also be obtained in a numerically stable way from a homotopy equation ....

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F. H. Branin (1972), `Widely convergent method for finding multiple solutions of simultaneous nonlinear equations', IBM J. Res. Develop. 16, 504--522.

....algorithm. SHOOT can be viewed as an algorithm that uses different start points and also as an algorithm that in essence in the worst case enumerates all the local minima. The idea of using different start points (multistart algorithms) 7] or the naive approach of enumerating all the local minima [8] is not new. The major difference is the manner in which SHOOT generates these new start points. Random start points with uniform distribution give dismal results (hundreds of thousands of function evaluations on a 20 dimension problem) and we have verified this for the test function. Exhaustive ....

Branin, F. H., "Widely convergent methods for finding multiple solutions of simultaneous nonlinear equations, IBM Journal of Research and Development, pp. 504-522, 1972. Derivative and function evaluations min. sol. found

....operating points has a long history, see Chao and Saeks [6] However the problem of developing continuation methods guaranteed to find all operating points has received relatively little study. The idea of finding several zeros of F (x) along one curve was made in the early 1970 s in Branin [5] and Chua and Ushida [12] In some of their examples, there are zeros of F (x) on several connected components of Gamma(H ) It is natural in pursuing this approach to try to get al..l zeros 1 Some authors use the term dc equilibrium point for a solution to the network equations F (x) 0, and ....

F. H. Branin, Widely convergent method for finding multiple solutions of simultaneous nonlinear equations, IBM J. Research Develop. 16 (1972), 504--522.

....objective functions used for comparing the global optimization algorithms. The first part of our study uses functions selected from GO literature and algorithm demonstrations in order to reveal their relative merits. RAST is a scaled Rastrigin function [4] HUMP is the six hump camelback function [2]. G P is the Goldstein Price function [2] GW1 and GW100 are 6 dimensional Griewank functions with bounds of each dimension [ Gamma1; 1] and [ Gamma100; 100] respectively [4] SWISS is a 4 D paraboloid with a lattice of many circular pits [12] CMMR is a 4 D paraboloid with a grid of deep ....

....global optimization algorithms. The first part of our study uses functions selected from GO literature and algorithm demonstrations in order to reveal their relative merits. RAST is a scaled Rastrigin function [4] HUMP is the six hump camelback function [2] G P is the Goldstein Price function [2]. GW1 and GW100 are 6 dimensional Griewank functions with bounds of each dimension [ Gamma1; 1] and [ Gamma100; 100] respectively [4] SWISS is a 4 D paraboloid with a lattice of many circular pits [12] CMMR is a 4 D paraboloid with a grid of deep troughs [3] GW100, SWISS, and CMMR have many ....

F.H. Branin. Widely convergent method for finding multiple solutions of simultaneous nonlinear equations. I.B.M. J. R&D., Sept 1972.

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Branin, F.H., Widely Convergent Methods for Finding Multiple Solutions of Simultaneous Nonlinear Equations, IBM Journal of Research Developments, 504 (1972).

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F. H. Branin, A widely convergent method for finding multiple solutions of simultaeneous nonlinear equations, IBM J. Res. Develop. 16 (1972) 504--522.

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F. H. Branin, Jr., Widely Convergent Method for Finding Multiple Solutions of Simultaneous Nonlinear Equations, IBM J. Res. Develop. (Sept. 1992) 504--522.

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Branin, F.H. (1972). Widely convergent methods for finding multiple solutions of simultaneous nonlinear equations. IBM Journal of Research Developments, 504-522.

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Branin, F.H., Widely Convergent Methods for Finding Multiple Solutions of Simultaneous Nonlinear Equations, IBM Journal of Research Developments, 504 (1972).

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Branin, F.H., Widely Convergent Methods for Finding Multiple Solutions of Simultaneous Nonlinear Equations, IBM Journal of Research Developments, 504 (1972).

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