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Generalized disjunctive programming: A framework for formulation and alternative algorithms for minlp optimization
 In Mixed Integer Nonlinear Programming
, 2012
"... Generalized disjunctive programming (GDP) is an extension of the disjunctive programming paradigm developed by Balas. The GDP formulation involves Boolean and continuous variables that are specified in algebraic constraints, disjunctions and logic propositions, which is an alternative representatio ..."
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Generalized disjunctive programming (GDP) is an extension of the disjunctive programming paradigm developed by Balas. The GDP formulation involves Boolean and continuous variables that are specified in algebraic constraints, disjunctions and logic propositions, which is an alternative representation to the traditional algebraic mixedinteger programming formulation. After providing a brief review of MINLP optimization, we present an overview of GDP for the case of convex functions emphasizing the quality of continuous relaxations of alternative reformulations that include the bigM and the hull relaxation. We then review disjunctive branch and bound as well as logicbased decomposition methods that circumvent some of the limitations in traditional MINLP optimization. We next consider the case of linear GDP problems to show how a hierarchy of relaxations can be developed by performing sequential intersection of disjunctions. Finally, for the case when the GDP problem involves nonconvex functions, we propose a scheme for tightening the lower bounds for obtaining the global optimum using a combined disjunctive and spatial branch and bound search. We illustrate the application of the theoretical concepts and algorithms on several engineering and OR
An Extended Mathematical Programming Framework
, 2009
"... Extended mathematical programs are collections of functions and variables joined together using specific optimization and complementarity primitives. This paper outlines a mechanism to describe such an extended mathematical program by means of annotating the existing relationships within a model to ..."
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Cited by 5 (1 self)
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Extended mathematical programs are collections of functions and variables joined together using specific optimization and complementarity primitives. This paper outlines a mechanism to describe such an extended mathematical program by means of annotating the existing relationships within a model to facilitate higher level structure identification. The structures, which often involve constraints on the solution sets of other models or complementarity relationships, can be exploited by modern large scale mathematical programming algorithms for efficient solution. A specific implementation of this framework is outlined that communicates structure from the GAMS modeling system to appropriate solvers in a computationally beneficial manner. Example applications are taken from chemical engineering.
Review of mixedinteger nonlinear and generalized disjunctive programming applications in Process Systems Engineering
, 2014
"... In this chapter we present some of the applications of MINLP and generalized disjunctive programming (GDP) in process systems engineering (PSE). For a comprehensive review of mixedinteger nonlinear optimization we refer the reader to the work by Belotti et al.[1]. Bonami et al.[2] review convex MI ..."
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Cited by 4 (3 self)
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In this chapter we present some of the applications of MINLP and generalized disjunctive programming (GDP) in process systems engineering (PSE). For a comprehensive review of mixedinteger nonlinear optimization we refer the reader to the work by Belotti et al.[1]. Bonami et al.[2] review convex MINLP algorithms and software in more detail. Tawarmalani and Sahinidis[3] describe global optimization theory,
Generalized Disjunctive Programming as a Systematic Modeling Framework to Derive Scheduling Formulations
"... We propose generalized disjunctive programming models for the shortterm scheduling problem of single stage batch plants with parallel units. Three different concepts of continuoustime representation are explored, immediate and general precedence, as well as multiple time grids. The GDP models are ..."
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We propose generalized disjunctive programming models for the shortterm scheduling problem of single stage batch plants with parallel units. Three different concepts of continuoustime representation are explored, immediate and general precedence, as well as multiple time grids. The GDP models are then reformulated using both bigM and convex hull reformulations, and the resulting mixedinteger linear programming models compared to the solution of a set of example problems. We show that two general precedence models from the literature can be derived using a bigM reformulation for a set of disjunctions and a convex hull reformulation for another. The best performer is, however, a multiple time grid model which can be derived from the convex hull reformulation followed by simple algebraic manipulations to eliminate the disaggregated variables and reduce the sets of constraints, thus leading to a more compact and efficient formulation.
Algorithmic approach for improved mixedinteger reformulations of convex Generalized Disjunctive Programs
, 2013
"... In this work, we propose an algorithmic approach to improve mixedinteger models that are originally formulated as convex Generalized Disjunctive Programs (GDP). The algorithm seeks to obtain an improved continuous relaxation of the MILP/MINLP reformulation of the GDP, while limiting the growth in t ..."
