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Small and strong formulations for unions of convex sets from the Cayley embedding

MATHEMATICAL programming 2019年第1-2期177卷 21-53页

作者： Vielma, Juan Pablo MIT Sloan Sch Management 77 Massachusetts Ave Cambridge MA 02139 USA

There is often a significant trade-off between formulation strength and size in mixed integer programming. When modeling convex disjunctive constraints (e.g. unions of convex sets), adding auxiliary continuous variables can sometimes help resolve this trade-off. However, standard formulations that use such auxiliary continuous variables can have a worse-than-expected computational effectiveness, which is often attributed precisely to these auxiliary continuous variables. For this reason, there has been considerable interest in constructing strong formulations that do not use continuous auxiliary variables. We introduce a technique to construct formulations without these detrimental continuous auxiliary variables. To develop this technique we introduce a natural non-polyhedral generalization of the Cayley embedding of a family of polytopes and show it inherits many geometric properties of the original embedding. We then show how the associated formulation technique can be used to construct small and strong formulation for a wide range of disjunctive constraints. In particular, we show it can recover and generalize all known strong formulations without continuous auxiliary variables.

关键词： mixed integer nonlinear programming mixed integer programming formulations Disjunctive constraints

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Competitive facility location problem with attractiveness adjustment of the follower A bilevel programming model and its solution

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2011年第3期208卷 206-220页

作者： Kucukaydin, Hande Aras, Necati Altinel, I. Kuban Bogazici Univ Dept Ind Eng Istanbul Turkey

We are concerned with a problem in which a firm or franchise enters a market by locating new facilities where there are existing facilities belonging to a competitor The firm aims at finding the location and attractiveness of each facility to be opened so as to maximize its profit The competitor on the other hand can react by adjusting the attractiveness of its existing facilities with the objective of maximizing its own profit The demand is assumed to be aggregated at certain points in the plane and the facilities of the firm can be located at predetermined candidate sites We employ Huff's gravity-based rule in modeling the behavior of the customers where the fraction of customers at a demand point that visit a certain facility is proportional to the facility attractiveness and Inversely proportional to the distance between the facility site and demand point We formulate a bilevel mixed-integer nonlinear programming model where the firm entering the market is the leader and the competitor is the follower In order to find the optimal solution of this model we convert It Into an equivalent one-level mixed integer nonlinear program so that it can be solved by global optimization methods Apart from reporting computational results obtained on a set of randomly generated instances we also compute the benefit the leader firm derives from anticipating the competitor s reaction of adjusting the attractiveness levels of its facilities The results on the test Instances indicate that the benefit is 58 33% on the average (C) 2010 Elsevier B V All rights reserved

关键词： Competitive facility location Bilevel programming mixed integer nonlinear programming Variable facility attractiveness

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Interval based MINLP superstructure synthesis of combined heat and mass exchanger networks

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CHEMICAL ENGINEERING RESEARCH & DESIGN 2009年第11A期87卷 1536-1542页

作者： Isafiade, A. J. Fraser, D. M. Univ Cape Town Dept Chem Engn ZA-7701 Cape Town South Africa

An interval based superstructure approach for combining the synthesis of heat and mass exchanger networks is presented in this paper. The technique involves combining the interval based MINLP superstructure (IBMS) for the synthesis of heat exchanger networks (HENS) with that of mass exchanger networks (MENs). The two networks are made to interact through the lean stream in the mass exchange network. The new approach involves the use of the lean substream concept to explore potential mass exchange temperatures. An example which involves a one-lean stream mass exchange problem alongside regeneration and hot and cold utilities is presented. (C) 2009 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

关键词： Heat exchange network synthesis Mass exchange network synthesis mixed integer nonlinear programming

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Optimal design of multiple-objective Lot Quality Assurance Sampling (LQAS) plans

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BIOMETRICS 2019年第2期75卷 572-581页

作者： Duarte, Belmiro P. M. Wong, Weng Kee Univ Coimbra Dept Chem Engn Ctr Invest Proc Quim & Prod Floresta Polo 2R Silvio Lima P-3030790 Coimbra Portugal Polytech Inst Coimbra ISEC Dept Chem & Biol Engn R Pedro Nunes P-3030199 Coimbra Portugal Univ Calif Los Angeles Fielding Sch Publ Hlth Dept Biostat 10833 Le Conte Ave Los Angeles CA 90095 USA

Lot Quality Assurance Sampling (LQAS) plans are widely used for health monitoring purposes. We propose a systematic approach to design multiple-objective LQAS plans that meet user-specified type 1 and 2 error rates and targets for selected diagnostic accuracy metrics. These metrics may include sensitivity, specificity, positive predictive value, and negative predictive value in high or low anticipated prevalence rate populations. We use mixed integer nonlinear programming (MINLP) tools to implement our design methodology. Our approach is flexible in that it can directly generate classic LQAS plans that control error rates only and find optimal LQAS plans that meet multiple objectives in terms of diagnostic metrics. We give examples, compare results with the classic LQAS and provide an application using a malaria outcome indicator survey in Mozambique.

