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Review of Nonlinear Mixed-Integer and disjunctive programming Techniques

OPTIMIZATION AND ENGINEERING 2002年第3期3卷 227-252页

作者： Grossmann, Ignacio E. Carnegie Mellon Univ Dept Chem Engn Pittsburgh PA 15213 USA

This paper has as a major objective to present a unified overview and derivation of mixed-integer nonlinear programming (MINLP) techniques, Branch and Bound, Outer-Approximation, 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.

关键词： mixed-integer programming disjunctive programming nonlinear programming

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Model Predictive Control of Discrete-Continuous Energy Systems via Generalized disjunctive programming

Model Predictive Control of Discrete-Continuous Energy Syste...

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Modeling, Estimation and Control Conference (MECC)

作者： Bhattacharya, Arnab Ma, Xu Vrabie, Draguna L. Pacific Northwest Natl Lab Richland WA 99352 USA

Generalized disjunctive programming (GDP) provides an alternative framework to model optimization problems with both discrete and continuous variables. The key idea behind GDP involves the use of logical disjunctions to represent discrete decisions in the continuous space, and logical propositions to denote algebraic constraints in the discrete space. Compared to traditional mixed-integer programming (MIP), the logic structure in GDP yields tighter relaxations that are exploited by global branch and bound algorithms to improve solution quality. We present a general GDP model for optimal control of hybrid systems that exhibit both discrete and continuous dynamics. Specifically, we use GDP to formulate a model predictive control (MPC) model for piecewise-affine systems with implicit switching logic. As an example, the GDP-based MPC approach is used as a supervisory control to improve energy efficiency in residential buildings with binary on/off, relay-based thermostats. A simulation study is used to demonstrate the efficacy of the proposed approach compared to existing MIP-based approaches. Copyright (C) 2021 The Authors.

关键词： Model Predictive Control Mixed-Integer Optimization Energy Systems disjunctive programming

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Intersection cuts from multiple rows: a disjunctive programming approach

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EURO JOURNAL ON COMPUTATIONAL OPTIMIZATION 2013年第1-2期1卷 3-49页

作者： Balas, Egon Qualizza, Andrea Carnegie Mellon Univ Tepper Sch Business Pittsburgh PA 15213 USA Amazon Res Seattle WA 98109 USA

We address the issue of generating cutting planes for mixed integer programs from multiple rows of the simplex tableau with the tools of disjunctive programming. A cut from q rows of the simplex tableau is an intersection cut from a q-dimensional parametric cross-polytope, which can also be viewed as a disjunctive cut from a 2q-term disjunction. We define the disjunctive hull of the q-row problem, describe its relation to the integer hull, and show how to generate its facets. For the case of binary basic variables, we derive cuts from the stronger disjunctions whose terms are equations. We give cut strengthening procedures using the integrality of the nonbasic variables for both the integer and the binary case. Finally, we discuss some computational experiments.

关键词： Mixed integer programming disjunctive programming Intersection cuts 0-1 programming

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Finite disjunctive programming methods for general mixed integer linear programs

Finite disjunctive programming methods for general mixed int...

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作者： Chen, Binyuan The University of Arizona

学位级别：Ph.D.

In this dissertation, a finitely convergent disjunctive programming procedure, the Convex Hull Tree (CHT) algorithm, is proposed to obtain the convex hull of a general mixed-integer linear program with bounded integer variables. The CHT algorithm constructs a linear program that has the same optimal solution as the associated mixed-integer linear program. The standard notion of sequential cutting planes is then combined with ideas underlying the CHT algorithm to help guide the choice of disjunctions to use within a new cutting plane method, the Cutting Plane Tree (CPT) algorithm. We show that the CPT algorithm converges to an integer optimal solution of the general mixed-integer linear program with bounded integer variables in finitely many steps. We also enhance the CPT algorithm with several techniques including a “round-of-cuts” approach and an iterative method for solving the cut generation linear program (CGLP). Two normalization constraints are discussed in detail for solving the CGLP. For moderately sized instances, our study shows that the CPT algorithm provides significant gap closures with a pure cutting plane method. Key words: Mixed-integer linear program, disjunctive programming, convex hull, cutting plane, finite convergence.

关键词： cutting plane finite convergence convex hull disjunctive programming mixed-integer linear program

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Assembly planning by disjunctive programming and geometrical reasoning

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COMPUTERS & OPERATIONS RESEARCH 2022年 138卷 105603-105603页

作者： Horvath, Marko Kis, Tamas Kovacs, Andras Fekula, Mark Inst Comp Sci & Control SZTAKI Kende Str 13-17 H-1111 Budapest Hungary

The challenge in modeling and solving assembly planning problems lies in integrating combinatorial optimization techniques for finding efficient task sequences and resource assignments with geometrical reasoning to ensure the geometrical and technological feasibility of the assembly plans. This paper proposes a Benders decomposition approach that separates the macro-level planning problem, responsible for task sequencing and resource assignment, from micro-level validation on detailed geometrical and technological models. Feedback from the micro to the macro level is provided in the form of disjunctive constraints generated during search, which precludes the repeated occurrence of the collisions encountered in earlier iterations. A disjunctive programming approach is proposed to solve the macro-level planning problem with the added constraints. The efficiency of the approach is demonstrated both in industrial case studies and in computational experiments on generated problem instances.

