R.B. Kearfott, Empirical Evaluation of Innovations in Interval Branch and Bound Algorithms for Nonlinear Algebraic Systems, SIAM J. Sci. Comput. 18 (1997), 574-594.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....so called cluster e ect, and how to reduce the cluster e ect by de ning exclusion regions around each zero found, that are guaranteed to contain no other zero and hence can safely be discarded. Such exclusion boxes are the basis for the backboxing strategy by van Iwaarden [22] see also Kearfott [6, 7]) that eliminates the cluster e ect near well conditioned zeros. Exclusion regions are traditionally constructed using uniqueness tests based on the Krawczyk operator (see, e.g. Neumaier [14, Chapter 5] or the Kantorovich theorem (see, e.g. Ortega Rheinboldt [17, Theorem 12.6.1] both ....

R.B. Kearfott, Empirical Evaluation of Innovations in Interval Branch and Bound Algorithms for Nonlinear Algebraic Systems, SIAM J. Sci. Comput. 18 (1997), 574-594.

....a compact domain. The present paper discusses a new class of tests for such algorithms in the context of global optimization, and presents complexity results concerning the resulting algorithms. 1 Introduction Exclusion algorithms are a well known tool in the area of interval analysis, see, e.g. [6, 7], for finding all solutions of a system of nonlinear equations or for finding the global minimum of a function over a compact domain. They also have been introduced in [11, 12] from a slightly di#erent viewpoint. In particular, such algorithms seem to be very useful for finding all solutions of ....

....level, and this leads to significant numerical ine#ciency. In the area of interval analysis, the idea of exclusion is exploited in interval branch and bound algorithms which are used to find all the zero points of a nonlinear system of equations, or also to minimize functions, see, e.g. Kearfott [7] and the bibliography cited there, and the software package GlobSol accompanying the book [6] From an interval analysis viewpoint, a simple exclusion test could be designed in the following way: T f (#) 1 : ## 0 # [f ] #) where [f ] #) is the interval obtained from # by applying f in an ....

R. B. Kearfott. Empirical evaluation of innovations in interval branch and bound algorithms for nonlinear algebraic systems. SIAM Journal on Scientific Computing, 18(2):574--594, 1997. 13

.... the E#ciency of Exclusion Algorithms Kurt Georg 134 Colorado State University 2 November 2000 1 Introduction Exclusion algorithms are a well known tool in the area of interval analysis, see, e.g. [5, 6], for finding all solutions of a system of nonlinear equations. They also have been introduced in [14, 15] from a slightly di#erent viewpoint. In particular, such algorithms seem to be very useful for finding all solutions of low dimensional, but highly nonlinear systems which have many solutions. ....

....level, and this leads to significant numerical ine#ciency. In the area of interval analysis, the idea of exclusion is exploited in interval branch and bound algorithms which are used to find all the zero points of a nonlinear system of equations, or also to minimize functions, see, e.g. Kearfott [6] and the bibliography cited there, and the software package GlobSol accompanying the book [5] From an interval analysis viewpoint, a simple exclusion test could be designed in the following way: T f (#) 1 : ## 0 # [f ] #) where [f ] #) is the interval obtained from # by applying f in an ....

R. B. Kearfott. Empirical evaluation of innovations in interval branch and bound algorithms for nonlinear algebraic systems. SIAM Journal on Scientific Computing, 18(2):574--594, 1997.

....10 1281:47944x 9 283:4435875x 8 202:6270915x 7 (1. 3) 16:17913459x 6 8:88303902x 5 1:575580173x 4 0:1245990848x 3 0:03589148622x 2 0:0001951095576x 0:0002274682229; x 2 x: This example can be treated by careful tesselation and use of point evaluations, as explained in [Kea97]. However, a more elegant way would be if 1. Taylor Series Models in Deterministic Global Optimization 3 sharper bounds on the range could be easily obtained. Taylor models have shown promise for this. 2 Taylor Models and Global Optimization Oversimplifying, Taylor models are models of the form ....

R. B. Kearfott. Empirical evaluation of innovations in interval branch and bound algorithms for nonlinear algebraic systems. SIAM J. Sci. Comput., 18(2):574-594, March 1997.

....is that it took far more effort to perturb the approximate feasible point than to verify that a feasible point existed, once the approximate feasible point was perturbed. Most of the CPU time in such perturbation steps was spent in the constrained optimizer DAUGLG. See [9] for details. Also, see [8, 12] for a discussion of efficiency in the Fortran 90 system [11] The alternative to selecting square subsystems and perturbing approximate feasible points, using the Fritz John system, was unsuccessful in preliminary experiments, and is not reported in these tables. Table 3: Interval constraint ....

R. B. Kearfott. Empirical evaluation of innovations in interval branch and bound algorithms for nonlinear algebraic systems, 1994. Accepted for publication in SIAM J. Sci. Comput.

