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37
Review of nonlinear mixedinteger and disjunctive programming techniques
 Optimization and Engineering
, 2002
"... This paper has as a major objective to present a unified overview and derivation of mixedinteger nonlinear programming (MINLP) techniques, Branch and Bound, OuterApproximation, Generalized Benders and Extended Cutting Plane methods, as applied to nonlinear discrete optimization problems that are ex ..."
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Cited by 55 (15 self)
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This paper has as a major objective to present a unified overview and derivation of mixedinteger nonlinear programming (MINLP) techniques, Branch and Bound, OuterApproximation, Generalized Benders and Extended Cutting Plane methods, as applied to nonlinear discrete optimization problems that are expressed in algebraic form. The solution of MINLP problems with convex functions is presented first, followed by a brief discussion on extensions for the nonconvex case. The solution of logic based representations, known as generalized disjunctive programs, is also described. Theoretical properties are presented, and numerical comparisons on a small process network problem.
Filter Pattern Search Algorithms for Mixed Variable Constrained Optimization Problems
 SIAM Journal on Optimization
, 2004
"... A new class of algorithms for solving nonlinearly constrained mixed variable optimization problems is presented. This class combines and extends the AudetDennis Generalized Pattern Search (GPS) algorithms for bound constrained mixed variable optimization, and their GPSfilter algorithms for gene ..."
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Cited by 37 (8 self)
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A new class of algorithms for solving nonlinearly constrained mixed variable optimization problems is presented. This class combines and extends the AudetDennis Generalized Pattern Search (GPS) algorithms for bound constrained mixed variable optimization, and their GPSfilter algorithms for general nonlinear constraints. In generalizing existing algorithms, new theoretical convergence results are presented that reduce seamlessly to existing results for more specific classes of problems. While no local continuity or smoothness assumptions are required to apply the algorithm, a hierarchy of theoretical convergence results based on the Clarke calculus is given, in which local smoothness dictate what can be proved about certain limit points generated by the algorithm. To demonstrate the usefulness of the algorithm, the algorithm is applied to the design of a loadbearing thermal insulation system. We believe this is the first algorithm with provable convergence results to directly target this class of problems.
New Algorithms for Nonlinear Generalized Disjunctive Programming
 Computers and Chemical Engineering Journal
, 2000
"... Generalized Disjunctive Programming (GDP) has been introduced recently as an alternative model to MINLP for representing discrete/continuous optimization problems. The basic idea of GDP consists of representing discrete decisions in the continuous space with disjunctions, and constraints in the disc ..."
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Cited by 22 (17 self)
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Generalized Disjunctive Programming (GDP) has been introduced recently as an alternative model to MINLP for representing discrete/continuous optimization problems. The basic idea of GDP consists of representing discrete decisions in the continuous space with disjunctions, and constraints in the discrete space with logic propositions. In this paper, we describe a new convex nonlinear relaxation of the nonlinear GDP problem that relies on the use of the convex hull of each of the disjunctions involving nonlinear inequalities. The proposed nonlinear relaxation is used to reformulate the GDP problem as a tight MINLP problem, and for deriving a branch and bound method. Properties of these methods are given, and the relation of this method with the Logic Based OuterApproximation method is established. Numerical results are presented for problems in jobshop scheduling, synthesis of process networks, optimal positioning of new products and batch process design.
A lifted linear programming branchandbound algorithm for mixed integer conic quadratic programs
, 2007
"... This paper develops a linear programming based branchandbound algorithm for mixed integer conic quadratic programs. The algorithm is based on a higher dimensional or lifted polyhedral relaxation of conic quadratic constraints introduced by BenTal and Nemirovski. The algorithm is different from o ..."
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Cited by 19 (0 self)
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This paper develops a linear programming based branchandbound algorithm for mixed integer conic quadratic programs. The algorithm is based on a higher dimensional or lifted polyhedral relaxation of conic quadratic constraints introduced by BenTal and Nemirovski. The algorithm is different from other linear programming based branchandbound algorithms for mixed integer nonlinear programs in that, it is not based on cuts from gradient inequalities and it sometimes branches on integer feasible solutions. The algorithm is tested on a series of portfolio optimization problems. It is shown that it significantly outperforms commercial and open source solvers based on both linear and nonlinear relaxations. Key words: nonlinear integer programming; branch and bound; portfolio optimization History: February 2007. 1.
Global Optimization of MINLP Problems in Process Synthesis and Design
 Computers & Chemical Engineering
, 1997
"... : Two new methodologies for the global optimization of MINLP models, the Special structure Mixed Integer Nonlinear ffBB, SMINffBB, and the General structure Mixed Integer Nonlinear ffBB, GMINffBB, are presented. Their theoretical foundations provide guarantees that the global optimum solution of ..."
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Cited by 16 (6 self)
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: Two new methodologies for the global optimization of MINLP models, the Special structure Mixed Integer Nonlinear ffBB, SMINffBB, and the General structure Mixed Integer Nonlinear ffBB, GMINffBB, are presented. Their theoretical foundations provide guarantees that the global optimum solution of MINLPs involving twicedifferentiable nonconvex functions in the continuous variables can be identified. The conditions imposed on the functionality of the binary variables differ for each method : linear and mixed bilinear terms can be treated with the SMINffBB; mixed nonlinear terms whose continuous relaxation is twicedifferentiable are handled by the GMINffBB. While both algorithms use the concept of a branch & bound tree, they rely on fundamentally different bounding and branching strategies. In the GMINffBB algorithm, lower (upper) bounds at each node result from the solution of convex (nonconvex) MINLPs derived from the original problem. The construction of convex lower bound...
