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A new box selection criterion in interval Bernstein global optimization algorithm for MINLPs

GRANULAR COMPUTING 2020年第2期5卷 155-168页

作者： Patil, Bhagyesh V. Zidna, Ahmed Cambridge Ctr Adv Res & Educ Singapore Ltd 1 Create Way CREATE Campus Singapore 138602 Singapore Univ Lorraine Lab Gnie Informat Prod & Maintenance LGIPM Lorraine France

mixed-integer nonlinear programming (MINLP) problems are known to be challenging due to the involvement of nonlinear mathematical relations and combinatorial complexities. In this paper, an interval form-based Bernstein global optimization algorithm is presented to solve polynomial MINLP problems. The major contribution of this paper is a new box selection criterion for this interval form-based Bernstein global optimization algorithm. This new box selection criterion promotes fast convergence of the Bernstein algorithm. Moreover, the new box selection criterion allows the construction of a hybrid branch-and-bound framework for the Bernstein algorithm, wherein a local search method is used to obtain a good upper bound on the global minimum, and the Bernstein form is employed to obtain a valid lower bound on the global minimum. We show with numerical results that it is possible to obtain a significant reduction in the number of subdivisions required with this new box selection criterion. Furthermore, the Bernstein algorithm with this new box selection criterion is evaluated numerically on a variety of small- to medium-size MINLP problem instances and its performance is compared with the previously reported Bernstein global optimization algorithm as well as with a few state-of-the-art MINLP solvers. The results of the numerical studies demonstrate satisfactory performance in terms of the chosen performance metrics. Finally, the trim-loss minimization problem from the paper industry is solved to demonstrate the practical applicability of the Bernstein algorithm.

关键词： Bernstein polynomial form Global optimization Local search mixed-integer nonlinear programming Trim-loss minimization problem

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Price-and-verify: a new algorithm for recursive circle packing using Dantzig-Wolfe decomposition

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ANNALS OF OPERATIONS RESEARCH 2020年第2期284卷 527-555页

作者： Gleixner, Ambros Maher, Stephen J. Mueller, Benjamin Pedroso, Joao Pedro Zuse Inst Berlin Takustr 7 D-14195 Berlin Germany Univ Lancaster Dept Management Sci Lancaster LA1 4YX England Univ Porto Rua Campo Alegre P-4169007 Porto Portugal

Packing rings into a minimum number of rectangles is an optimization problem which appears naturally in the logistics operations of the tube industry. It encompasses two major difficulties, namely the positioning of rings in rectangles and the recursive packing of rings into other rings. This problem is known as the Recursive Circle Packing Problem (RCPP). We present the first dedicated method for solving RCPP that provides strong dual bounds based on an exact Dantzig-Wolfe reformulation of a nonconvex mixed-integer nonlinear programming formulation. The key idea of this reformulation is to break symmetry on each recursion level by enumerating one-level packings, i.e., packings of circles into other circles, and by dynamically generating packings of circles into rectangles. We use column generation techniques to design a "price-and-verify" algorithm that solves this reformulation to global optimality. Extensive computational experiments on a large test set show that our method not only computes tight dual bounds, but often produces primal solutions better than those computed by heuristics from the literature.

关键词： mixed-integer nonlinear programming Dantzig-Wolfe decomposition Symmetry breaking Global optimization Recursive circle packing

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Simultaneous synthesis of mass exchange networks for waste minimization

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COMPUTERS & CHEMICAL ENGINEERING 2005年第7期29卷 1561-1576页

作者： Chen, CL Hung, PS Natl Taiwan Univ Dept Chem Engn Taipei 10617 Taiwan

The paper deals with the synthesis problem of mass exchange networks (MEN's) for waste minimization by adopting a mathematical programming approach based on the stage-wise superstructure representation of the MEN's, analogous to the one introduced by [Yee, T. F., & Grossmann, I. E. (1990a). Simultaneous optimization models for heat integration. I. Area and energy targeting and modeling of multi-stream exchangers. Computers and Chemical Engineering 14, 1151;Yee, T. R, & Grossmann, I. E. (1990b). Simultaneous optimization models for heat integration. II. Heat exchanger network synthesis. Computers and Chemical Engineering 14, 1165] for synthesis of heat exchange networks (HEN's). This stage-wise superstructure-based representation cannot only handle multiple transferable components and reactive separating agents directly, but also be extended to include regeneration networks straightforwardly. Not using any heuristics that are based on the concept of pinch points, the proposed superstructure-based representation for MEN's is formulated as a mixed-integer nonlinear programming (MINLP) optimization model, and therefore the operating cost for the external lean mass separating agents as well as the regenerating agents and the annualized equipment cost for exchange units can be minimized simultaneously. Four benchmark examples from literatures-including those with single recovery component, multiple waste components, reactive mass separating agent, and regenerating streams-are examined to illustrate the applicability of proposed approach for synthesis of various MEN's. (c) 2004 Elsevier Ltd. All fights reserved.

