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Rigorous filtering using linear relaxations
, 2010
"... This paper presents rigorous filtering methods for continuous constraint satisfaction problems based on linear relaxations. Filtering or pruning stands for reducing the search space of constraint satisfaction problems. Discussed are old and new approaches for rigorously enclosing the solution set o ..."
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Cited by 5 (3 self)
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This paper presents rigorous filtering methods for continuous constraint satisfaction problems based on linear relaxations. Filtering or pruning stands for reducing the search space of constraint satisfaction problems. Discussed are old and new approaches for rigorously enclosing the solution set of linear systems of inequalities, as well as different methods for computing linear relaxations. This allows custom combinations of relaxation and filtering. Care is taken to ensure that all methods correctly account for rounding errors in the computations. Although most of the results apply more generally, strong emphasis is given to relaxing and filtering quadratic constraints, as implemented in the GloptLab environment, which internally exploits a quadratic structure. Demonstrative examples and tests comparing the different linear relaxation methods are also presented.
The Optimization Test Environment
"... Testing is a crucial part of software development in general, and hence also in mathematical programming. Unfortunately, it is often a time consuming and little exciting activity. This naturally motivated us to increase the e ciency in testing solvers for optimization problems and to automatize as m ..."
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Cited by 4 (4 self)
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Testing is a crucial part of software development in general, and hence also in mathematical programming. Unfortunately, it is often a time consuming and little exciting activity. This naturally motivated us to increase the e ciency in testing solvers for optimization problems and to automatize as much of the procedure as possible. Keywords: test environment, optimization, solver benchmarking, solver comparison The testing procedure typically consists of three basic tasks: a) organize test problem sets, also called test libraries; b) solve selected test problems with selected solvers; c) analyze, check and compare the results. The Test Environment is a graphical user interface (GUI) that enables to manage the tasks a) and b) interactively, and task c) automatically. The Test Environment is particularly designed for users who seek to 1. adjust solver parameters, or 2. compare solvers on single problems, or 3. evaluate solvers on suitable test sets.
Improving interval enclosures
, 2009
"... This paper serves as background information for the Vienna proposal for interval standardization, explaining what is needed in practice to make competent use of the interval arithmetic provided by an implementation of the standard to be. Discussed are methods to improve the quality of interval encl ..."
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Cited by 3 (0 self)
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This paper serves as background information for the Vienna proposal for interval standardization, explaining what is needed in practice to make competent use of the interval arithmetic provided by an implementation of the standard to be. Discussed are methods to improve the quality of interval enclosures of the range of a function over a box, considerations of possible hardware support facilitating the implementation of such methods, and the results of a simple interval challenge that I had posed to the reliable computing mailing list on November 26, 2008. Also given is an example of a bound constrained global optimization problem in 4 variables that has a 2dimensional continuum of global minimizers. This makes standard branch and bound codes extremely slow, and therefore may serve as a useful degenerate test problem.
Rigorous enclosures of ellipsoids and directed cholesky factorizations
, 2009
"... This paper discusses the rigorous enclosure of an ellipsoid by a rectangular box, its interval hull, providing a convenient preprocessing step for constrained optimization problems. A quadratic inequality constraint with a positive definite Hessian defines an ellipsoid. The Cholesky factorization ca ..."
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This paper discusses the rigorous enclosure of an ellipsoid by a rectangular box, its interval hull, providing a convenient preprocessing step for constrained optimization problems. A quadratic inequality constraint with a positive definite Hessian defines an ellipsoid. The Cholesky factorization can be used to transform a strictly convex quadratic constraint into a norm inequality, for which the interval hull is easy to compute analytically. In exact arithmetic, the Cholesky factorization of a nonsingular symmetric matrix exists iff the matrix is positive definite. However, to cope efficiently with rounding errors in inexact arithmetic is nontrivial. Numerical tests show that even nearly singular problems can be handled successfully by our techniques. To rigorously account for the rounding errors involved in the computation of the interval hull and to handle quadratic inequality constraints having uncertain coefficients, we define the concept of a directed Cholesky factorization, and give two algorithms for computing one. We also discuss how a directed Cholesky factorization can be used for testing positive definiteness. Some numerical test are given in order to exploit the features and boundaries of the directed Cholesky factorization methods.
The Optimization Test Environment  User manual
, 2010
"... The Test Environment is an interface to efficiently test different optimization solvers. It is designed as a tool for both developers of solver software and practitioners who just look for the best solver for their specific problem class. It enables users to: • Choose and compare diverse solver ro ..."
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The Test Environment is an interface to efficiently test different optimization solvers. It is designed as a tool for both developers of solver software and practitioners who just look for the best solver for their specific problem class. It enables users to: • Choose and compare diverse solver routines; • Organize and solve large test problem sets; • Select interactively subsets of test problem sets; • Perform a statistical analysis of the results, automatically produced as LATEX and PDF output. The Test Environment is free to use for research purposes.
Noname manuscript No. (will
"... be inserted by the editor) Certificates of infeasibility via nonsmooth optimization ..."
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be inserted by the editor) Certificates of infeasibility via nonsmooth optimization
On solving mixedinteger constraint satisfaction problems
"... with unbounded variables ..."
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