Global Optimization of MixedInteger Nonlinear Problems
 AIChE J
"... Two novel deterministic global optimization algorithms for nonconvex mixedinteger problems (MINLPs) are proposed, using the advances of the ffBB algorithm for nonconvex NLPs Adjiman et al. (1998a). The Special Structure MixedInteger ffBB algorithm (SMINffBB addresses problems with nonconvexities ..."
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Cited by 14 (2 self)
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Two novel deterministic global optimization algorithms for nonconvex mixedinteger problems (MINLPs) are proposed, using the advances of the ffBB algorithm for nonconvex NLPs Adjiman et al. (1998a). The Special Structure MixedInteger ffBB algorithm (SMINffBB addresses problems with nonconvexities in the continuous variables and linear and mixedbilinear participation of the binary variables. The General Structure MixedInteger ffBB algorithm (GMINffBB), is applicable to a very general class of problems for which the continuous relaxation is twice continuously differentiable. Both algorithms are developed using the concepts of branchandbound, but they differ in their approach to each of the required steps. The SMINffBB algorithm is based on the convex underestimation of the continuous functions while the GMINffBB algorithm is centered around the convex relaxation of the entire problem. Both algorithms rely on optimization or interval based variable bound updates to enhance effici...
Generalized Convex Disjunctive Programming: Nonlinear Convex Hull Relaxation
 Computational Optimization and Applications
, 2001
"... Generalized Disjunctive Programming (GDP) has been introduced recently as an alternative to mixedinteger programming for represent ing discrete/continuous optimization problems. The basic idea of GDP consists of representing these problems in terms of sets of disjunctions in the continuous spa ..."
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Cited by 12 (3 self)
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Generalized Disjunctive Programming (GDP) has been introduced recently as an alternative to mixedinteger programming for represent ing discrete/continuous optimization problems. The basic idea of GDP consists of representing these problems in terms of sets of disjunctions in the continuous space, and logic propositions in terms of Boolean variables. In this paper we consider GDP problems involving convex nonlinear inequalities in the disjunctions. Based on the work by Stubbs and Mehrotra [19] and Ceria and Soares [5], we propose a con vex nonlinear relaxation of the nonlinear convex GDP problem that relies on the convex hull of each of the disjunctions that is obtained by variable disaggregation and reformulation of the inequalities. The proposed nonlinear relaxation is used to formulate the GDP problem as a MixedInteger Nonlinear Programming (MINLP) problem that is shown to be tighter than the conventional "bigM" formulation. A disjunctive branch and bound method is also presented, and numerical results are given for a set of test problems.
Algorithms and software for convex mixed integer nonlinear programs, IMA Volumes
"... Abstract. This paper provides a survey of recent progress and software for solving convex mixed integer nonlinear programs (MINLP)s, where the objective and constraints are defined by convex functions and integrality restrictions are imposed on a subset of the decision variables. Convex MINLPs have ..."
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Cited by 10 (2 self)
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Abstract. This paper provides a survey of recent progress and software for solving convex mixed integer nonlinear programs (MINLP)s, where the objective and constraints are defined by convex functions and integrality restrictions are imposed on a subset of the decision variables. Convex MINLPs have received sustained attention in recent years. By exploiting analogies to wellknown techniques for solving mixed integer linear programs and incorporating these techniques into software, significant improvements have been made in the ability to solve these problems. Key words. Mixed Integer Nonlinear Programming; Branch and Bound; AMS(MOS) subject classifications.
Barttfeld M, “Optimal synthesis of complex distillation columns using rigorous models”, Computers and Chemical Engineering 29
 1203–1215, 2005. 32 Grossmann I.E. and Ruiz J.P., "Generalized Disjunctive Programming: A Framework for Formulation and Alternative Algorithms for MINLP Optimization," IMA Volume 154, Mixed Integer Nonlinear Programming (eds., Jon Lee and Sven Leyffer
, 2011
"... The synthesis of complex distillation columns has remained a major challenge since the pioneering work by Sargent and Gaminibanadara that was reported in 1976. In this paper we first provide a review of recent work for the optimal design of distillation of individual columns using traybytray model ..."
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Cited by 9 (2 self)
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The synthesis of complex distillation columns has remained a major challenge since the pioneering work by Sargent and Gaminibanadara that was reported in 1976. In this paper we first provide a review of recent work for the optimal design of distillation of individual columns using traybytray models. We examine the impact of different representations and models, NLP, MINLP and GDP, as well as the importance of appropriate initialization schemes. We next provide a review of the synthesis of complex column configurations for zeotropic mixtures and discuss different superstructure representations as well as decomposition schemes for tackling these problems. Finally, we briefly discuss extensions for handling azeotropic mixtures, reactive distillation columns and integration in process flowsheets. Numerical examples are presented to demonstrate that effective computational strategies are emerging that are based on disjunctive programming models that are coupled with thermodynamic initialization models and integrated through hierarchical decomposition techniques. 1.