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14
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...
Exclusion Regions for Systems of Equations
 SIAM J. NUM. ANALYSIS
, 2003
"... Branch and bound methods for nding all zeros of a nonlinear system of equations in a box frequently have the diculty that subboxes containing no solution cannot be easily eliminated if there is a nearby zero outside the box. This has the eect that near each zero, many small boxes are created by repe ..."
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Cited by 11 (7 self)
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Branch and bound methods for nding all zeros of a nonlinear system of equations in a box frequently have the diculty that subboxes containing no solution cannot be easily eliminated if there is a nearby zero outside the box. This has the eect that near each zero, many small boxes are created by repeated splitting, whose processing may dominate the total work spent on the global search. This paper discusses the reasons for the occurrence of this socalled cluster eect, and how to reduce the cluster eect 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 regions are traditionally constructed using uniqueness tests based on the Krawczyk operator or the Kantorovich theorem. These results are reviewed; moreover, re nements are proved that signi cantly enlarge the size of the exclusion region. Existence and uniqueness tests are also given.
New Interval Methodologies for Reliable Chemical Process Modeling
 COMPUT. CHEM. ENG. 2002
, 2002
"... The use of interval methods, in particular intervalNewton/generalizedbisection techniques, provides an approach that is mathematically and computationally guaranteed to reliably solve difficult nonlinear equation solving and global optimization problems, such as those that arise in chemical proces ..."
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Cited by 10 (8 self)
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The use of interval methods, in particular intervalNewton/generalizedbisection techniques, provides an approach that is mathematically and computationally guaranteed to reliably solve difficult nonlinear equation solving and global optimization problems, such as those that arise in chemical process modeling. The most significant drawback of the currently used interval methods is the potentially high computational cost that must be paid to obtain the mathematical and computational guarantees of certainty. New methodologies are described here for improving the efficiency of the interval approach. In particular, a new hybrid preconditioning strategy, in which a simple pivoting preconditioner is used in combination with the standard inversemidpoint method, is presented, as is a new scheme for selection of the real point used in formulating the intervalNewton equation. These techniques can be implemented with relatively little computational overhead, and lead to a large reduction in the number of subintervals that must be tested during the intervalNewton procedure. Tests on a variety of problems arising in chemical process modeling have shown that the new methodologies lead to substantial reductions in computation time requirements, in many cases by multiple orders of magnitude.
Improving the efficiency of exclusion algorithms
 Advances in Geometry
, 2001
"... Exclusion algorithms are a wellknown 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 different viewpoint. In particular, such algorithms seem to be very... ..."
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Cited by 9 (1 self)
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Exclusion algorithms are a wellknown 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 different viewpoint. In particular, such algorithms seem to be very...
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...
Globsol: History, composition, and advice on use
 In Global Optimization and Constraint Satisfaction, Lecture Notes in Computer Science
, 2003
"... Abstract. The GlobSol software package combines various ideas from interval analysis, automatic differentiation, and constraint propagation to provide verified solutions to unconstrained and constrained global optimization problems. After briefly reviewing some of these techniques and GlobSol’s deve ..."
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Cited by 8 (4 self)
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Abstract. The GlobSol software package combines various ideas from interval analysis, automatic differentiation, and constraint propagation to provide verified solutions to unconstrained and constrained global optimization problems. After briefly reviewing some of these techniques and GlobSol’s development history, we provide the first overall description of GlobSol’s algorithm. Giving advice on use, we point out strengths and weaknesses in GlobSol’s approaches. Through examples, we show how to configure and use GlobSol.
On Proving Existence of Feasible Points in Equality Constrained Optimization Problems
 Mathematical Programming
, 1995
"... Various algorithms can compute approximate feasible points or approximate solutions to equality and bound constrained optimization problems. In exhaustive search algorithms for global optimizers and other contexts, it is of interest to construct bounds around such approximate feasible points, then t ..."
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Cited by 8 (5 self)
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Various algorithms can compute approximate feasible points or approximate solutions to equality and bound constrained optimization problems. In exhaustive search algorithms for global optimizers and other contexts, it is of interest to construct bounds around such approximate feasible points, then to verify (computationally but rigorously) that an actual feasible point exists within these bounds. Hansen and others have proposed techniques for proving the existence of feasible points within given bounds, but practical implementations have not, to our knowledge, previously been described. Various alternatives are possible in such an implementation, and details must be carefully considered. Also, in addition to Hansen's technique for handling the underdetermined case, it is important to handle the overdetermined case, when the approximate feasible point corresponds to a point with many active bound constraints. The basic ideas, along with experimental results from an actual implementation...
On Verifying Feasibility in Equality Constrained Optimization Problems
, 1996
"... Techniques for verifying feasibility of equality constraints are presented. The underlying verification procedures are similar to a proposed algorithm of Hansen, but various possibilities, as well as additional procedures for handling bound constraints, are investigated. The overall scheme differs f ..."
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Cited by 7 (3 self)
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Techniques for verifying feasibility of equality constraints are presented. The underlying verification procedures are similar to a proposed algorithm of Hansen, but various possibilities, as well as additional procedures for handling bound constraints, are investigated. The overall scheme differs from some algorithms in that it rigorously verifies exact (rather than approximate) feasibility. The scheme starts with an approximate feasible point, then constructs a box (i.e. a set of tolerances) about this point within which it is rigorously verified that a feasible point exists. Alternate ways of proceeding are compared, and numerical results on a set of test problems appear.
On the Complexity of Exclusion Algorithms for Optimization
 J. Complexity
, 2000
"... Exclusion algorithms are a wellknown tool in the area of interval analysis for finding all solutions of a system of nonlinear equations or for finding the global minimum of a function over a compact domain. The present paper discusses a new class of tests for such algorithms in the context of globa ..."
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Cited by 6 (1 self)
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Exclusion algorithms are a wellknown tool in the area of interval analysis for finding all solutions of a system of nonlinear equations or for finding the global minimum of a function over 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 wellknown 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 lowdimensional, but highly nonlinear systems which have many solutions, or for finding the global minimum of a function of a few variables in the presence of many local minima. Such systems occur, e.g., in me...