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Global optimization of disjunctive programs

JOURNAL OF GLOBAL OPTIMIZATION 2017年第2期69卷 283-307页

作者： Kirst, Peter Rigterink, Fabian Stein, Oliver Karlsruhe Inst Technol Inst Operat Res Karlsruhe Germany Univ Newcastle Sch Math & Phys Sci Newcastle NSW Australia

We propose a new branch-and-bound framework for global optimization of disjunctive programs with general logical expressions. We do not assume the logical expressions to be in any normal form, and, under slightly stronger assumptions, we allow the use of negations and implications. In contrast to the widely used reformulation as a mixed-integer program, we compute the lower bounds and evaluate the logical expression in one step. Thus, we reduce the size of the problem and work exclusively with continuous variables, which is computationally advantageous. We present preliminary numerical results as proof of concept.

关键词： Global optimization disjunctive programming generalized disjunctive programming Branch-and-bound

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Optimal plate design problem in steel production

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INTERNATIONAL JOURNAL OF PRODUCTION RESEARCH 2023年第5期61卷 1575-1590页

作者： Zhao, Shengnan Meng, Ying Su, Lijie Liu, Jiyin Tang, Lixin Northeastern Univ Minist Educ Key Lab Data Analyt & Optimizat Smart Ind Shenyang Peoples R China Northeastern Univ Natl Frontiers Sci Ctr Ind Intelligence & Syst Op Shenyang 110819 Peoples R China Northeastern Univ Liaoning Engn Lab Data Analyt & Optimizat Smart I Shenyang Peoples R China Northeastern Univ Liaoning Key Lab Mfg Syst & Logist Optimizat Shenyang Peoples R China Loughborough Univ Sch Business Econ Loughborough Leics England

In this paper, we address a new optimal plate design problem in steel production, in which slab selection is jointly considered. On the basis of the underlying features, the problem is formulated as a mixed integer nonlinear programming (MINLP) model with generalized disjunctive programming (GDP) constraints. A logic-based outer approximation (L-OA) algorithm is proposed to solve the problem. Specifically, a two-stage heuristic method is designed to initialise the L-OA algorithm. Numerical results are presented to demonstrate that the proposed L-OA algorithm and the heuristic method are effective and computationally efficient.

关键词： Production planning Iron and Steel Industry plate design generalized disjunctive programming logic-based outer approximation algorithm

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Modeling of discrete/continuous optimization problems: characterization and formulation of disjunctions and their relaxations

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COMPUTERS & CHEMICAL ENGINEERING 2003年第3期27卷 433-448页

作者： Vecchietti, A Lee, S Grossmann, IE Carnegie Mellon Univ Dept Chem Engn Pittsburgh PA 15213 USA UTN INGAR Inst Desarrollo & Diseno Fac Reg Santa Fe Santa Fe Argentina

This paper addresses the relaxations in alternative models for disjunctions, big-M and convex hull model, in order to develop guidelines and insights when formulating Mixed-Integer Non-Linear programming (MINLP), generalized disjunctive programming (GDP), or hybrid models. Characterization and properties are presented for various types of disjunctions. An interesting result is presented for improper disjunctions where results in the continuous space differ from the ones in the mixed-integer space. A cutting plane method is also proposed that avoids the explicit generation of equations and variables of the convex hull. Several examples are presented throughout the paper, as well as a small process synthesis problem, which is solved with the proposed cutting plane method. (C) 2002 Elsevier Science Ltd. All rights reserved.

关键词： discrete-continuous optimization mixed-integer nonlinear programming generalized disjunctive programming big-M relaxation convex hull relaxation

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Optimal design of multiproduct batch plants considering duplication of units in series

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CHEMICAL ENGINEERING RESEARCH & DESIGN 2009年第11A期87卷 1497-1508页

作者： Moreno, Marta S. Iribarren, Oscar A. Montagna, Jorge M. Consejo Nacl Invest Cient & Tecn INGAR Inst Desarrollo & Diseno RA-1033 Buenos Aires DF Argentina Univ Tecnol Nacl Unidad Acad Reconquista Buenos Aires DF Argentina Univ Tecnol Nacl Fac Reg Santa Fe Buenos Aires DF Argentina

A generalized disjunctive programming (GDP) model for the optimal design of multiproduct batch plants is presented. This general model manages the duplication of units in series to perform a given operation in the process, which is an alternative that has not been considered in previous general approaches. Unlike duplication in parallel, duplication in series is only applicable to some operations which present trade-offs between duplication and other cost-impacting elements in the batch process. In order to use a fixed time and size factor model some assumptions had to be made in the operations that allow the duplication in series. To show the effectiveness of this approach, a plant that produces multiple recombinant proteins is presented and solved. (C) 2009 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

