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41
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 36 (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 16 (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 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...
Nonlinear integer programming
 DISC. OPTIM
, 2009
"... Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject to integrality requirements for the variables. This chapt ..."
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Cited by 13 (5 self)
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Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject to integrality requirements for the variables. This chapter is dedicated to this topic. The primary goal is a study of a simple version of general nonlinear integer problems, where all constraints are still linear. Our focus is on the computational complexity of the problem, which varies significantly with the type of nonlinear objective function in combination with the underlying combinatorial structure. Numerous boundary cases of complexity emerge, which sometimes surprisingly lead even to polynomial time algorithms. We also cover recent successful approaches for more general classes of problems. Though no positive theoretical efficiency results are available, nor are they likely to ever be available, these seem to be the currently most successful and interesting approaches for solving practical problems. It is our belief that the study of algorithms motivated by theoretical considerations
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.
Disjuntive multiperiod optimization methods for design and planning of chemical process systems
 Computers and Chemical Engineering
, 1999
"... In this paper, we present a general disjunctive multiperiod nonlinear optimization model, which incorporates design, as well as operation and expansion planning, and takes into account the corresponding costs incurred in each time period. This model is formulated with the use of logic and disjunctiv ..."
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Cited by 8 (3 self)
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In this paper, we present a general disjunctive multiperiod nonlinear optimization model, which incorporates design, as well as operation and expansion planning, and takes into account the corresponding costs incurred in each time period. This model is formulated with the use of logic and disjunctive programming, and includes Boolean variables for design, operation planning and expansion planning. We propose two algorithms for the solution of these problems. The first is a logicbased Outer Approximation (OA) algorithm for multiperiod problems. The second is a bilevel decomposition algorithm, that exploits the problem structure by decomposing it into an upper level design problem and a lower level operation and expansion planning problem, each of which is solved with the logicbased OA algorithm. Applications are considered in the areas of design and planning of process networks, as well as retrofit design for multiproduct batch plants. The results show that the disjunctive logicbased OA algorithm performs best for small problems, while the disjunctive bilevel decomposition algorithm is superior for larger problems. In both cases, a significant decrease in MILP solution time and total solution time is achieved compared to DICOPT++. Results also show that problems with a significant number of time periods can be solved in
MixedInteger Nonlinear Optimization in Process Synthesis
, 1998
"... The use of networks allows the representation of a variety of important engineering problems. The treatment of a particular class of network applications, the process synthesis problem, is exposed in this paper. Process Synthesis seeks to develop systematically process flowsheets that convert raw ma ..."
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Cited by 7 (0 self)
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The use of networks allows the representation of a variety of important engineering problems. The treatment of a particular class of network applications, the process synthesis problem, is exposed in this paper. Process Synthesis seeks to develop systematically process flowsheets that convert raw materials into desired products. In recent years, the optimization approach to process synthesis has shown promise in tackling this challenge. It requires the development of a network of interconnected units, the process superstructure, that represents the alternative process flowsheets. The mathematical modeling of the superstructure has a mixed set of binary and continuous variables and results in a mixedinteger optimization model. Due to the nonlinearity of chemical models, these problems are generally classified as MixedInteger Nonlinear Programming (MINLP) problems. A number of local optimization algorithms, developed for the solution of this class of problems, are presented in this pap...