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27
Numerica: a Modeling Language for Global Optimization
, 1997
"... Introduction Many science and engineering applications require the user to find solutions to systems of nonlinear constraints over real numbers or to optimize a nonlinear function subject to nonlinear constraints. This includes applications such the modeling of chemical engineering processes and of ..."
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Cited by 170 (11 self)
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Introduction Many science and engineering applications require the user to find solutions to systems of nonlinear constraints over real numbers or to optimize a nonlinear function subject to nonlinear constraints. This includes applications such the modeling of chemical engineering processes and of electrical circuits, robot kinematics, chemical equilibrium problems, and design problems (e.g., nuclear reactor design). The field of global optimization is the study of methods to find all solutions to systems of nonlinear constraints and all global optima to optimization problems. Nonlinear problems raise many issues from a computation standpoint. On the one hand, deciding if a set of polynomial constraints has a solution is NPhard. In fact, Canny [ Canny, 1988 ] and Renegar [ Renegar, 1988 ] have shown that the problem is in PSPACE and it is not known whether the problem lies in NP. Nonlinear programming problems can be so hard that some methods are designed only to solve probl
Solving Polynomial Systems Using a Branch and Prune Approach
 SIAM Journal on Numerical Analysis
, 1997
"... This paper presents Newton, a branch & prune algorithm to find all isolated solutions of a system of polynomial constraints. Newton can be characterized as a global search method which uses intervals for numerical correctness and for pruning the search space early. The pruning in Newton consists in ..."
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Cited by 101 (7 self)
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This paper presents Newton, a branch & prune algorithm to find all isolated solutions of a system of polynomial constraints. Newton can be characterized as a global search method which uses intervals for numerical correctness and for pruning the search space early. The pruning in Newton consists in enforcing at each node of the search tree a unique local consistency condition, called boxconsistency, which approximates the notion of arcconsistency wellknown in artificial intelligence. Boxconsistency is parametrized by an interval extension of the constraint and can be instantiated to produce the HansenSegupta's narrowing operator (used in interval methods) as well as new operators which are more effective when the computation is far from a solution. Newton has been evaluated on a variety of benchmarks from kinematics, chemistry, combustion, economics, and mechanics. On these benchmarks, it outperforms the interval methods we are aware of and compares well with stateoftheart continuation methods. Limitations of Newton (e.g., a sensitivity to the size of the initial intervals on some problems) are also discussed. Of particular interest is the mathematical and programming simplicity of the method.
Solving Geometric Constraints By Homotopy
 IEEE Trans on Visualization and Computer Graphics
, 1996
"... Geometric modeling by constraints yields systems of equations. They are classically solved by NewtonRaphson's iteration, from a starting guess interactively provided by the designer. However, this method may fail to converge, or may converge to an unwanted solution after a `chaotic' behaviour. This ..."
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Cited by 22 (1 self)
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Geometric modeling by constraints yields systems of equations. They are classically solved by NewtonRaphson's iteration, from a starting guess interactively provided by the designer. However, this method may fail to converge, or may converge to an unwanted solution after a `chaotic' behaviour. This paper claims that, in such cases, the homotopic method is much more satisfactory. 1 INTRODUCTION In CAD, geometric modeling by constraints enables users to describe geometric objects such as points, lines, circles, conics, B'ezier curves, etc in 2D and planes, quadrics, tori, B'ezier patches, etc in 3D, by geometric constraints, ie distances or angles between elements, incidence or tangency relations : : : This modeling yields large systems of equations, typically algebraic ones. The problem is then to solve such constraint systems. Since the seminal work of Sutherland [Sut63], a lot of research has been done on this topic. We roughly classify resolution methods for constraint systems in...
Solving the 6R inverse position problem using a genericcase solution methodology
 Mech. Mach. Theory
, 1991
"... AlmU'netThis paper considers the computation of all solutions to the inverse position problem for general sixrevolutejoint manipulators. Instead of reducing the problem to one highly complicated inputoutput equation, we work with a system of I I very simple polynomial equations. Although the to ..."
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Cited by 21 (1 self)
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AlmU'netThis paper considers the computation of all solutions to the inverse position problem for general sixrevolutejoint manipulators. Instead of reducing the problem to one highly complicated inputoutput equation, we work with a system of I I very simple polynomial equations. Although the total degree of the system is large (1024), using the "method of the generic case " we show numerically that the generic number of solutions is 16, in agreement with previous works. Moreover. we present an elEcient nun~rical method for finding all 16 solutions, based on coe~icientparameter polynomial continuation. We present a set of 41 test problems, on which the algorithm used an average of le ~ than l0 s of CPU time on an IBM 3703090 in double precision FORTRAN. The methodology applies equally well to other problems in kinematics that can be formulated as polynomial systems. I.
Advances in Polynomial Continuation for Solving Problems in Kinematics
, 2004
"... For many mechanical systems, including nearly all robotic manipulators, the set of possible configurations that the links may assume can be described by a system of polynomial equations. Thus, solving such systems is central to many problems in analyzing the motion of a mechanism or in designing a m ..."
