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45
Deterministic global optimization for parameter estimation of dynamic systems
 Industrial and Engineering Chemistry Research
, 2006
"... A method is presented for deterministic global optimization in the estimation of parameters in models of dynamic systems. The method can be implemented as an ɛglobal algorithm, or, by use of the intervalNewton method, as an exact algorithm. In the latter case, the method provides a mathematically ..."
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A method is presented for deterministic global optimization in the estimation of parameters in models of dynamic systems. The method can be implemented as an ɛglobal algorithm, or, by use of the intervalNewton method, as an exact algorithm. In the latter case, the method provides a mathematically guaranteed and computationally validated global optimum in the goodness of fit function. A key feature of the method is the use of a new validated solver for parametric ODEs, which is used to produce guaranteed bounds on the solutions of dynamic systems with intervalvalued parameters, as well as on the first and secondorder sensitivities of the state variables with respect to the parameters. The computational efficiency of the method is demonstrated using several benchmark problems.
Deterministic global optimization of nonlinear dynamic systems
 Eng
"... Author to whom all correspondence should be addressed. ..."
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Author to whom all correspondence should be addressed.
A kriging based method for the solution of mixedinteger nonlinear programs containing blackbox functions
, 2009
"... ..."
On convex relaxations of quadrilinear terms
, 2009
"... The best known method to find exact or at least εapproximate solutions to polynomial programming problems is the spatial BranchandBound algorithm, which rests on computing lower bounds to the value of the objective function to be minimized on each region that it explores. These lower bounds are o ..."
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The best known method to find exact or at least εapproximate solutions to polynomial programming problems is the spatial BranchandBound algorithm, which rests on computing lower bounds to the value of the objective function to be minimized on each region that it explores. These lower bounds are often computed by solving convex relaxations of the original program. Although convex envelopes are explicitly known (via linear inequalities) for bilinear and trilinear terms on arbitrary boxes, such a description is unknown, in general, for multilinear terms of higher order. In this paper, we study convex relaxations of quadrilinear terms. We exploit associativity to rewrite such terms as products of bilinear and trilinear terms. Using a general technique, we establish that, any relaxation for klinear terms that employs a successive use of relaxing bilinear terms (via the bilinear convex envelope) can be improved by employing instead a relaxation of a trilinear term (via the trilinear convex envelope). We present a computational analysis which helps establish which relaxations are strictly tighter, and we apply our findings to two wellstudied applications: the Molecular Distance Geometry Problem and the HartreeFock Problem.
Strong valid inequalities for orthogonal disjunctions and bilinear covering sets
 Mathematical Programming
"... In this paper, we develop a convexification tool that enables the construction of convex hulls for orthogonal disjunctive sets using convex extensions and disjunctive programming techniques. A distinguishing feature of our technique is that, unlike most applications of disjunctive programming, it do ..."
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In this paper, we develop a convexification tool that enables the construction of convex hulls for orthogonal disjunctive sets using convex extensions and disjunctive programming techniques. A distinguishing feature of our technique is that, unlike most applications of disjunctive programming, it does not require the introduction of new variables in the relaxation. We develop and apply a toolbox of results that help in checking the technical assumptions under which the convexification tool can be employed. We demonstrate its applicability in integer programming by deriving the intersection cut for mixedinteger polyhedral sets and the convex hull of certain mixed/pureinteger bilinear sets. We then develop a key result that extends the applicability of the convexification tool to relaxing nonconvex inequalities, which are not naturally disjunctive, by providing sufficient conditions for establishing the convex extension property over the nonnegative orthant. We illustrate the utility of this result by deriving the convex hull of a continuous bilinear covering set over the nonnegative orthant. 1
Challenges in the new millennium: product discovery and design, enterprise and supply chain optimization, global life cycle assessment
 Computers and Chemical Engineering, in print (2004
, 2003
"... Abstract. This paper first provides an overview of the financial state of the process industry, major issues it currently faces, and job placement of chemical engineers in the U.S. These facts combined with an expanded role of Process Systems Engineering, are used to argue that to support the “value ..."
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Abstract. This paper first provides an overview of the financial state of the process industry, major issues it currently faces, and job placement of chemical engineers in the U.S. These facts combined with an expanded role of Process Systems Engineering, are used to argue that to support the “value preservation ” and “value growth ” industry three major future research challenges need to be addressed: Product Discovery and Design, Enterprise and Supply Chain Optimization, and Global Life Cycle Assessment. We provide a brief review of the progress that has been made in these areas, as well as the supporting methods and tools for tackling these problems. Finally, we provide some concluding remarks.
Feasibilitybased bounds tightening via fixed points
"... Abstract. The search tree size of the spatial BranchandBound algorithm for MixedInteger Nonlinear Programming depends on many factors, one of which is the width of the variable ranges at every tree node. A range reduction technique often employed is called Feasibility Based Bounds Tightening, whi ..."
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Abstract. The search tree size of the spatial BranchandBound algorithm for MixedInteger Nonlinear Programming depends on many factors, one of which is the width of the variable ranges at every tree node. A range reduction technique often employed is called Feasibility Based Bounds Tightening, which is known to be practically fast, and is thus deployed at every node of the search tree. From time to time, however, this technique fails to converge to its limit point in finite time, thereby slowing the whole BranchandBound search considerably. In this paper we propose a polynomial time method, based on solving a linear program, for computing the limit point of the Feasibility Based Bounds Tightening algorithm applied to linear equality and inequality constraints. Keywords: global optimization, MINLP, spatial BranchandBound, range reduction, constraint programming. 1
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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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
Comparison of convex relaxations of quadrilinear terms
"... In this paper we compare four different ways to compute a convex linear relaxation of a quadrilinear monomial on a box, analyzing their relative tightness. We computationally compare the quality of the relaxations, and we provide a general theorem on pairwisecomparison of relaxation strength, which ..."
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In this paper we compare four different ways to compute a convex linear relaxation of a quadrilinear monomial on a box, analyzing their relative tightness. We computationally compare the quality of the relaxations, and we provide a general theorem on pairwisecomparison of relaxation strength, which applies to some of our pairs of relaxations for quadrilinear monomials. Our results can be used to configure a spatial BranchandBound global optimization algorithm. We apply our results to the Molecular Distance Geometry Problem, demonstrating the usefulness of the present study. quadrilinear; convex relaxation; reformulation; global optimization, spatial Branch
A Sequential Parametric Convex Approximation Method with Applications to Nonconvex Truss Topology Design Problems
, 2009
"... We describe a general scheme for solving nonconvex optimization problems, where in each iteration the nonconvex feasible set is approximated by an inner convex approximation. The latter is defined using an upper bound on the nonconvex constraint functions. Under appropriate conditions on this upper ..."
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We describe a general scheme for solving nonconvex optimization problems, where in each iteration the nonconvex feasible set is approximated by an inner convex approximation. The latter is defined using an upper bound on the nonconvex constraint functions. Under appropriate conditions on this upper bounding convex function, a monotone convergence to a KKT point is established. The scheme is applied to Truss Topology Design (TTD) problems, where the nonconvex constraints are associated with bounds on displacements and stresses. It is shown that the approximate convex problem solved at each inner iteration can be cast as a conic quadratic programming problem, hence large scale TTD problems can be efficiently solved by the proposed method. 1