关键词： Assembly planning Combinatorial optimization disjunctive programming Geometrical reasoning

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Model Predictive Control of Discrete-Continuous Energy Systems via Generalized disjunctive programming

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IFAC-PapersOnLine 2021年第20期54卷 913-918页

作者： Arnab Bhattacharya Xu Ma Draguna L. Vrabie Pacific Northwest National Laboratory Richland WA 99352 USA

关键词： Model Predictive Control Mixed-Integer Optimization Energy Systems disjunctive programming

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A study on optimality and duality theorems of nonlinear generalized disjunctive fractional programming

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MATHEMATICAL AND COMPUTER MODELLING 2008年第1-2期48卷 69-82页

作者： Ammar, E. E. Tanta Univ Dept Math Fac Sci Tanta Egypt

This paper is concerned with the study of necessary and sufficient optimality conditions for convex - concave generalized fractional disjunctive programming problems for which the decision set is the union of a family of convex sets. The Lagrangian function for such problems is defined and the Kuhn - Tucker Saddle and Stationary points are characterized. In addition, some important theorems related to the Kuhn - Tucker problem for saddle and stationary points are established. Moreover, a general dual problem is formulated and weak, strong and converse duality theorems are proved. Throughout the presented paper illustrative examples are given to clarify and implement the developed theory. (C) 2007 Elsevier Ltd. All rights reserved.

关键词： generalized fractional programming disjunctive programming convexity concavity optimality duality

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A disjunctive CUTTING PLANE ALGORITHM FOR BILINEAR programming

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SIAM JOURNAL ON OPTIMIZATION 2024年第4期34卷 3286-3313页

作者： Rahimian, Hamed Mehrotra, Sanjay Clemson Univ Dept Ind Engn Clemson SC 29634 USA Northwestern Univ Dept Ind Engn & Management Sci Evanston IL 60208 USA

In this paper, we present and analyze a finitely convergent disjunctive cutting plane algorithm to obtain an \epsilon -optimal solution or detect the infeasibility of a general nonconvex continuous bilinear program. While the cutting planes are obtained like Saxena, Bonami, and Lee [Math. Prog., the algorithm that guarantees finite convergence is exploring near-optimal extreme point solutions to a current relaxation at each iteration. In this sense, the presented algorithm and its analysis extend the work Owen and Mehrotra [Math. Prog., 89 (2001), pp. 437--448] for solving mixed-integer linear programs to the general bilinear programs.

关键词： rogramming nonconvex programming disjunctive programming global optimization cutting planes

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Monoidal strengthening of simple V-polyhedral disjunctive cuts

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MATHEMATICAL programming 2025年第1期210卷 567-590页

作者： Kazachkov, Aleksandr M. Balas, Egon Univ Florida Ind & Syst Engn Gainesville FL 32611 USA Carnegie Mellon Univ Tepper Sch Business Pittsburgh PA 15213 USA

disjunctive cutting planes can tighten a relaxation of a mixed-integer linear program. Traditionally, such cuts are obtained by solving a higher-dimensional linear program, whose additional variables cause the procedure to be computationally prohibitive. Adopting a V-polyhedral perspective is a practical alternative that enables the separation of disjunctive cuts via a linear program with only as many variables as the original problem. The drawback is that the classical approach of monoidal strengthening cannot be directly employed without the values of the extra variables appearing in the extended formulation, which constitute a certificate of validity of the cut. We derive how to compute this certificate from a solution to the linear program generating V-polyhedral disjunctive cuts. We then present computational experiments with monoidal strengthening of cuts from disjunctions with as many as 64 terms. Some instances are dramatically impacted, with strengthening increasing the gap closed by the cuts from 0 to 100%. However, for larger disjunctions, monoidal strengthening appears to be less effective, for which we identify a potential cause. Lastly, the certificates of validity also enable us to verify which disjunctive cuts are equivalent to intersection cuts, which happens increasingly rarely for larger disjunctions.

关键词： Mixed-integer linear programming Cutting planes disjunctive programming Monoidal strengthening

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Comparison of global optimization algorithms for the design of water-using networks

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COMPUTERS & CHEMICAL ENGINEERING 2013年 52卷 249-261页

作者： Castro, Pedro M. Teles, Joao P. Lab Nacl Energia & Geol Unidade Modelacao & Optimizacao Sistemas Energet P-1649038 Lisbon Portugal

We address a special class of bilinear process network problems with global optimization algorithms iterating between a lower bound provided by a mixed-integer linear programming (MILP) formulation and an upper bound given by the solution of the original nonlinear problem (NLP) with a local solver. Two conceptually different relaxation approaches are tested, piecewise McCormick envelopes and multiparametric disaggregation, each considered in two variants according to the choice of variables to partition/parameterize. The four complete MILP formulations are derived from disjunctive programming models followed by convex hull reformulations. The results on a set of test problems from the literature show that the algorithm relying on multiparametric disaggregation with parameterization of the concentrations is the best performer, primarily due to a logarithmic as opposed to linear increase in problem size with the number of partitions. The algorithms are also compared to the commercial solvers BARON and GloMIQO through performance profiles. (C) 2013 Elsevier Ltd. All rights reserved.

关键词： Mathematical modeling Nonlinear programming Integer programming disjunctive programming Process design Water minimization

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