关键词： diagnostic measures disease monitoring programs Lot Quality Assurance Sampling mixed integer nonlinear programming optimal design public health policy

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A multi-period and multi-criterion optimization model integrating multiple input configurations, reuse, and disposal options for a wastewater treatment facility

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JOURNAL OF CLEANER PRODUCTION 2019年 231卷 1437-1449页

作者： Ang, Michael S. Duyag, Jensen Tee, Kimberly C. Sy, Charlie L. De La Salle Univ Ind Engn Dept 2401 Taft Ave Manila 0922 Philippines

Rapid global industrial development has led to a significant increase in waste generation, including wastewater. Improper disposal of wastewater leads to the degradation of water bodies, endangering marine life and posing health hazards to the nearby communities. This study addresses the current lack of integration in the design of wastewater treatment plants and the challenge presented by the conflicting criteria of economic impact and environmental cost. A multi-period and multi-criterion non-linear programming model for a wastewater treatment plant that simultaneously considers economic and environmental tradeoffs, alternative plant configurations, disposal and reuse options is then developed. The study considers the variability of inputs in the form of water quality and quantity in order to demonstrate the natural variations presented by wastewater sources across a planning horizon. The proposed model is applied to a case study of an actual water utility company in the Philippines. It was seen that the integration of disposal and reuse options had facilitated the realization of improved economic and environmental benefits as it was able to match the effluent water quality to the best suited option. The model enabled the improvement of the treatment process of wastewater inputs by considering alternative methods of entry such as series and parallel configurations instead of just having a mixed input configuration. This significantly improved relevant metrics such as processing time and operational costs. (C) 2019 Elsevier Ltd. All rights reserved.

关键词： Multi-period optimization Environment mixed integer nonlinear programming Multi-objective optimization Wastewater treatment

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Relax-tighten-round algorithm for optimal placement and control of valves and chlorine boosters in water networks

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2021年第2期295卷 690-698页

作者： Pecci, Filippo Stoianov, Ivan Ostfeld, Avi Imperial Coll London Dept Civil & Environm Engn InfraSense Labs London SW7 2BB England Israel Inst Technol Fac Civil & Environm Engn Technion IL-32000 Haifa Israel

In this paper, a new mixed integer nonlinear programming formulation is proposed for optimally placing and operating pressure reducing valves and chlorine booster stations in water distribution networks. The objective is the minimization of average zone pressure, while penalizing deviations from a target chlorine concentration. We propose a relax-tighten-round algorithm based on tightened polyhedral relaxations and a rounding scheme to compute feasible solutions, with bounds on their optimality gaps. This is because off-the-shelf global optimization solvers failed to compute feasible solutions for the considered non-convex mixed integer nonlinear program. The implemented algorithm is evaluated using three benchmarking water networks, and they are shown to outperform off-the-shelf solvers, for these case studies. The proposed heuristic has enabled the computation of good quality feasible solutions in most instances, with bounds on the optimality gaps that are comparable to the order of uncertainty observed in operational water network models. (c) 2021 Elsevier B.V. All rights reserved.

关键词： Global optimization mixed integer nonlinear programming Water networks Pressure management Water quality

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Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption

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COMPUTERS & CHEMICAL ENGINEERING 2002年第11期26卷 1581-1593页

作者： Björk, KM Westerlund, T Abo Akad Univ Proc Design Lab FIN-20500 Turku Finland

In this paper, a new global optimization approach for the synthesis of heat exchanger networks is presented. The Synheat model by [Comp. Chem. Eng. 1 (1990) 1165] uses a promising superstructure that includes the most common heat exchanger structures, and optimizes utility costs, the number of units and heat exchanger areas simultaneously. With some minor modifications of the model, it is possible to apply a new global optimization strategy to the problem. The heart of the strategy is to convexify signomial terms, and create approximate convexified subproblems. If the Synheat model is extended in an appropriate way, the isothermal mixing assumption can be removed. Applying the new global optimization strategy to the model allowing non-isothermal mixing makes it possible to find truly optimal network configurations for the case of constant heat-capacity flow rates and heat transfer coefficients. Some examples to illustrate the global optimization strategy and to illustrate that the Synheat model can exclude optimal configurations due to the isothermal mixing assumption are also given in this paper. (C) 2002 Elsevier Science Ltd. All rights reserved.

