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disjunctive programming and relaxations of polyhedra

MATHEMATICAL programming 2014年第1-2期144卷 307-314页

作者： Conforti, Michele Del Pia, Alberto Univ Padua Dipartimento Matemat Pura Applicata I-35121 Padua Italy Swiss Fed Inst Technol Dept Math IFOR CH-8092 Zurich Switzerland

Given a polyhedron with facets, whose interior contains no integral points, and a polyhedron , recent work in integer programming has focused on characterizing the convex hull of minus the interior of . We show that to obtain such a characterization it suffices to consider all relaxations of defined by at most among the inequalities defining . This extends a result by Andersen, Cornu,jols, and Li.

关键词： Mixed integer programming disjunctive programming Polyhedral relaxations

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disjunctive programming for Multiobjective Discrete Optimisation

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INFORMS JOURNAL ON COMPUTING 2018年第4期30卷 625-633页

作者： Bektas, Tolga Univ Southampton CORMSIS Southampton Business Sch Southampton SO17 1BJ Hants England

In this paper, I view and present the multiobjective discrete optimisation problem as a particular case of disjunctive programming where one seeks to identify efficient solutions from within a disjunction formed by a set of systems. The proposed approach lends itself to a simple yet effective iterative algorithm that is able to yield the set of all nondominated points, both supported and nonsupported, for a multiobjective discrete optimisation problem. Each iteration of the algorithm is a series of feasibility checks and requires only one formulation to be solved to optimality that has the same number of integer variables as that of the single objective formulation of the problem. The application of the algorithm shows that it is particularly effective when solving constrained multiobjective discrete optimisation problem instances.

关键词： multiobjective optimisation disjunctive programming integer programming

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disjunctive programming and the generalized Leontief input-output model

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APPLIED MATHEMATICS AND COMPUTATION 2008年第2期198卷 551-558页

作者： Ebiefung, Aniekan Kostreva, Michael Majumdar, Ishita Univ Tennessee Chattanooga TN 37403 USA Clemson Univ Clemson SC 29634 USA

This paper considers a generalization of the Leontief input-output model that is useful in modeling the concept of choice of technology. It is shown that a disjunctive programming problem, together with its dual problem, may be used to effectively solve the new input-output model. Various theoretical results are presented together with an illustrative example. Three production sectors, each having two possible choices of technology, are provided in the example and only one linear program is required for finding the solution. (c) 2007 Elsevier Inc. All rights reserved.

关键词： Leontief models input-output disjunctive programming linear programming choice of technology

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Parametric disjunctive programming: One-sided differentiability of the value function

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 1997年第2期92卷 285-310页

作者： Gugat, M Research Assistant Department of Mathematics University of Trier Trier Germany

We consider a countable family of one-parameter convex programs and give sufficient conditions for the one-sided differentiability of its optimal value function. The analysis is based on the Borwein dual problem for a family of convex programs (a convex disjunctive program). We give conditions that assure stability of the situation of perfect duality in the Borwein theory. For the reader's convenience, we start with a review of duality results for families of convex programs. A parametric family of dual problems is introduced that contains the dual problems of Balas and Borwein as special cases. In addition, a vector optimization problem is defined as a dual problem. This generalizes a result by Helbig about families of linear programs.

关键词： disjunctive programming families of convex programs parametric family of dual problems efficiency sensitivity optimal value function mixed integer programming

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A disjunctive programming APPROACH FOR MOTION PLANNING OF MOBILE ROUTER NETWORKS

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INTERNATIONAL JOURNAL OF ROBOTICS & AUTOMATION 2011年第1期26卷 13-25页

作者： Bezzo, Nicola Fierro, Rafael Swingler, Ashleigh Ferrari, Silvia Univ New Mexico Sch Elect & Comp Engn MARHES Lab Albuquerque NM 87131 USA Duke Univ Dept Mech Engn & Mat Sci Lab Intelligent Syst & Controls Durham NC 27708 USA

