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Global solutions of nonconvex standard quadratic programs via mixed integer linear programming reformulations

JOURNAL OF GLOBAL OPTIMIZATION 2021年第2期81卷 293-321页

作者： Gondzio, Jacek Yildirim, E. Alper Univ Edinburgh Sch Math James Clerk Maxwell Bldg Edinburgh EH9 3FD Midlothian Scotland

A standard quadratic program is an optimization problem that consists of minimizing a (nonconvex) quadratic form over the unit simplex. We focus on reformulating a standard quadratic program as a mixed integer linear programming problem. We propose two alternative formulations. Our first formulation is based on casting a standard quadratic program as a linear program with complementarity constraints. We then employ binary variables to linearize the complementarity constraints. For the second formulation, we first derive an overestimating function of the objective function and establish its tightness at any global minimizer. We then linearize the overestimating function using binary variables and obtain our second formulation. For both formulations, we propose a set of valid inequalities. Our extensive computational results illustrate that the proposed mixed integer linear programming reformulations significantly outperform other global solution approaches. On larger instances, we usually observe improvements of several orders of magnitude.

关键词： Nonconvex optimization Quadratic programming mixed integer linear programming Global optimization

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An Effective Problem Decomposition Method for Scheduling of Diffusion Processes Based on mixed integer linear programming

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IEEE TRANSACTIONS ON SEMICONDUCTOR MANUFACTURING 2014年第3期27卷 357-363页

作者： Jung, Chihyun Pabst, Detlef Ham, Myoungsoo Stehli, Marcel Rothe, Marcel GLOBALFOUNDRIES Inc Malta NY 12020 USA Liberty Univ Sch Engn & Computat Sci Lynchburg VA 24515 USA GLOBALFOUNDRIES Inc D-10109 Dresden Germany

Diffusion processes in semiconductor fabrication facilities (Fabs) refer to the series of processes from wafer cleaning processes to furnace processes. Most furnace tools are batch tools, with large batch sizes, and have relatively long process times, when compared to the other processes. Strict time window constraints link cleaning processes with furnace processes for quality control. Those operational requirements for diffusion processes make their scheduling very difficult. This paper proposes an advanced scheduling approach based on a rolling horizon scheduling concept. Due to the combinatorial nature of the scheduling problem, the complexity of the problem increases exponentially, when the number of jobs and tools increase. However, the computation time allowed for the scheduler is limited in practice, because the variability in most Fabs requires schedulers to update the schedule in short intervals. We suggest an mixed integer linear programming model for diffusion processes, and propose an effective decomposition method to deal with this complexity problem. The decomposition method repeats multiple scheduling iterations, as it gradually extends the number of runs on tools, enabling the scheduler to generate near-optimal schedules in limited time intervals. The scheduler could make large improvements on key performance indicators, such as time window violation rates, batch sizes, throughput, etc. The software architecture of the scheduler implementation is also addressed in this paper.

关键词： Diffusion processes scheduling batch tools time window constraints mixed integer linear programming decomposition method

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Optimal placement of capacitors in a radial network using conic and mixed integer linear programming

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ELECTRIC POWER SYSTEMS RESEARCH 2008年第6期78卷 941-948页

作者： Jabr, R. A. Notre Dame Univ Elect Comp & Commun Engn Dept Zouk Mosbeh Lebanon

This paper considers the problem of optimally placing fixed and switched type capacitors in a radial distribution network. The aim of this problem is to minimize the costs associated with capacitor banks, peak power, and energy losses whilst satisfying a pre-specified set of physical and technical constraints. The proposed solution is obtained using a two-phase approach. In phase-I, the problem is formulated as a conic program in which all nodes are candidates for placement of capacitor banks whose sizes are considered as continuous variables. A global solution of the phase-I problem is obtained using an interior-point based conic programming solver. Phase-II seeks a practical optimal solution by considering capacitor sizes as discrete variables. The problem in this phase is formulated as a mixed integer linear program based on minimizing the L1-norm of deviations from the phase-I state variable values. The solution to the phase-II problem is obtained using a mixed integer linear programming solver. The proposed method is validated via extensive comparisons with previously published results. (C) 2007 Elsevier B.V. All rights reserved.