关键词： heat exchanger network synthesis global optimization mixed integer nonlinear programming

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Global solution of optimization problems with signomial parts

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DISCRETE OPTIMIZATION 2008年第1期5卷 108-120页

作者： Porn, Ray Bjork, Kaj-Mikael Westerlund, Tapio Abo Akad Univ IAMSR FIN-20540 Turku Finland Swedish Polytechn Sector Technol & Commun FIN-65201 Vaasa Finland Abo Akad Univ Proc Design Lab FIN-20500 Turku Finland

In this paper a new approach for the global solution of nonconvex MINLP (mixed integer nonlinear programming) problems that contain signomial (generalized geometric) expressions is proposed and illustrated. By applying different variable transformation techniques and a discretization scheme a lower bounding convex MINLP problem can be derived. The convexified MINLP problem can be solved with standard methods. The key element in this approach is that all transformations are applied termwise. In this way all convex parts of the problem are left unaffected by the transformations. The method is illustrated by four example problems. (C) 2007 Elsevier B.V. All rights reserved.

关键词： convexification global optimization mixed integer nonlinear programming signomials variable transformations

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A Lagrangian heuristic and GRASP for the hub-and-spoke network system with economies-of-scale and congestion

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TRANSPORTATION RESEARCH PART C-EMERGING TECHNOLOGIES 2019年第May期102卷 249-273页

作者： Alkaabneh, Faisal Diabat, Ali Elhedhli, Samir Cornell Univ Sch Civil & Environm Engn Ithaca NY 14853 USA New York Univ Abu Dhabi Div Engn Abu Dhabi 129188 U Arab Emirates NYU Tandon Sch Engn Dept Civil & Urban Engn Brooklyn NY 11201 USA Univ Waterloo Dept Management Sci 200 Univ Ave West Waterloo ON N2L 3G1 Canada

We consider a hub-and-spoke network design problem with inter-hub economies-of-scale and hub congestion. We explicitly model the economies-of-scale as a concave piece-wise linear function and congestion as a convex function. The problem is modeled as a nonlinear mixed integer program that is difficult to solve directly since the objective function has both convex and concave nonlinear terms and hence finding an optimal solution may not be easy. A Lagrangian approach is proposed to obtain tight upper and lower bounds. The Lagrangian decomposition exploits the structure of the problem and decomposes it to convex and concave subproblems. Furthermore, we add some valid inequalities to accelerate the convergence rate of the Lagrangian heuristic. To tackle large instances, a Greedy Randomized Adaptive Search Procedure (GRASP) is developed. Both the Lagrangian heuristic and GRASP provide high-quality solutions within reasonable computational time. The optimal designs of hub-and-spoke networks with nonlinear inter-hub economies-of-scale and congestion at hub locations are analyzed in comparison with other models in the literature to demonstrate the significant impact of modeling nonlinearity in economies-of-scale and congestion simultaneously rather than independently.

关键词： Hub-and-Spoke design network Congestion Economies-of-scale mixed integer nonlinear programming Lagrangian relaxation GRASP

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Towards a lifecycle oriented design of infrastructure by mathematical optimization

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OPTIMIZATION AND ENGINEERING 2019年第1期20卷 215-249页

作者： Kufner, T. Leugering, G. Martin, A. Medgenberg, J. Schelbert, J. Schewe, L. Stingl, M. Strohmeyer, C. Walther, M. Friedrich Alexander Univ Erlangen Nurnberg FAU Dept Math Cauerstr 11 D-91058 Erlangen Germany Aenergen GmbH Lange Rotterstr 66 D-68167 Mannheim Germany

Today's infrastructures are mainly designed heuristically using state-of-the-art simulation software and engineering approaches. However, due to complexity, only part of the restrictions and costs that show up during the lifecycle can be taken into account. In this paper, we focus on a typical and important class of infrastructure problems, the design of high-pressure steam pipes in power plants, and describe a holistic approach taking all design, physical, and technical constraints and the costs over the full lifecycle into account. The problem leads to a large-scale mixed-integer optimization problem with partial differential equation (PDE) constraints which will be addressed hierarchically. The hierarchy consists of a combinatorial and a PDE-constrained optimization problem. The final design is evaluated with respect to damage, using beam models that are nonlinear with respect to kinematics as well as constitutive law. We demonstrate the success of our approach on a real-world instance from our industrial partner Bilfinger SE.

关键词： Structural optimization Lifecycle oriented design mixed integer nonlinear programming

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