In this paper we develop a framework based on disjunctive programming for motion planning of robotic networks. Although the methodology presented in this paper can be applied to general motion planning problems we focus on coordinating a team of mobile routers to maintain connectivity between a fixed base station mid a mobile user within a walled environment. This connectivity management problem is decomposed into three steps: (i) a feasible line-of-sight path between tie base station and the mobile user is computed;(ii) the number of required routers and their goal locations are determined;and (iii) the motion planning with obstacle and inter-vehicle collision avoidance problem is solved. To illustrate the flexibility of the proposed approach we also formulate a novel motion planning algorithm for a team of mobile robots as a disjunctive program. Cell decomposition is used to take into account the size and orientation of the robots. In both cases, connectivity and motion planning, the mixed-integer optimization problems are solved rising CPLEX. Moreover, the proposed approach can easily accommodate input and other constraints and mission objectives. Simulation results show the applicability of the proposed strategy.

关键词： Motion planning communications mobile routers disjunctive programming

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Generalized convex disjunctive programming: Nonlinear convex hull relaxation

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2003年第1期26卷 83-100页

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

Generalized disjunctive programming (GDP) has been introduced recently as an alternative to mixed-integer programming for representing 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 [ 21] and Ceria and Soares [ 6], we propose a convex 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 Mixed-Integer Nonlinear programming (MINLP) problem that is shown to be tighter than the conventional "big-M" formulation. A disjunctive branch and bound method is also presented, and numerical results are given for a set of test problems.

关键词： disjunctive programming convex programming mixed integer nonlinear programming convex hull

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A disjunctive programming approach for the optimal design of reactive distillation columns

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COMPUTERS & CHEMICAL ENGINEERING 2001年第11-12期25卷 1661-1673页

作者： Jackson, JR Grossmann, IE Carnegie Mellon Univ Dept Chem Engn Pittsburgh PA 15213 USA

A generalized disjunctive programming formulation is presented for the optimal design of reactive distillation columns using tray-by-tray, phase equilibrium and kinetic based models. The proposed formulation uses disjunctions for conditional trays to apply the MESH and reaction kinetics equations for only the selected trays in order to reduce the size of the nonlinear programming subproblems. Solution of the model yields the optimal feed tray locations, number of trays, reaction zones, and operating and design parameters. The disjunctive program is solved using a logic-based outer-approximation algorithm where the MILP master problem is based on the big-M formulation of disjunctions, and where a special initialization scheme is used to reduce the number of initial NLP subproblems that need to be solved. Two examples are presented that include reactive distillation for the metathesis reaction of 2-pentene and for the production of ethylene glycol. The results show that the proposed method can effectively handle these difficult nonlinear optimization problems. (C) 2001 Elsevier Science Ltd. All rights reserved.

关键词： reactive distillation disjunctive programming optimal distillation column design

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A cutting plane method for solving linear generalized disjunctive programming problems

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COMPUTERS & CHEMICAL ENGINEERING 2005年第9期29卷 1891-1913页

作者： Sawaya, NW Grossmann, IE Carnegie Mellon Univ Dept Chem Engn Pittsburgh PA 15213 USA

Raman and Grossmann [Raman, R., & Grossmann, I.E. (1994). Modeling and computational techniques for logic based integer programming. Computers and Chemical Engineering, 18(7), 563-578] and Lee and Grossmann [Lee, S., & Grossmann, I.E. (2000). New algorithms for nonlinear generalized disjunctive programming. Computers and Chemical Engineering, 24, 2125-2141] have developed a reformulation of Generalized disjunctive programming (GDP) problems that is based on determining the convex hull of each disjunction. Although the relaxation of the reformulated problem using this method will often produce a significantly tighter lower bound when compared with the traditional big-M reformulation, the limitation of this method is that the representation of the convex hull of every disjunction requires the introduction of new disaggregated variables and additional constraints that can greatly increase the size of the problem. In order to circumvent this issue, a cutting plane method that can be applied to linear GDP problems is proposed in this paper. The method relies on converting the GDP problem into an equivalent big-M reformulation that is successively strengthened by cuts generated from an LP or QP separation problem. The sequence of problems is repeatedly solved, either until the optimal integer solution is found, or else until there is no improvement within a specified tolerance, in which case one switches to a branch and bound method. The strip-packing, retrofit planning and zero-wait job-shop scheduling problems are presented as examples to illustrate the performance of the proposed cutting plane method. (c) 2005 Elsevier Ltd. All rights reserved.