关键词： Duplication in series Multiproduct batch plants generalized disjunctive programming

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Block-oriented modeling of superstructure optimization problems

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COMPUTERS & CHEMICAL ENGINEERING 2013年 57卷 10-23页

作者： Friedman, Zev Ingalls, Jack Siirola, John D. Watson, Jean-Paul Sandia Natl Labs Discrete Math & Complex Syst Dept Albuquerque NM 87185 USA

We present a novel software framework for modeling large-scale engineered systems as mathematical optimization problems. A key motivating feature in such systems is their hierarchical, highly structured topology. Existing mathematical optimization modeling environments do not facilitate the natural expression and manipulation of hierarchically structured systems. Rather, the modeler is forced to "flatten" the system description, hiding structure that may be exploited by solvers, and obfuscating the system that the modeling environment is attempting to represent. To correct this deficiency, we propose a Python-based "block-oriented" modeling approach for representing the discrete components within the system. Our approach is an extension of the Pyomo library for specifying mathematical optimization problems. Through the use of a modeling components library, the block-oriented approach facilitates a clean separation of system superstructure from the details of individual components. This approach also naturally lends itself to expressing design and operational decisions as disjunctive expressions over the component blocks. By expressing a mathematical optimization problem in a block-oriented manner, inherent structure (e.g., multiple scenarios) is preserved for potential exploitation by solvers. In particular, we show that block-structured mathematical optimization problems can be straightforwardly manipulated by decomposition-based multi-scenario algorithmic strategies, specifically in the context of the PySP stochastic programming library. We illustrate our block-oriented modeling approach using a case study drawn from the electricity grid operations domain: unit commitment with transmission switching and N - 1 reliability constraints. Finally, we demonstrate that the overhead associated with block-oriented modeling only minimally increases model instantiation times, and need not adversely impact solver behavior. (C) 2013 Elsevier Ltd. All rights reserved.

关键词： Superstructure optimization generalized disjunctive programming Stochastic programming Algebraic modeling language

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A novel MINLP-based representation of the original complex model for predicting gasoline emissions

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COMPUTERS & CHEMICAL ENGINEERING 2008年第12期32卷 2857-2876页

作者： Furman, Kevin C. Androulakis, Ioannis P. ExxonMobil Res & Engn Co Corp Strateg Res Annandale NJ 08801 USA Rutgers State Univ Dept Biomed Engn Piscataway NJ 08854 USA

The Environmental Protection Agency (EPA) introduced Reformulated Gasoline (RFG) requirements as a measure to reduce emissions from gasoline-powered vehicles in certain geographic areas. As part of this effort, the EPA developed empirical models for predicting emissions as a function of gasoline properties and established statutory baseline emissions from a representative set of gasolines. All reformulated gasoline requires certification via this model, known as the Complex Model, and all refiners and importers calculate emissions performance reductions from the statutory baseline gasoline. The current representation of the Complex Model is extremely difficult to implement within refinery operations models or to use in combination with models for designer gasoline. RFG and boutique fuels are key driving forces in the North American refining industry. The RFG models introduce increasingly complex constraints with the major limitation that they are implicitly defined through a series of complicated disjunctions assembled by the EPA in the form of spreadsheets. This implicit and cumbersome representation of the emissions predictive models renders rigorous optimization and sensitivity analysis very difficult to address directly. In this paper, we discuss how the federal government requirements for reformulated gasoline can be restated as a set of mixed-integer nonlinear programming (MINLP) constraints with the aid of disjunctive programming techniques. We illustrate the use of this model with two simple example fuel blending problems. (C) 2008 Elsevier Ltd. All rights reserved.

关键词： Reformulated gasoline Mixed-integer nonlinear programming generalized disjunctive programming

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Tightening piecewise McCormick relaxations for bilinear problems

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COMPUTERS & CHEMICAL ENGINEERING 2015年 72卷 300-311页

作者： Castro, Pedro M. Univ Lisbon Fac Ciencias Ctr Invest Operac P-1749016 Lisbon Portugal Lab Nacl Energia & Geol P-1649038 Lisbon Portugal

We address nonconvex bilinear problems where the main objective is the computation of a tight lower bound for the objective function to be minimized. This can be obtained through a mixed-integer linear programming formulation relying on the concept of piecewise McCormick relaxation. It works by dividing the domain of one of the variables in each bilinear term into a given number of partitions, while considering global bounds for the other. We now propose using partition-dependent bounds for the latter so as to further improve the quality of the relaxation. While it involves solving hundreds or even thousands of linear bound contracting problems in a pre-processing step, the benefit from having a tighter formulation more than compensates the additional computational time. Results for a set of water network design problems show that the new algorithm can lead to orders of magnitude reduction in the optimality gap compared to commercial solvers. (C) 2014 Elsevier Ltd. All rights reserved.