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In this work, we propose an algorithmic approach to improve mixedinteger models that are originally formulated as convex Generalized Disjunctive Programs (GDP). The algorithm seeks to obtain an improved continuous relaxation of the MILP/MINLP reformulation of the GDP, while limiting the growth in the problem size. There are three main stages that form the basis of the algorithm. The first one is a presolve, consequence of the logic nature of GDP, which allows us to reduce the problem size, find good relaxation bounds and identify properties that help us determine where to apply a basic step. The second stage is the iterative application of basic steps, selecting where to apply them, and monitoring the improvement of the formulation. Finally, we use a hybrid reformulation of GDP that seeks to exploit both of the advantages attributed to the two common GDPtoMILP/MINLP transformations, the BigM and Hull reformulation. We illustrate the application of this algorithm with several examples. The results show the improvement in the problem formulations by generating models with improved relaxed solutions and relatively small growth in the number of continuous variables and constraints. The algorithm generally leads to reduction in the solution times. 1
A MIXEDINTEGER MODEL PREDICTIVE CONTROL FORMULATION FOR LINEAR SYSTEMS
"... Most industrial model predictive controllers (MPC) use the traditional twolayer structure developed in the early 1980’s, where the upper layer defines optimal steadystate targets for inputs and outputs, while the lower layer calculates the control moves that drive the system towards these steadys ..."
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Most industrial model predictive controllers (MPC) use the traditional twolayer structure developed in the early 1980’s, where the upper layer defines optimal steadystate targets for inputs and outputs, while the lower layer calculates the control moves that drive the system towards these steadystate targets. Typically both layers use continuous quadratic programming (QP) formulations to derive the optimal solutions. On the other hand, advances in mixedinteger programming (MIP) algorithms and their successful application to solve large scheduling problems in reasonable time show that MIP formulations have the potential of being applied advantageously to the multivariable model predictive control problem. In this paper, we present mixedinteger quadratic programming (MIQP) formulations for both layers and show that several difficulties faced in the MPC practical implementation can be overcome with this approach. In particular, it is possible to set explicit priorities for inputs and outputs, define minimum moves to overcome hysteresis, and deal with digital or integer inputs. The proposed formulation is applied to 2 benchmark problems and to a simulated industrial system and the results compared with those achieved by a traditional continuous MPC. The solutions of the MIQP problems are derived by a computer implementation of the Outer Approximation method (OA) also developed as part of this work.
Disjunctive Programs
"... manuscript Authors are encouraged to submit new papers to INFORMS journals by means of a style file template, which includes the journal title. However, use of a template does not certify that the paper has been accepted for publication in the named journal. INFORMS journal templates are for the ex ..."
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manuscript Authors are encouraged to submit new papers to INFORMS journals by means of a style file template, which includes the journal title. However, use of a template does not certify that the paper has been accepted for publication in the named journal. INFORMS journal templates are for the exclusive purpose of submitting to an INFORMS journal and should not be used to distribute the papers in print or online or to submit the papers to another publication. Cutting planes algorithm for convex Generalized
Improved BigM Reformulation for Generalized Disjunctive Programs
"... In this work, we present a new BigM reformulation for Generalized Disjunctive Programs. The proposed MINLP reformulation is stronger than the traditional BigM, and it does not require additional variables or constraints. We present the new BigM, and analyze the strength in its continuous relaxat ..."
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In this work, we present a new BigM reformulation for Generalized Disjunctive Programs. The proposed MINLP reformulation is stronger than the traditional BigM, and it does not require additional variables or constraints. We present the new BigM, and analyze the strength in its continuous relaxation compared to that of the traditional BigM. The new formulation is tested by solving several instances of process networks and muliproduct batch plant problems with an NLPbased branch and bound method. The results show that, in most cases, the new reformulation requires fewer nodes and less time to find the optimal solution. 1
A MIXEDINTEGER MODEL PREDICTIVE CONTROL FORMULATION FOR LINEAR SYSTEMS
"... A mixedinteger model predictive control formulation for linear systems ..."
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