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LP Strategy for IntervalNewton Method in Deterministic Global Optimization
, 2004
"... A strategy is described for using linear programming (LP) to bound the solution set of the linear interval equation system that must be solved in the context of the intervalNewton method for deterministic global optimization. An implementation of this technique is described in detail, and several i ..."
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Cited by 9 (3 self)
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A strategy is described for using linear programming (LP) to bound the solution set of the linear interval equation system that must be solved in the context of the intervalNewton method for deterministic global optimization. An implementation of this technique is described in detail, and several important issues are considered. These include selection of the interval corner required by the LP strategy, and determination of rigorous bounds on the solutions of the LP problems. The impact of using a local minimizer for updating the upper bound on the global minimum in this context is also considered. The procedure based on these techniques, LISS LP, is demonstrated using several global optimization problems, with focus on problems arising in chemical engineering. Problems with a very large number of local optima can be effectively solved, as well as problems with a relatively large number of variables.
Advances in Interval Methods for Deterministic Global Optimization in Chemical Engineering
, 2003
"... In recent years, it has been shown that strategies based on an intervalNewton approach can be used to reliably solve a variety of nonlinear equation solving and optimization problems in chemical process engineering, including problems in parameter estimation and in the computation of phase behavior ..."
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Cited by 6 (4 self)
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In recent years, it has been shown that strategies based on an intervalNewton approach can be used to reliably solve a variety of nonlinear equation solving and optimization problems in chemical process engineering, including problems in parameter estimation and in the computation of phase behavior. These strategies provide a mathematical and computational guarantee either that all solutions have been located in an equation solving problem or that the global optimum has been found in an optimization problem. The primary drawback to this approach is the potentially high computational cost. In this paper, we consider strategies for bounding the solution set of the linear interval equation system that must be solved in the context of the intervalNewton method. Recent preconditioning techniques for this purpose are reviewed, and a new bounding approach based on the use of linear programming (LP) techniques is presented. Using this approach it is possible to determine the desired bounds exactly (within round out), leading to significant overall improvements in computational efficiency. These techniques will be demonstrated using several global optimization problems, with focus on problems arising in chemical engineering, including parameter estimation and molecular modeling. These problems range in size from under ten variables to over two hundred, and are solved deterministically using the interval methodology.
Reliable Computation of Equilibrium States and Bifurcations in Food Chain Models
"... Food chains and webs in the environment can be modeled by systems of ordinary differential equations that approximate species or functional feeding group behavior with a variety of functional responses. We present here a new methodology for computing all equilibrium states and bifurcations of equili ..."
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Cited by 4 (2 self)
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Food chains and webs in the environment can be modeled by systems of ordinary differential equations that approximate species or functional feeding group behavior with a variety of functional responses. We present here a new methodology for computing all equilibrium states and bifurcations of equilibria in food chain models. The methodology used is based on interval analysis, in particular an intervalNewton/generalizedbisection algorithm that provides a mathematical and computational guarantee that all roots of a nonlinear equation system are enclosed. The procedure is initializationindependent, and thus requires no a priori insights concerning the number of equilibrium states and bifurcations of equilibria or their approximate locations. The technique is tested using several example problems involving tritrophic food chains.
Interval Methods for Nonlinear Equation Solving Applications
"... Interval analysis provides techniques that make it possible to determine all solutions to a nonlinear algebraic equation system and to do so with mathematical and computational certainty. Such methods are based on the processing of granules in the form of intervals and thus can be regarded as one f ..."
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Cited by 2 (0 self)
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Interval analysis provides techniques that make it possible to determine all solutions to a nonlinear algebraic equation system and to do so with mathematical and computational certainty. Such methods are based on the processing of granules in the form of intervals and thus can be regarded as one facet of granular computing. We review here some of the key concepts used in these methods and then focus on some specific application areas, namely ecological modeling, transition state analysis, and the modeling of phase equilibrium. 1
estimation using errorinvariables
"... global optimization for data reconciliation and parameter ..."
LPBASED STRATEGY FOR MODELING AND OPTIMIZATION USING INTERVAL METHODS
"... The intervalNewton approach provides the power to solve nonlinear equation solving and global optimization problems with complete mathematical and computational certainty. The primary drawback to this approach is that computation time requirements may become quite high. In this paper, a strategy fo ..."
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The intervalNewton approach provides the power to solve nonlinear equation solving and global optimization problems with complete mathematical and computational certainty. The primary drawback to this approach is that computation time requirements may become quite high. In this paper, a strategy for using linear programming (LP) techniques to improve computational efficiency is considered. In particular, an LP strategy is used to determine exact (within round out) bounds on the solution set of the linear interval equation system that must be solved in the context of the intervalNewton method. The strategy is tested using global optimization problems arising in parameter estimation based on the errorinvariables approach.
Global optimization in the 21st century: Advances and challenges
, 2005
"... This paper presents an overview of the research progress in global optimization during the last 5 years (1998–2003), and a brief account of our recent research contributions. The review part covers the areas of (a) twice continuously differentiable nonlinear optimization, (b) mixedinteger nonlinear ..."
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This paper presents an overview of the research progress in global optimization during the last 5 years (1998–2003), and a brief account of our recent research contributions. The review part covers the areas of (a) twice continuously differentiable nonlinear optimization, (b) mixedinteger nonlinear optimization, (c) optimization with differentialalgebraic models, (d) optimization with greybox/blackbox/nonfactorable models, and (e) bilevel nonlinear optimization. Our research contributions part focuses on (i) improved convex underestimation approaches that include convex envelope results for multilinear functions, convex relaxation results for trigonometric functions, and a piecewise quadratic convex underestimator for twice continuously differentiable functions, and (ii) the recently proposed novel generalized �BB framework. Computational studies will illustrate the potential of these advances.