Results 11  20
of
539
Some tests of generalized bisection
 ACM Trans. Math. Software
, 1987
"... This paper addresses the task of reliably finding approximations to all solutions to a system of nonlinear equations within a region defined by bounds on each of the individual coordinates. Various forms of generalized bisection were proposed some time ago for this task. This paper systematically co ..."
Abstract

Cited by 49 (2 self)
 Add to MetaCart
This paper addresses the task of reliably finding approximations to all solutions to a system of nonlinear equations within a region defined by bounds on each of the individual coordinates. Various forms of generalized bisection were proposed some time ago for this task. This paper systematically compares such generalized bisection algorithms to themselves, to continuation methods, and to hybrid steepest descent/quasiNewton methods. A specific algorithm containing novel “expansion ” and “exclusion ” steps is fully described, and the effectiveness of these steps is evaluated. A test problem consisting of a small, highdegree polynomial system that is appropriate for generalized bisection, but very difticult for continuation methods, is presented. This problem forms part of a set of 17 test problems from published literature on the methods being compared; this test set is fully described here.
A Fortran 90 Environment for Research and Prototyping of Enclosure Algorithms for Nonlinear Equations and Global Optimization
"... An environment for general research into and prototyping of algorithms for reliable constrained and unconstrained global nonlinear optimization and reliable enclosure of all roots of nonlinear systems of equations, with or without inequality constraints, is being developed. This environment should b ..."
Abstract

Cited by 40 (19 self)
 Add to MetaCart
An environment for general research into and prototyping of algorithms for reliable constrained and unconstrained global nonlinear optimization and reliable enclosure of all roots of nonlinear systems of equations, with or without inequality constraints, is being developed. This environment should be portable, easy to learn, use, and maintain, and sufficiently fast for some production work. The motivation, design principles, uses, and capabilities for this environment are outlined. The environment includes an interval data type, a symbolic form of automatic differentiation to obtain an internal representation for functions, a special technique to allow conditional branches with operator overloading and interval computations, and generic routines to give interval and noninterval function and derivative information. Some of these generic routines use a special version of the backward mode of automatic differentiation. The package also includes dynamic data structures for exhaustive sear...
Robust Semidefinite Programming” – in
 Handbook on Semidefinite Programming, Kluwer Academis Publishers
"... In this paper, we consider semidefinite programs where the data is only known to belong to some uncertainty set U. Following recent work by the authors, we develop the notion of robust solution to such problems, which are required to satisfy the (uncertain) constraints whatever the value of the data ..."
Abstract

Cited by 38 (18 self)
 Add to MetaCart
In this paper, we consider semidefinite programs where the data is only known to belong to some uncertainty set U. Following recent work by the authors, we develop the notion of robust solution to such problems, which are required to satisfy the (uncertain) constraints whatever the value of the data in U. Even when the decision variable is fixed, checking robust feasibility is in general NPhard. For a number of uncertainty sets U, we show how to compute robust solutions, based on a sufficient condition for robust feasibility, via SDP. We detail some cases when the sufficient condition is also necessary, such as linear programming or convex quadratic programming with ellipsoidal uncertainty. Finally, we provide examples, taken from interval computations and truss topology design. 1
Engineering Design Calculations with Fuzzy Parameters. Fuzzy Sets and Systems
, 1992
"... Uncertainty in engineering analysis usually pertains to stochastic uncertainty, i.e.,variance in product or process parameters characterized by probability (uncertainty in truth). Methods for calculating under stochastic uncertainty are well documented. It has been proposed by the authors that other ..."
Abstract

Cited by 35 (13 self)
 Add to MetaCart
Uncertainty in engineering analysis usually pertains to stochastic uncertainty, i.e.,variance in product or process parameters characterized by probability (uncertainty in truth). Methods for calculating under stochastic uncertainty are well documented. It has been proposed by the authors that other forms of uncertainty exist in engineering design. Imprecision, or the concept of uncertainty in choice, is one such form. This paper considers realtime techniques for calculating with imprecise parameters. These techniques utilize interval mathematics and the notion of αcuts from the fuzzy calculus. The extremes or anomalies of the techniques are also investigated, particularly the evaluation of singular or multivalued functions. It will be shown that realistic engineering functions can be used in imprecision calculations, with reasonable computational performance.
A Review Of Techniques In The Verified Solution Of Constrained Global Optimization Problems
, 1996
"... Elements and techniques of stateoftheart automatically verified constrained global optimization algorithms are reviewed, including a description of ways of rigorously verifying feasibility for equality constraints and a careful consideration of the role of active inequality constraints. Previousl ..."
Abstract

Cited by 25 (6 self)
 Add to MetaCart
Elements and techniques of stateoftheart automatically verified constrained global optimization algorithms are reviewed, including a description of ways of rigorously verifying feasibility for equality constraints and a careful consideration of the role of active inequality constraints. Previously developed algorithms and general work on the subject are also listed. Limitations of present knowledge are mentioned, and advice is given on which techniques to use in various contexts. Applications are discussed. 1 INTRODUCTION, BASIC IDEAS AND LITERATURE We consider the constrained global optimization problem minimize OE(X) subject to c i (X) = 0; i = 1; : : : ; m (1.1) a i j x i j b i j ; j = 1; : : : ; q; where X = (x 1 ; : : : ; xn ) T . A general constrained optimization problem, including inequality constraints g(X) 0 can be put into this form by introducing slack variables s, replacing by s + g(X) = 0, and appending the bound constraint 0 s ! 1; see x2.2. 2 Chapter 1 W...
Comparison of Interval Methods for Plotting Algebraic Curves
 Comput. Aided Geom. Des
, 2002
"... This paper compares the performance and e#ciency of di#erent function range interval methods for plotting f(x, y) = 0 on a rectangular region based on a subdivision scheme, where f(x, y) is a polynomial. The solution of this problem has many applications in CAGD. ..."
Abstract

