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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.
Valid inequalities for mixed integer linear programs
 Mathematical Programming B
, 2006
"... Abstract. This tutorial presents a theory of valid inequalities for mixed integer linear sets. It introduces the necessary tools from polyhedral theory and gives a geometric understanding of several classical families of valid inequalities such as liftandproject cuts, Gomory mixed integer cuts, mi ..."
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Cited by 31 (0 self)
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Abstract. This tutorial presents a theory of valid inequalities for mixed integer linear sets. It introduces the necessary tools from polyhedral theory and gives a geometric understanding of several classical families of valid inequalities such as liftandproject cuts, Gomory mixed integer cuts, mixed integer rounding cuts, split cuts and intersection cuts, and it reveals the relationships between these families. The tutorial also discusses computational aspects of generating the cuts and their strength. Key words: mixed integer linear program, liftandproject, split cut, Gomory cut, mixed integer rounding, elementary closure, polyhedra, union of polyhedra 1.
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.
Convex Programming for Disjunctive Convex Optimization
, 1998
"... Given a finite number of closed convex sets whose algebraic representation is known, we study the problem of optimizing a convex function over the closure of the convex hull of the union of these sets. We derive an algebraic characterization of the feasible region in a higherdimensional space and p ..."
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Cited by 20 (0 self)
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Given a finite number of closed convex sets whose algebraic representation is known, we study the problem of optimizing a convex function over the closure of the convex hull of the union of these sets. We derive an algebraic characterization of the feasible region in a higherdimensional space and propose a solution procedure akin to the interiorpoint approach for convex programming. Research partly supported by NSF HPCC Grant DMS 9527124. Author's address: 417 Uris Hall, Graduate School of Business, Columbia University, New York NY 10027. Email sebas@cumparsita.gsb.columbia.edu. Also affiliated with Computational Optimization Research Center (CORC), Columbia University. y Supported by Subprograma Ciencia e Tecnologia do 2 o Quadro comunit'ario de Apoio grant Praxis XXI/BD/2831/94. Author's address: 804 Uris Hall, Graduate School of Business, Columbia University, New York NY 10027. Email jsoares@corc.ieor.columbia.edu. 1 Introduction The literature in optimality condition...
A Class of Stochastic Programs with Decision Dependent Uncertainty
 MATHEMATICAL PROGRAMMING
, 2005
"... The standard approach to formulating stochastic programs is based on the assumption that the stochastic process is independent of the optimization decisions. We address a class of problems where the optimization decisions influence the time of information discovery for a subset of the uncertain para ..."
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Cited by 20 (9 self)
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The standard approach to formulating stochastic programs is based on the assumption that the stochastic process is independent of the optimization decisions. We address a class of problems where the optimization decisions influence the time of information discovery for a subset of the uncertain parameters. We extend the standard modeling approach by presenting a disjunctive programming formulation that accommodates stochastic programs for this class of problems. A set of theoretical properties that lead to reduction in the size of the model is identified. A Lagrangean duality based branch and bound algorithm is also presented.
A New General ContinuousTime State Task Network Formulation for ShortTerm Scheduling of Multipurpose Batch Plants
 INDUSTRIAL AND ENGINEERING CHEMISTRY RESEARCH
, 2003
"... A new continuous time MILP model for the shortterm scheduling of multipurpose batch plants is presented. The proposed model relies on the State Task Network (STN) and addresses the general problem of batch scheduling, accounting for resource (utility) constraints, variable batch sizes and proces ..."
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Cited by 17 (13 self)
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A new continuous time MILP model for the shortterm scheduling of multipurpose batch plants is presented. The proposed model relies on the State Task Network (STN) and addresses the general problem of batch scheduling, accounting for resource (utility) constraints, variable batch sizes and processing times, various storage policies (UIS/FIS/NIS/ZW), batch mixing/splitting, and sequence dependent changeover times. The key features of the proposed model are the following: (a) a continuous time representation is used, common for all units, (b) assignment constraints are expressed using binary variables that are defined only for tasks, not for units, (c) start times of tasks are eliminated; thus, time matching constraints are used only for the finish times of tasks, and (d) a new class of valid inequalities that improves the LP relaxation is added to the MILP formulation. Compared to
A Stochastic Programming Approach to Planning of Offshore Gas Field Developments under Uncertainty in Reserves
 Computers and Chemical Engineering
, 2004
"... In this work, we consider the investment and operational planning of gas field developments under uncertainty in gas reserves. The resolution of uncertainty in gas reserves, and hence the shape of the scenario tree associated with the problem depends on the investment decisions. A novel stochastic p ..."
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Cited by 16 (10 self)
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In this work, we consider the investment and operational planning of gas field developments under uncertainty in gas reserves. The resolution of uncertainty in gas reserves, and hence the shape of the scenario tree associated with the problem depends on the investment decisions. A novel stochastic programming model that incorporates the decisiondependence of the scenario tree is presented. A decomposition based approximation algorithm for the solution of this model is also proposed. We show that the proposed approach yields solutions with significantly higher Expected Net Present Value (ENPV) than that of solutions obtained using a deterministic approach. For a small sized example, the proposed approximation algorithm is shown to yield the optimal solution with more than one order of magnitude reduction in solution time, as compared to the full space method. "Good" solutions to larger problems, that require up to 165,000 binary variables in full space, are obtained in a few hours using the proposed approach.
LiftandProject for Mixed 01 Programming: Recent Progress
, 1999
"... Contents 1 Introduction 2 Disjunctive programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Two basic ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Compact Representation of the Convex Hull 3 Projection and polarity . . . . . . . . . ..."
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Cited by 15 (1 self)
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Contents 1 Introduction 2 Disjunctive programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Two basic ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Compact Representation of the Convex Hull 3 Projection and polarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Adjacency on the higher dimensional polyhedron . . . . . . . . . . . . . . . . . . 5 3 Sequential Convexication 7 Disjunctive rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Fractionality of intermediate points . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4 Another Derivation of the Basic Results 12 5 Generating Cuts 13 Deepest cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Cut lifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Cut strengthening . . . . . . . .
Global optimization for the synthesis of integrated water systems in chemical processes
 Comp. Chem. Eng
"... In this paper, we address the problem of optimal synthesis of an integrated water system, where water using processes and water treatment operations are combined into a single network such that the total cost of obtaining freshwater for use in the water using operations, and treating wastewater is m ..."
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Cited by 13 (5 self)
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In this paper, we address the problem of optimal synthesis of an integrated water system, where water using processes and water treatment operations are combined into a single network such that the total cost of obtaining freshwater for use in the water using operations, and treating wastewater is minimized. A superstructure, which incorporates all feasible design alternatives for water treatment, reuse and recycle, is proposed. We formulate this structure as a nonconvex NonLinear Programming (NLP) problem, which is solved to global optimality. The problem takes the form of a nonconvex Generalized Disjunctive Program (GDP) if there is a flexibility of choosing different treatment technologies for the removal of the various contaminants in the wastewater streams. A new deterministic spatial branch and contract algorithm is proposed for optimizing such systems, in which piecewise under and overestimators are used to approximate the nonconvex terms in the original model to obtain a convex relaxation whose solution gives a lower bound on the global optimum. These lower bounds are made to converge to the solution within a branch and bound procedure. Several examples are presented to illustrate the optimization of these integrated networks using the proposed algorithm.
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.