关键词： mass exchange network synthesis superstructure mixed-integer nonlinear programming

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Delay-Constrained Shortest Paths: Approximation Algorithms and Second-Order Cone Models

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2015年第3期164卷 1051-1077页

作者： Frangioni, Antonio Galli, Laura Scutella, Maria Grazia Univ Pisa Dipartimento Informat Largo B Pontecorvo 3 I-56127 Pisa Italy

Routing real-time traffic with maximum packet delay in contemporary telecommunication networks requires not only choosing a path but also reserving transmission capacity along its arcs, as the delay is a nonlinear function of both components. The problem is known to be solvable in polynomial time under quite restrictive assumptions, i.e., equal rate allocations (all arcs are reserved the same capacity) and identical reservation costs, whereas the general problem is -hard. We first extend the approaches to the equal rate allocation (ERA) version to a pseudo-polynomial Dynamic programming one for integer arc costs and a FPTAS for the case of general arc costs. We then show that the general problem can be formulated as a mixed-integer Second-Order Cone (SOCP) program and therefore solved with off-the-shelf technology. We compare two formulations: one based on standard big-M constraints and one where Perspective Reformulation techniques are used to tighten the continuous relaxation. Extensive computational experiments on both real-world networks and randomly generated realistic ones show that the ERA approach is fast and provides an effective heuristic for the general problem whenever it manages to find a solution at all, but it fails for a significant fraction of the instances that the SOCP models can solve. We therefore propose a three-pronged approach that combines the fast running time of the ERA algorithm and the effectiveness of the SOCP models, and show that it is capable of solving realistic-sized instances with high accuracy at different levels of network load in a time compatible with real-time usage in an operating environment.

关键词： Delay-constrained routing Approximation algorithms mixed-integer nonlinear programming Second-order cone model Perspective reformulation

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A unique optimization model for deterministic bundle pricing of two products with limited stock

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INTERNATIONAL JOURNAL OF SYSTEM ASSURANCE ENGINEERING AND MANAGEMENT 2017年第2期8卷 1154-1160页

作者： Azadeh, Ali Songhori, Hassan Salehi, Nima Univ Tehran Sch Ind & Syst Engn Ctr Excellence Intelligent Based Expt Mech Coll Engn Tehran Iran Univ Tehran Dept Engn Optimizat Res Coll Engn Tehran Iran Univ Michigan Dept Ind & Operat Engn Ann Arbor MI 48109 USA

In this study a firm in a monopoly environment is considered. The firm sells two types of perishable products using a commodity bundling practice. This study aims to find the optimal strategy of selling these products. We determine whether they should be offered separately or in a bundle, define their optimal prices and determine the initial amount of the bundle which should be made from components with limited stocks. A benefit-lost cost in a case that the customer segment does not find their desirable product is considered along with a shortage cost in a case that the customer segment selects a product or the bundle but there is not enough of it to satisfy all of the customer segment demands. In this study several different customer segments with different behaviors and reservation prices are assumed. The problem is solved using the mixed-integer Non-Linear programming solver in LINGO software and a Genetic Algorithm. Finally, the superiority of obtained model and results are presented. This is the first study to locate optimal strategy of selling two types of perishable products using a commodity bundling practice. This study introduces a new mathematical optimization model based on previous gaps. Moreover, it covers the significant gaps of previous studies with respect to both practical and theoretical aspects.

关键词： Bundle pricing Limited stocks Degree of contingency Customer demands Reservation prices Customer segments Optimization mixed-integer nonlinear programming Genetic algorithm

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Solving a mathematical model integrating unequal-area facilities layout and part scheduling in a cellular manufacturing system by a genetic algorithm

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SPRINGERPLUS 2016年第1期5卷 1页

作者： Ebrahimi, Ahmad Kia, Reza Komijan, Alireza Rashidi Islamic Azad Univ Firoozkooh Branch Dept Ind Engn Firoozkooh Iran

In this article, a novel integrated mixed-integer nonlinear programming model is presented for designing a cellular manufacturing system (CMS) considering machine layout and part scheduling problems simultaneously as interrelated decisions. The integrated CMS model is formulated to incorporate several design features including part due date, material handling time, operation sequence, processing time, an intra-cell layout of unequal-area facilities, and part scheduling. The objective function is to minimize makespan, tardiness penalties, and material handling costs of inter-cell and intra-cell movements. Two numerical examples are solved by the Lingo software to illustrate the results obtained by the incorporated features. In order to assess the effects and importance of integration of machine layout and part scheduling in designing a CMS, two approaches, sequentially and concurrent are investigated and the improvement resulted from a concurrent approach is revealed. Also, due to the NP-hardness of the integrated model, an efficient genetic algorithm is designed. As a consequence, computational results of this study indicate that the best solutions found by GA are better than the solutions found by B&B in much less time for both sequential and concurrent approaches. Moreover, the comparisons between the objective function values (OFVs) obtained by sequential and concurrent approaches demonstrate that the OFV improvement is averagely around 17 % by GA and 14 % by B&B.