关键词： Optimization Mathematical modeling Nonlinear programming generalized disjunctive programming Water minimization

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***: an ecosystem for logic based modeling and optimization development

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OPTIMIZATION AND ENGINEERING 2022年第1期23卷 607-642页

作者： Chen, Qi Johnson, Emma S. Bernal, David E. Valentin, Romeo Kale, Sunjeev Bates, Johnny Siirola, John D. Grossmann, Ignacio E. Carnegie Mellon Univ Ctr Adv Proc Decis Making Dept Chem Engn Pittsburgh PA 15213 USA Sandia Natl Labs Albuquerque NM 87123 USA

We present three core principles for engineering-oriented integrated modeling and optimization tool sets-intuitive modeling contexts, systematic computer-aided reformulations, and flexible solution strategies-and describe how new developments in *** for generalized disjunctive programming (GDP) advance this vision. We describe a new logical expression system implementation for *** allowing for a more intuitive description of logical propositions. The logical expression system supports automated reformulation of these logical constraints to linear constraints. We also describe two new logic-based global optimization solver implementations built on *** that exploit logical structure to avoid "zero-flow" numerical difficulties that arise in nonlinear network design problems when nodes or streams disappear. These new solvers also demonstrate the capability to link to external libraries for expanded functionality within an integrated implementation. We present these new solvers in the context of a flexible array of solution paths available to GDP models. Finally, we present results on a new library of GDP models demonstrating the value of multiple solution approaches.

关键词： Mathematical programming generalized disjunctive programming MINLP

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Optimal synthesis of work and heat exchangers networks considering unclassified process streams at sub and above-ambient conditions

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APPLIED ENERGY 2018年 224卷 567-581页

作者： Onishi, Viviani C. Quirante, Natalia Ravagnani, Mauro A. S. S. Caballero, Jose A. Univ Coimbra Dept Elect & Comp Engn Polo 2 P-3030290 Coimbra Portugal Inst Syst Engn & Comp Coimbra INESC Coimbra Polo 2 P-3030290 Coimbra Portugal Univ Alicante Dept Chem Engn Ap Correos 99 E-03080 Alicante Spain Univ Estadual Maringa Dept Chem Engn Ave Colombo 5790 BR-87020900 Maringa PR Brazil

Work and heat exchanger networks have recently drawn increasing attention due to their paramount importance in achieving energy savings. In this work, we introduce a new optimization model for the cost-effective synthesis and energy integration of work and heat exchanger networks considering unclassified process streams (ie., streams whose classification as hot or cold streams cannot be defined a priori). Our innovative modelling approach combines mathematical programming techniques and the pinch location method to obtain an optimal network design with minimal cost, while adjusting pressure and temperature levels of unclassified streams. We propose disjunctive operators for the selection of pressure manipulation equipment, and streams identity classification depending on energy requirements and process operating conditions. In addition, our approach addresses previous shortcomings by eliminating the need for: (i) assigning a specific route of pressure manipulation;and, (ii) classifying streams as low or high-pressure streams;which provides further flexibility to the system. Our methodology is also able to effectively deal with variable inlet and outlet streams temperatures to reach specific optimization goals. The model is solved to global optimality through the minimization of the process total annualized cost. Besides improved computational performance, results from energy analyses reveal that streams classification during process optimization can be greatly advantageous for both subambient and above-ambient applications. In the liquefied natural gas process, it reduces up to 89% the energy demand when compared to literature records.

关键词： Simultaneous optimization generalized disjunctive programming Mixed-integer nonlinear programming Work and heat integration Pinch location method Variable temperatures

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Logic-based modeling and solution of nonlinear discrete/continuous optimization problems

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ANNALS OF OPERATIONS RESEARCH 2005年第1期139卷 267-288页

作者： Lee, S Grossmann, IE Carnegie Mellon Univ Dept Chem Engn Pittsburgh PA 15213 USA

This paper presents a review of advances in the mathematical programming approach to discrete/continuous optimization problems. We first present a brief review of MILP and MINLP for the case when these problems are modeled with algebraic equations and inequalities. Since algebraic representations have some limitations such as difficulty of formulation and numerical singularities for the nonlinear case, we consider logic-based modeling as an alternative approach, particularly generalized disjunctive programming (GDP), which the authors have extensively investigated over the last few years. Solution strategies for GDP models are reviewed, including the continuous relaxation of the disjunctive constraints. Also, we briefly review a hybrid model that integrates disjunctive programming and mixed-integer programming. Finally, the global optimization of nonconvex GDP problems is discussed through a two-level branch and bound procedure.

关键词： mixed-integer programming generalized disjunctive programming global optimization logic-based programming

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