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16
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.
The Minpack2 Test Problem Collection
, 1991
"... The Army High Performance Computing Research Center at the University of Minnesota and the Mathematics and Computer Science Division at Argonne National Laboratory are collaborating on the development of the software package MINPACK2. As part of the MINPACK2 project we are developing a collection ..."
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Cited by 47 (5 self)
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The Army High Performance Computing Research Center at the University of Minnesota and the Mathematics and Computer Science Division at Argonne National Laboratory are collaborating on the development of the software package MINPACK2. As part of the MINPACK2 project we are developing a collection of significant optimization problems to serve as test problems for the package. This report describes the problems in the preliminary version of this collection. 1 Introduction The Army High Performance Computing Research Center at the University of Minnesota and the Mathematics and Computer Science Division at Argonne National Laboratory have initiated a collaboration for the development of the software package MINPACK2. As part of the MINPACK2 project, we are developing a collection of significant optimization problems to serve as test problems for the package. This report describes some of the problems in the preliminary version of this collection. Optimization software has often bee...
Universally Quantified Interval Constraints
 PROCEEDINGS OF THE 6TH INTERNATIONAL CONFERENCE ON PRINCIPLES AND PRACTICE OF CONSTRAINT PROGRAMMING
, 2000
"... Nonlinear real constraint systems with universally and/or existentially quantified variables often need be solved in such contexts as control design or sensor planning. To date, these systems are mostly handled by computing a quantifierfree equivalent form by means of Cylindrical Algebraic Decompo ..."
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Cited by 46 (0 self)
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Nonlinear real constraint systems with universally and/or existentially quantified variables often need be solved in such contexts as control design or sensor planning. To date, these systems are mostly handled by computing a quantifierfree equivalent form by means of Cylindrical Algebraic Decomposition (CAD). However, CAD restricts its input to be conjunctions and disjunctions of polynomial constraints with rational coefficients, while some applications such as camera control involve systems with arbitrary forms where time is the only universally quantified variable. In this paper, the handling of universally quantified variables is first related to the computation of innerapproximation of real relations.
Finding All Solutions of Nonlinearly Constrained Systems of Equations
 Journal of Global Optimization
, 1995
"... . A new approach is proposed for finding all fflfeasible solutions for certain classes of nonlinearly constrained systems of equations. By introducing slack variables, the initial problem is transformed into a global optimization problem(P) whose multiple global minimum solutionswith a zero object ..."
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Cited by 30 (14 self)
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. A new approach is proposed for finding all fflfeasible solutions for certain classes of nonlinearly constrained systems of equations. By introducing slack variables, the initial problem is transformed into a global optimization problem(P) whose multiple global minimum solutionswith a zero objectivevalue (if any)correspond to all solutionsof the initial constrainedsystem of equalities. All fflglobally optimal points of (P) are then localized within a set of arbitrarily small disjoint rectangles. This is based on a branch and bound type global optimization algorithm which attains finite fflconvergence to each of the multiple global minima of (P) through the successive refinement of a convexrelaxation of the feasible region and the subsequent solution of a series of nonlinear convex optimization problems. Based on the form of the participating functions, a number of techniques for constructing this convex relaxation are proposed. By taking advantage of the properties of products o...
Global optimization by continuous GRASP
 Optimization Letters
"... ABSTRACT. We introduce a novel global optimization method called Continuous GRASP (CGRASP) which extends Feo and Resende’s greedy randomized adaptive search procedure (GRASP) from the domain of discrete optimization to that of continuous global optimization. This stochastic local search method is s ..."
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Cited by 22 (9 self)
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ABSTRACT. We introduce a novel global optimization method called Continuous GRASP (CGRASP) which extends Feo and Resende’s greedy randomized adaptive search procedure (GRASP) from the domain of discrete optimization to that of continuous global optimization. This stochastic local search method is simple to implement, is widely applicable, and does not make use of derivative information, thus making it a wellsuited approach for solving global optimization problems. We illustrate the effectiveness of the procedure on a set of standard test problems as well as two hard global optimization problems. 1.
Homotopy Continuation Methods For Solving Polynomial Systems
, 1996
"... Homotopy continuation methods have been proven to be reliable for computing numerically approximations to all isolated solutions of polynomial systems. The performance of... ..."
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Cited by 16 (1 self)
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Homotopy continuation methods have been proven to be reliable for computing numerically approximations to all isolated solutions of polynomial systems. The performance of...
UniCalc, a novel approach to solving systems of algebraic equations
 Proceedings of ALP’96, 5th International Conference on Algebraic and Logic Programming
, 1993
"... This paper describes a novel approach to solving systems of algebraic equations and inequalities that is based on subdefinite calculations. The use of these methods makes it possible to solve overdetermined and underdetermined systems, as well as systems with imprecise and incomplete data. The appro ..."
