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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.
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 ..."
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Cited by 25 (6 self)
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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...
Safe Bounds in Linear and MixedInteger Programming
 Math. Prog
"... Current mixedinteger linear programming solvers are based on linear programming routines that use floating point arithmetic. Occasionally, this leads to wrong solutions, even for problems where all coefficients and all solution components are small integers. It is shown how, using directed rounding ..."
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Cited by 22 (2 self)
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Current mixedinteger linear programming solvers are based on linear programming routines that use floating point arithmetic. Occasionally, this leads to wrong solutions, even for problems where all coefficients and all solution components are small integers. It is shown how, using directed rounding and interval arithmetic, cheap pre and postprocessing of the linear programs arising in a branchandcut framework can guarantee that no solution is lost, at least for mixedinteger programs in which all variables can be bounded rigorously by bounds of reasonable size.
Numerical Validation of Solutions of Linear Complementarity Problems
 Numer. Math
, 1997
"... This paper proposes a validation method for solutions of linear complementarity problems. The validation procedure consists in two sufficient conditions that can be tested on a digital computer. If the first condition is satisfied then a given multidimensional interval centered at an approximate sol ..."
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Cited by 14 (8 self)
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This paper proposes a validation method for solutions of linear complementarity problems. The validation procedure consists in two sufficient conditions that can be tested on a digital computer. If the first condition is satisfied then a given multidimensional interval centered at an approximate solution of the problem is guaranteed to contain an exact solution. If the second condition is satisfied then the multidimensional interval is guaranteed to contain no exact solution. This study is based on the mean value theorem for absolutely continuous functions and the reformulation of linear complementarity problems as nonsmooth nonlinear systems of equations. 1 Introduction Linear Complementarity Problems (LCP) model many important problems in engineering, management and economics. Furthermore linear and quadratic programming problems can be written as LCP. Several algorithms have been developed for solving LCP [11, 21, 22, 25, 26, 31], but few validation methods have been studied to giv...
On Symmetric Solution Sets
, 2002
"... Given an n × n interval matrix [A] and an interval vector [b] with n components we present an overview on existing results on the solution set S sym of linear systems of equations Ax = b with symmetric matrices A 2 [A] and vectors b 2 [b]. Similarly we consider the set E sym of eigenpairs asso ..."
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Cited by 13 (10 self)
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Given an n × n interval matrix [A] and an interval vector [b] with n components we present an overview on existing results on the solution set S sym of linear systems of equations Ax = b with symmetric matrices A 2 [A] and vectors b 2 [b]. Similarly we consider the set E sym of eigenpairs associated with the symmetric matrices A 2 [A]. We report on characterizations of S sym by means of inequalities, by means of intersection of sets, and by an approach which is generalizable to more general dependencies of the entries. We also recall two methods for enclosing S sym by means of interval vectors, and we mention a characterization of E sym .
On the Shape of the Symmetric, Persymmetric and SkewSymmetric Solution Set
, 1997
"... We present a characterization of the solution set S, the symmetric solution set Ssym , the persymmetric solution set Sper and the skewsymmetric solution set S skew of real linear systems Ax = b with the n × n coefficient matrix A varying between a lower bound A and an upper bound A, and with ..."
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Cited by 12 (5 self)
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We present a characterization of the solution set S, the symmetric solution set Ssym , the persymmetric solution set Sper and the skewsymmetric solution set S skew of real linear systems Ax = b with the n × n coefficient matrix A varying between a lower bound A and an upper bound A, and with b similarly varying between b; b. We show that in each orthant the sets Ssym , Sper and S skew , respectively, are the intersection of S with sets the boundaries of which are quadrics.
Safe bounds in linear and mixedinteger linear programming
 Math. Programming
, 2004
"... Abstract. Current mixedinteger linear programming solvers are based on linear programming routines that use floatingpoint arithmetic. Occasionally, this leads to wrong solutions, even for problems where all coefficients and all solution components are small integers. An example is given where many ..."
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Cited by 12 (0 self)
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Abstract. Current mixedinteger linear programming solvers are based on linear programming routines that use floatingpoint arithmetic. Occasionally, this leads to wrong solutions, even for problems where all coefficients and all solution components are small integers. An example is given where many stateoftheart MILP solvers fail. It is then shown how, using directed rounding and interval arithmetic, cheap pre and postprocessing of the linear programs arising in a branchandcut framework can guarantee that no solution is lost, at least for mixedinteger programs in which all variables can be bounded rigorously by bounds of reasonable size.
Guaranteed Error Bounds for Ordinary Differential Equations
 In Theory of Numerics in Ordinary and Partial Differential Equations
, 1994
"... Hamming once said, "The purpose of computing is insight, not numbers." If that is so, then the speed of our computers should be measured in insights per year, not operations per second. One key insight we wish from nearly all computing in engineering and scientific applications is, "How accurate is ..."
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Cited by 10 (0 self)
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Hamming once said, "The purpose of computing is insight, not numbers." If that is so, then the speed of our computers should be measured in insights per year, not operations per second. One key insight we wish from nearly all computing in engineering and scientific applications is, "How accurate is the answer?" Standard numerical analysis has developed techniques of forward and backward error analysis to help provide this insight, but even the best codes for computing approximate answers can be fooled. In contrast, validated computation ffl checks that the hypotheses of appropriate existence and uniqueness theorems are satisfied, ffl uses interval arithmetic with directed rounding to capture truncation and rounding errors in computation, and ffl organizes the computations to obtain as tight an enclosure of the answer as possible. These notes for a series of lectures at the VIth SERC Numerical Analysis Summer School, Leicester University, apply the principles of validated computatio...
Test Results for an Interval Branch and Bound Algorithm for EqualityConstrained Optimization
 In: Computational Methods and Applications, Kluwer
, 1995
"... . Various techniques have been proposed for incorporating constraints in interval branch and bound algorithms for global optimization. However, few reports of practical experience with these techniques have appeared to date. Such experimental results appear here. The underlying implementation includ ..."
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Cited by 8 (1 self)
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. Various techniques have been proposed for incorporating constraints in interval branch and bound algorithms for global optimization. However, few reports of practical experience with these techniques have appeared to date. Such experimental results appear here. The underlying implementation includes use of an approximate optimizer combined with a careful tesselation process and rigorous verification of feasibility. The experiments include comparison of methods of handling bound constraints and comparison of two methods for normalizing Lagrange multipliers. Selected test problems from the Floudas / Pardalos monograph are used, as well as selected unconstrained test problems appearing in reports of interval branch and bound methods for unconstrained global optimization. Keywords: constrained global optimization, verified computations, interval computations, bound constraints, experimental results 1. Introduction We consider the constrained global optimization problem minimize OE(X) s...