关键词： MIP disjunctive programming cutting planes mixed integer linear programming strip-packing retrofit planning job-shop scheduling

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A disjunctive programming model and a rolling horizon algorithm for optimal multiperiod capacity expansion in a multiproduct batch plant

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COMPUTERS & CHEMICAL ENGINEERING 2012年 46卷 29-38页

作者： Garcia-Ayala, Gabriela Rios-Mercado, Roger Z. Chacon-Mondragon, Oscar L. Tecnol Monterrey Ctr Qual & Mfg Monterrey 64849 NL Mexico Univ Autonoma Nuevo Leon Grad Program Syst Engn San Nicol 66450 NL Mexico

This paper addresses a multiperiod mixed integer nonlinear programming problem for the capacity expansion of multiproduct batch plants. Given a certain batch plant with its current configuration, product recipes, and growing production targets, modular expansions are wanted so that new demand can be met. Unlike most work for the batch retrofit problem, a multiperiod disjunctive model is presented, such that long term investments and expansions can be planned out in advance. To solve the model we propose a rolling horizon algorithm that further exploits the advantages of a disjunctive programming model. Empirical work shows that the rolling horizon algorithm is very effective on finding near optimal solutions to large instances with a considerable number of time periods. Furthermore, the solution found by the proposed algorithm can be used as a starting solution for the direct method to the original problem delivering a global optimal solution. (c) 2012 Elsevier Ltd. All rights reserved.

关键词： Batch retrofit Multiproduct batch plants Multiperiod MINLP model disjunctive programming Rolling horizon algorithm

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Transmission Network Investment Using Incentive Regulation: A disjunctive programming Approach

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NETWORKS & SPATIAL ECONOMICS 2020年第4期20卷 1029-1068页

作者： Khastieva, D. Hesamzadeh, M. R. Vogelsang, I. Rosellon, J. KTH Royal Inst Technol Sch Elect Engn & Comp Sci Teknikringen 33 S-11428 Stockholm Sweden Boston Univ Dept Econ Boston MA 02215 USA CIDE Div Econ Carretera Mexico Toluca 3655 Lomas De Santa Fe 01210 Mexico DIW Berlin Dept Energy Transportat Environm Mohrenstr 58 D-10117 Berlin Germany Rice Univ Ctr Energy Studies Baker Inst Publ Policy Houston TX USA Univ Panamer Ctr Regulac Energet & Econ Desarrollo CREED UP Campus Mexico Mexico City DF Mexico

A well-planned electric transmission infrastructure is the foundation of a reliable and efficient power system, especially in the presence of large scale renewable generation. However, the current electricity market designs lack incentive mechanisms which can guarantee optimal transmission investments and ensure reliable integration of renewable generation such as wind. This paper first proposes a stochastic bilevel disjunctive program for optimal transmission investment based on the newly proposed theoretical H-R-G-V incentive mechanism. The upper level is a profit-maximization problem of an independent transmission company (Transco), while the lower level is a welfare maximization problem. The revenue of the Transco is bounded by a regulatory constraint set by the regulator in order to induce socially optimal investments. The application of the H-R-G-V mechanism allows the regulator to ensure social maximum transmission investments and helps to reduce transmission congestion and wind power spillage. The transmission investment under the H-R-G-V mechanism is modeled as a stochastic bilevel disjunctive program. To solve the developed mathematical model we first propose a series of linearization and reformulation techniques to recast the original model as a stochastic mixed integer linear problem (MILP). We exploit the disjunctive nature of the reformulated stochastic MILP model and further propose a Bean decomposition algorithm to efficiently solve the stochastic MILP model. The proposed decomposition algorithm is also modified and accelerated to improve the computational performance. The computational performance of our MILP modeling approach and modified and accelerated Bean decomposition algorithm is studied through several examples in detail. The simulation results confirm a promising performance of both the modeling approach and its solution algorithm.

关键词： Transmission network investments Incentive regulation Bean decomposition algorithm disjunctive programming

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