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Cited by 16 (8 self)
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For many mechanical systems, including nearly all robotic manipulators, the set of possible configurations that the links may assume can be described by a system of polynomial equations. Thus, solving such systems is central to many problems in analyzing the motion of a mechanism or in designing a mechanism to achieve a desired motion. This paper describes techniques, based on polynomial continuation, for numerically solving such systems. Whereas in the past, these techniques were focused on finding isolated roots, we now address the treatment of systems having higherdimensional solution sets. Special attention is given to cases of exceptional mechanisms, which have a higher degree of freedom of motion than predicted by their mobility. In fact, such mechanisms often have several disjoint assembly modes, and the degree of freedom of motion is not necessarily the same in each mode. Our algorithms identify all such assembly modes, determine their dimension and degree, and give sample points on each.
Theory of globally convergent probabilityone homotopies for nonlinear programming
 SIAM Journal on Optimization
, 2000
"... Abstract. For many years globally convergent probabilityone homotopy methods have been remarkably successful on difficult realistic engineering optimization problems, most of which were attacked by homotopy methods because other optimization algorithms failed or were ineffective. Convergence theory ..."
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Cited by 16 (4 self)
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Abstract. For many years globally convergent probabilityone homotopy methods have been remarkably successful on difficult realistic engineering optimization problems, most of which were attacked by homotopy methods because other optimization algorithms failed or were ineffective. Convergence theory has been derived for a few particular problems, and considerable fixed point theory exists, but generally convergence theory for the homotopy maps used in practice for nonlinear constrained optimization has been lacking. This paper derives some probabilityone homotopy convergence theorems for unconstrained and inequality constrained optimization, for linear and nonlinear inequality constraints, and with and without convexity. Some insight is provided into why the homotopies used in engineering practice are so successful, and why this success is more than dumb luck. By presenting the theory as variations on a prototype probabilityone homotopy convergence theorem, the essence of such convergence theory is elucidated.
A list of matrix flows with applications
 in Hamiltonian and Gradients Flows, Algorithms and Control
, 1994
"... Many mathematical problems, such as existence questions, are studied by using an appropriate realization process, either iteratively or continuously. This article is a collection of di erential equations that have been proposed as special continuous realization processes. In some cases, there are re ..."
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Cited by 15 (1 self)
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Many mathematical problems, such as existence questions, are studied by using an appropriate realization process, either iteratively or continuously. This article is a collection of di erential equations that have been proposed as special continuous realization processes. In some cases, there are remarkable connections betweensmooth ows and discrete numerical algorithms. In other cases, the ow approach seems advantageous in tackling very di cult problems. The ow approach has potential applications ranging from new development ofnumerical algorithms to the theoretical solution of open problems. Various aspects of the recent development and applications of the ow approach are reviewed in this article. 1
Efficient Variable Elimination Using Resultants
, 1996
"... A new method for eliminating variables from polynomial equations is proposed, analyzed, evaluated and applied. Problems from many applications involve computation of resultants  polynomials obtained after eliminating variables from polynomials. Therefore effective elimination procedures require ef ..."
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Cited by 13 (1 self)
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A new method for eliminating variables from polynomial equations is proposed, analyzed, evaluated and applied. Problems from many applications involve computation of resultants  polynomials obtained after eliminating variables from polynomials. Therefore effective elimination procedures require efficient methods to compute resultants, especially those exploiting sparsity which typically exists in most realworld applications. A new method for computing resultants is proposed. This method exploits the sparsity in the problem, despite being based on a classical formulation by Dixon. It also extends the Dixon formulation for bidegree polynomials to compute resultants of most multivariate polynomial systems, while implicitly exploiting the sparse structure of the input polynomial system. Our work, for the first time, links the classical Dixon formulation to the modern line of sparsity analysis based on Newton polytopes. This link enables us to (i) devise an algorithm for directly inter...
An Approach for Solving Systems of Parametric Polynomial Equations
, 1993
"... An approach for solving nonlinear polynomial equations involving parameters is proposed. A distinction is made between parameters and variables. The objective is to generate from a system of parametric equations, solved forms from which solutions for specific values of parameters can be obtained wit ..."
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Cited by 11 (3 self)
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An approach for solving nonlinear polynomial equations involving parameters is proposed. A distinction is made between parameters and variables. The objective is to generate from a system of parametric equations, solved forms from which solutions for specific values of parameters can be obtained without much additional computations. It should be possible to analyze the parametrized solved forms so that it can be determined for different parameter values whether there are infinitely many solutions, finitely many solutions, or no solutions at all. The approach is illustrated for two different symbolic methods for solving parametric equations  Grobner basis computations and characteristic set computations. These methods are illustrated on a number of examples. 1.2 Introduction Many complex phenomena can be modeled using nonlinear polynomial equations. Examples include imaging transformations in computer vision, computing geometric invariants, geometric and solid modeling, constraintb...