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R. B. Kearfott. Empirical evaluation of innovations in interval branch and bound algorithms for nonlinear algebraic systems, accepted for publication in the SIAM J. Sci. Statist. Comput.

....branch and bound methods for nonlinear equations and for nonlinear optimization benefit from use of a floating point code to obtain approximate optimizers. This has been explored for nonlinear equations, first in the context of handling singular roots in [13] then more generally and thoroughly in [18]. There, a tesselation process was given for explicitly verifying uniqueness and removing regions around roots that had been quickly found by a local, floating point algorithm. The analogous technique for unconstrained optimization, using a floating point local optimizer, has perhaps been most ....

....X and X, that is, the smallest interval vector that contains both X and X, will be denoted by X[ X. If L is a list of boxes and X is a box, then L n X will denote a list the union of whose elements is the complement of X in the union of elements of L. Algorithms 7 and 8, and Figure 2 in [18] describe how to create L n X. The symbols OE(X) OE(X) OE(X) will denote an interval extension of OE over X. Consistent with the above, C(X) c 1 (X) c m (X) T = 0, C : R n R m , will denote the set of equality constraints, C(X) will denote the set of interval residuals of ....

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Kearfott, R. B., Empirical Evaluation of Innovations in Interval Branch and Bound Algorithms for Nonlinear Algebraic Systems, accepted for publication in SIAM J. Sci. Comput..

....and coupled with interval Newton methods to accelerate the search, can form practical algorithms for rigorously finding all roots of nonlinear systems or for global optimization. For introductions to such techniques, see [4] 6] or [21] while for test results for such algorithms, see [6] or [12], 26] and others. For an advanced introduction to the underlying techniques, see [7] while for classic introductions to interval computations, see [1] or [19] Traditionally, such methods have been implemented with subroutine packages for interval extensions, ad hoc packages, or special ....

....The Testing Software and Environment The Fortran 90 environment of [13] with the interval arithmetic package of [17] is used. The global optimization problems were tested essentially with the code of [14] while the problems involving nonlinear systems of equations were tested with the code of [12]; minor modifications had been made to these codes subsequent to the experiments in [12] and [14] In both the optimization and nonlinear equations codes, an approximate solution was computed (if possible) first 3 . If an approximate optimum or solution was found, a box was constructed around ....

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R. B. Kearfott. Empirical evaluation of innovations in interval branch and bound algorithms for nonlinear algebraic systems, 1994. Accepted for publication in SIAM J. Sci. Comput.

....the components of X to zero. This allows subregions to be rigorously searched without excessive tessellation. In addition, a compound algorithm can be used to show uniqueness when A is merely a slope matrix, and not a Lipschitz matrix. See [58] and the implementation description and experiments in [35]. Thus, performance of an interval Newton method can have differing goals ( existence, uniqueness, nonexistence, or reduction in the size of X) Also, interval Newton methods can be applied to different types of systems (general nonlinear system, Lagrange multiplier or Fritz John system, etc. ....

....and represent Lagrange multipliers for the equality constraints c i = 0. The last equation is a normalization condition suggested and justified in [20] ffl is on the order of the computational precision. 15 This is done in the unconstrained case in [3] and for general nonlinear systems in [35]. 30 Chapter 1 By not including the bound constraints a i j x i j b i j in this function, we reduce the size of the system by 2q. Furthermore, it is more flexible to include the bound constraints through the process reviewed in x2.3. Thus, equation (1.12) is applicable for points X when none ....

Kearfott, R. B., Empirical Evaluation of Innovations in Interval Branch and Bound Algorithms for Nonlinear Algebraic Systems, preprint, 1994.

....i j; 1g(ffl d =2) Additionally, Algorithm 1 can be executed iteratively: we adjust the boxes X in step 4 with an ffl inflation procedure 5 from iteration to iteration, to make it likely that the interval Newton method can verify the root. In fact, a modification of step 3 of Algorithm 3 in [13], as follows, can be used. However, our experience indicates that, if the tolerances for X and the coordinate widths of X are chosen as indicated above, feasibility is usually proven without inflation, if it is proven at all; see table 2. Algorithm 2 (ffl inflation: Repeat Algorithm 1 while ....

Kearfott, R. B., Empirical Evaluation of Innovations in Interval Branch and Bound Algorithms for Nonlinear Algebraic Systems , preprint, 1994.

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Kearfott, R. B., Empirical Evaluation of Innovations in Interval Branch and Bound Algorithms for Nonlinear Algebraic Systems , preprint, 1994.

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R.B. Kearfott, Empirical Evaluation of Innovations in Interval Branch and Bound Algorithms for Nonlinear Algebraic Systems, SIAM J. Sci. Comput. 18 (1997), 574-594.

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