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Cited by 15 (0 self)
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This paper describes a novel approach to solving systems of algebraic equations and inequalities that is based on subdefinite calculations. The use of these methods makes it possible to solve overdetermined and underdetermined systems, as well as systems with imprecise and incomplete data. The approach was implemented with the help of the methods of interval mathematics. The UniCalc solver, also described in this paper, was developed on the basis of this approach. To illustrate the capabilities of UniCalc, we give examples of problems solved with its help. UniCalc: новый подход к решению систем алгебраических уравнений
Newton: Constraint Programming over Nonlinear Constraints
 SCIENCE OF COMPUTER PROGRAMMING
, 1998
"... This paper is an introduction to Newton, a constraint programming language over nonlinear real constraints. Newton originates from an eort to reconcile the declarative nature of constraint logic programming (CLP) languages over intervals with advanced interval techniques developed in numerical analy ..."
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Cited by 8 (3 self)
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This paper is an introduction to Newton, a constraint programming language over nonlinear real constraints. Newton originates from an eort to reconcile the declarative nature of constraint logic programming (CLP) languages over intervals with advanced interval techniques developed in numerical analysis, such as the interval Newton method. Its key conceptual idea is to introduce the notion of boxconsistency, which approximates arcconsistency, a notion wellknown in articial intelligence. Boxconsistency achieves an eective pruning at a reasonable computation cost and generalizes some traditional interval operators. Newton has been applied to numerous applications in science and engineering, including nonlinear equationsolving, unconstrained optimization, and constrained optimization. It is competitive with continuation methods on their equationsolving benchmarks and outperforms the intervalbased methods we are aware of on optimization problems. Key words: Constraint Programming, Nonlinear Programming, Interval Reasoning 1 Introduction Many applications in science and engineering (e.g., chemistry, robotics, economics, mechanics) require nding all isolated solutions to a system of nonlinear real constraints or nding the minimum value of a nonlinear function subject to nonlinear constraints. These problems are dicult due to their inherent computational complexity (i.e., they are NPhard) and due to the numerical issues involved to guarantee correctness (i.e., nding all solutions or the global optimum) and to ensure termination. Preprint submitted to Elsevier Preprint 11 June 2001 Newton is a constraint programming language designed to support this class of applications. It originates from an attempt to reconcile the declarative nature of CLP(Intervals) languag...
A new approach for solving nonlinear equations systems
 IEEE Transactions on Systems, Man and Cybernetics Part A
"... Abstract—This paper proposes a new perspective for solving systems of complex nonlinear equations by simply viewing them as a multiobjective optimization problem. Every equation in the system represents an objective function whose goal is to minimize the difference between the right and left terms o ..."
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Cited by 7 (2 self)
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Abstract—This paper proposes a new perspective for solving systems of complex nonlinear equations by simply viewing them as a multiobjective optimization problem. Every equation in the system represents an objective function whose goal is to minimize the difference between the right and left terms of the corresponding equation. An evolutionary computation technique is applied to solve the problem obtained by transforming the system into a multiobjective optimization problem. The results obtained are compared with a very new technique that is considered as efficient and is also compared with some of the standard techniques that are used for solving nonlinear equations systems. Several wellknown and difficult applications (such as interval arithmetic benchmark, kinematic application, neuropsychology application, combustion application, and chemical equilibrium application) are considered for testing the performance of the new approach. Empirical results reveal that the proposed approach is able to deal with highdimensional equations systems. Index Terms—Computational intelligence, evolutionary multiobjective optimization, metaheuristics, nonlinear equation systems. I.
Three Cuts for Accelerated Interval Propagation
 MIT, Artif. Intell. Lab
, 1995
"... This paper addresses the problem of nonlinear multivariate root finding. In an earlier paper we describe a system called Newton which finds roots of systems of nonlinear equations using refinements of interval methods. The refinements are inspired by AI constraint propagation techniques. Newton is c ..."
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Cited by 4 (0 self)
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This paper addresses the problem of nonlinear multivariate root finding. In an earlier paper we describe a system called Newton which finds roots of systems of nonlinear equations using refinements of interval methods. The refinements are inspired by AI constraint propagation techniques. Newton is competitive with continuation methods on most benchmarks and can handle a variety of cases that are infeasible for continuation methods. This paper presents three "cuts" which we believe capture the essential theoretical ideas behind the success of Newton. This paper describes the cuts in a concise and abstract manner which, we believe, makes the theoretical content of our work more apparent. Any implementation will need to adopt some heuristic control mechanism. Heuristic control of the cuts is only briefly discussed here. Copyright c fl Massachusetts Institute of Technology, 1995 This report describes research done at the Artificial Intelligence Laboratory of the Massachusetts Institute of ...