关键词： Cellular manufacturing system Machine layout Part scheduling mixed-integer nonlinear programming Genetic algorithm

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Siting and Sizing of Facilities under Probabilistic Demands

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2011年第2期149卷 420-440页

作者： Fernandes, Luis M. Judice, Joaquim J. Sherali, Hanif D. Antunes, Antonio P. Inst Politecn Tomar Tomar Portugal Inst Telecomunicacoes Tomar Portugal Univ Coimbra Dept Matemat P-3000 Coimbra Portugal Univ Coimbra Dept Engn Civil P-3000 Coimbra Portugal Inst Telecomunicacoes Coimbra Portugal Virginia Polytech Inst & State Univ Grado Dept Ind & Syst Engn Blacksburg VA USA

In this paper a discrete location model for non-essential service facilities planning is described, which seeks the number, location, and size of facilities, that maximizes the total expected demand attracted by the facilities. It is assumed that the demand for service is sensitive to the distance from facilities and to their size. It is also assumed that facilities must satisfy a threshold level of demand (facilities are not economically viable below that level). A mixed-integer nonlinear programming (MINLP) model is proposed for this problem. A branch-and-bound algorithm is designed for solving this MINLP and its convergence to a global minimum is established. A finite procedure is also introduced to find a feasible solution for the MINLP that reduces the overall search in the binary tree generated by the branch-and-bound algorithm. Some numerical results using a GAMS/MINOS implementation of the algorithm are reported to illustrate its efficacy and efficiency in practice.

关键词： Facility location models mixed-integer nonlinear programming Discrete optimization Global optimization

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Constraint qualification failure in action

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OPERATIONS RESEARCH LETTERS 2016年第4期44卷 503-506页

作者： Hijazi, Hassan Liberti, Leo Australian Natl Univ CSIRO Data61 Canberra ACT 2601 Australia Ecole Polytech CNRS LIX F-91128 Palaiseau France

This note presents a theoretical analysis of disjunctive constraints featuring unbounded variables. In this framework, classical modeling techniques, including big-M approaches, are not applicable. We introduce a lifted second-order cone formulation of such on/off constraints and discuss related constraint qualification issues. A solution is proposed to avoid solvers' failure. (C) 2016 Elsevier B.V. All rights reserved.

关键词： mixed-integer nonlinear programming Disjunctive programming Second-order cone programming On/off constraints Constraint qualification

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Global optimization of general non-convex problems with intermediate bilinear substructures

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OPTIMIZATION METHODS & SOFTWARE 2014年第3期29卷 442-462页

作者： Zorn, Keith Sahinidis, Nikolaos V. Carnegie Mellon Univ Dept Chem Engn Pittsburgh PA 15213 USA

This work considers the global optimization of general non-convex nonlinear and mixed-integer nonlinear programming (MINLP) problems with underlying bilinear substructures. We combine reformulation-linearization techniques and advanced convex envelope construction techniques to produce tight subproblem formulations for these underlying structures. When incorporated as linear cutting planes, these relaxation strengthening strategies are highly effective at tightening standard linear programming relaxations generated by factorable programming techniques. Because the size of these augmented linear relaxations increases exponentially with the number of variables, we employ cut filtering and selection strategies to ensure that the tightened subproblems are solved efficiently. We introduce algorithms for bilinear substructure detection, cutting plane identification, cut filtering, and cut selection and embed the proposed implementation in Branch-and-Reduce Optimization Navigator at every node in the branch-and-bound tree. A computational study including problem instances from standard literature test libraries is included to assess the performance of the proposed implementation. Results show that underlying bilinear substructures are identified in 30% of the problems in GLOBALLib and MINLPLib and that the exploitation of these structures significantly reduces computational time, branch-and-bound tree size, and required memory.

关键词： branch-and-bound global optimization factorable polyhedral relaxation reformulation-linearization techniques mixed-integer nonlinear programming

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Modelling systems defined by RTD curves

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COMPUTERS & CHEMICAL ENGINEERING 2008年第12期32卷 3112-3120页

作者： Hocine, S. Pibouleau, L. Azzaro-Pantel, C. Domenech, S. ENSIACET Lab Genie Chim UMR CNRS INP UPS 5503 F-31077 Toulouse 4 France

The RTD theory is commonly used for describing flow patterns in a large class of applications, and particularly for ventilated enclosures. Experimental RTD curves are used for modelling these premises with an application in the nuclear industry for predicting the airborne pollutant transfers in order to prevent radiological risk. An approach based on a superstructure involving interlinked elementary flow patterns such as CSTRs, PFRs, recycles and by-passes is implemented. In order to propose a generic and easy to use tool, the associated large-scale MINLP problem is solved by using the GAMS package. After a validation phase on examples with known solutions, a laboratory enclosure, called MELANIE, used in the nuclear industry is modelled. The comparison between experimental RTD curve and the ones obtained from models extracted from superstructures shows good agreement. The superstructure-based solution procedure constitutes an efficient and intermediate way between numerical simulations using CFD codes and experimental determinations of characteristic parameters, which are both difficult to implement in the case of large and cluttered systems which are typical of the nuclear industry. (C) 2008 Elsevier Ltd. All rights reserved.

关键词： Air flow patterns Ventilation Residence time distribution Superstructure mixed-integer nonlinear programming

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