Cited by 24 (2 self)
 Add to MetaCart
This paper compares the performance and e#ciency of di#erent function range interval methods for plotting f(x, y) = 0 on a rectangular region based on a subdivision scheme, where f(x, y) is a polynomial. The solution of this problem has many applications in CAGD.
A method for analysis of C 1 continuity of subdivision surfaces
 SIAM J. Numer. Anal
, 1998
"... Abstract. A sufficient condition for C 1continuity of subdivision surfaces was proposed by Reif [Comput. Aided Geom. Design, 12(1995), pp. 153–174.] and extended to a more general setting in [D. Zorin, Constr. Approx., accepted for publication]. In both cases, the analysis of C 1continuity is redu ..."
Abstract

Cited by 24 (5 self)
 Add to MetaCart
Abstract. A sufficient condition for C 1continuity of subdivision surfaces was proposed by Reif [Comput. Aided Geom. Design, 12(1995), pp. 153–174.] and extended to a more general setting in [D. Zorin, Constr. Approx., accepted for publication]. In both cases, the analysis of C 1continuity is reduced to establishing injectivity and regularity of a characteristic map. In all known proofs of C 1continuity, explicit representation of the limit surface on an annular region was used to establish regularity, and a variety of relatively complex techniques were used to establish injectivity. We propose a new approach to this problem: we show that for a general class of subdivision schemes, regularity can be inferred from the properties of a sufficiently close linear approximation, and injectivity can be verified by computing the index of a curve. An additional advantage of our approach is that it allows us to prove C 1continuity for all valences of vertices, rather than for an arbitrarily large but finite number of valences. As an application, we use our method to analyze C 1continuity of most stationary subdivision schemes known to us, including interpolating butterfly and modified butterfly schemes, as well as the Kobbelt’s interpolating scheme for quadrilateral meshes.
Guaranteed Nonlinear Estimation Using Constraint Propagation on Sets
, 2001
"... Boundederror estimation is the estimation of the parameter or state vector of a model from experimental data, under the assumption that some suitabl y de...ned errors shoul d bel ong to some prior feasibl e sets. When the model outputs arel inear in the vector to be estimated, a number of methods a ..."
Abstract

Cited by 23 (12 self)
 Add to MetaCart
Boundederror estimation is the estimation of the parameter or state vector of a model from experimental data, under the assumption that some suitabl y de...ned errors shoul d bel ong to some prior feasibl e sets. When the model outputs arel inear in the vector to be estimated, a number of methods are avail#0 l e to encl ose al# estimates that are consistent with the data into simpl# sets such as el# ipsoids, orthotopes or paral#0xP90O es, thereby providing guaranteed set estimates. In the nonl#x]30 case, the situation is muchl#O4 devel#O ed and there are very few methods that produce such guaranteed estimates. In this paper, the discretetime probl em is cast into the more general framework of constraint satisfaction probl ems.Al# orithms rathercl assical in the area of interval constraint propagation are extended by repl acing interva l# by moregeneral subsets of real vector spaces. This makes it possibl# to propose a new al#9Oq30 m that contracts the feasibl e domains for each uncertain variabl# optimal#O (i.e., no smal# er domain coul d be obtained) and ecientl# . The resul ting methodol#03 isil#34 trated on discretetime nonl#O0O7 state estimation. The state at time k is estimated either from past measurement onl y or from al l measurements assumed to be avai l#bl# from the start. Even in the causal case, prior information on the future val# e of the state and output vectors, due for instance to physical constraints, is readil y taken into account.
A linearthne algorithm that locates local extrema of a function of one ~zriable from interval measurement results
 Interval Computations
, 1993
"... The problem of locating local maxima and minima of a function from approximate measurement results is vital for many physical applications: In spectral analysis, chemical species are identified by locating local maxima of the spectra. In radioastronomy, sources of celestial radio emission, and their ..."
Abstract

Cited by 23 (18 self)
 Add to MetaCart
The problem of locating local maxima and minima of a function from approximate measurement results is vital for many physical applications: In spectral analysis, chemical species are identified by locating local maxima of the spectra. In radioastronomy, sources of celestial radio emission, and their subcomponents, are identified by locating local maxima of the measured brightness of the radio sky. Elementary particles are identified by locating local maxima of the experimental curves. In mathematical terms, we know n numbers x1 < · · · < xn, n values y1,..., yn, value ε> 0, and we know that the values ¯ f(xi) of the unknown function ¯ f(x) at the points xi belong to the intervals Ii = [y − i, y+ i], i = 1,..., n, where y − i = yi − ε and y + i = yi + ε. The set F of all the functions f(x) that satisfy this property can be considered as a function interval (this definition was, in essence, first proposed by R. Moore). We say that an interval I locates a local maximum if all functions f ∈ F attain a local maximum at some point from I. So, the problem is to generate intervals I1,..., Ik that locate local maxima. Evidently, if I locates a local maximum, then any bigger interval J ⊃ I also locates this maximum. We want to find the smallest possible location I. We propose an algorithm that finds the smallest possible locations in linear time (i.e., in time that is ≤ Cn for some C).