关键词： capacitor compensated distribution lines mixed integer linear programming nonlinear programming optimization methods reactive power control

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Optimization Model of Key Equipment Maintenance Scheduling for an AC/DC Hybrid Transmission Network Based on mixed integer linear programming

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ENERGIES 2020年第4期13卷 1011页

作者： Cai, Jie Guo, Shuyu Liao, Shuang Chen, Xing Miao, Shihong Li, Yaowang State Grid Hubei Elect Power Co Ltd Econ Res Inst Wuhan 430062 Peoples R China Huazhong Univ Sci & Technol Sch Elect & Elect Engn State Key Lab Adv Electromagnet Engn & Technol Hubei Elect Power Secur & High Efficiency Key Lab Wuhan 430074 Peoples R China State Grid Xinjiang Elect Power Co Ltd Urumqi Power Supply Co Urumqi 830001 Peoples R China

The unbalanced distribution of resource and consuming centers in China has prompted the AC/DC hybrid transmission technology. The maintenance scheduling of an AC/DC hybrid transmission network is the key technology to ensure its safety and reliability. In this study, the mutual influence mechanism of an AC/DC system in a maintenance period was analyzed in detail. The overhead transmission line and transformer are key equipment within an AC/DC hybrid transmission network, and an optimization model of the key equipment maintenance scheduling was established. The objective of the model was to improve the system reliability during the maintenance scheduling. By considering the constraints of maintenance cost, maintenance resources, and maintenance workload, the maintenance scheduling of overhead transmission lines and transformer branches was obtained. The over-limit situation of power flow and the weakness of the system during the maintenance period was evaluated. The "double-layer substitution method" was adopted to convert the nonlinear constraints into its bilinear formulation such that it could then be solved. The random number sampling method was used to quantify the system reliability, and the commercial optimization software was used to solve the optimized scheduling. Based on the improved IEEE RTS-79 system and the Hubei Province electrical system, the simulation results showed the effectiveness of the proposed method.

关键词： AC DC hybrid transmission network mixed integer linear programming overhead transmission line and transformer optimization maintenance scheduling

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mixed integer linear programming and constraint programming models for the online printing shop scheduling problem

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COMPUTERS & OPERATIONS RESEARCH 2020年 123卷 105020-105020页

作者： Lunardi, Willian T. Birgin, Ernesto G. Laborie, Philippe Ronconi, Debora P. Voos, Holger Univ Luxembourg 29 John F Kennedy L-1855 Luxembourg Luxembourg Univ Sao Paulo Inst Math & Stat Dept Comp Sci Rua Matao 1010Cidade Univ BR-05508090 Sao Paulo SP Brazil IBM France 9 Rue VerdunBP 85 F-94253 Gentilly France Univ Sao Paulo Polytech Sch Dept Prod Engn Av Prof Luciano Cualberto 1380Cidade Univ BR-05508010 Sao Paulo SP Brazil

In this work, the online printing shop scheduling problem is considered. This challenging real problem, that appears in the nowadays printing industry, can be seen as a flexible job shop scheduling problem with sequence flexibility in which precedence constraints among operations of a job are given by an arbitrary directed acyclic graph. In addition, several complicating particularities such as periods of unavailability of the machines, resumable operations, sequence-dependent setup times, partial overlapping among operations with precedence constraints, release times, and fixed operations are also present in the addressed problem. In the present work, mixed integer linear programming and constraint programming models for the minimization of the makespan are presented. Modeling the problem is twofold. On the one hand, the problem is precisely defined. On the other hand, the capabilities and limitations of a commercial software for solving the models are analyzed. Extensive numerical experiments with small-, medium-, and large-sized instances are presented. Numerical experiments show that the commercial solver is able to optimally solve only a fraction of the small-sized instances when considering the mixed integer linear programming model;while all small-sized and a fraction of the mediumsized instances are optimally solved when considering the constraint programming formulation of the problem. Moreover, the commercial solver is able to deliver feasible solutions for the large-sized instances that are of the size of the instances that appear in practice. (C) 2020 Elsevier Ltd. All rights reserved.

关键词： Flexible job shop scheduling with sequence flexibility Resumable operations Unavailability of the machines Sequence-dependent setup time mixed integer linear programming Constraint programming

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A branch and bound method for the solution of multiparametric mixed integer linear programming problems

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JOURNAL OF GLOBAL OPTIMIZATION 2014年第2-3期59卷 527-543页

作者： Oberdieck, Richard Wittmann-Hohlbein, Martina Pistikopoulos, Efstratios N. ETH Inst Chem & Bioengn Dept Chem & Appl Biosci CH-8093 Zurich Switzerland Univ London Imperial Coll Sci Technol & Med Ctr Proc Syst Engn Dept Chem Engn London SW7 2BY England

In this paper, we present a novel algorithm for the solution of multiparametric mixed integer linear programming (mp-MILP) problems that exhibit uncertain objective function coefficients and uncertain entries in the right-hand side constraint vector. The algorithmic procedure employs a branch and bound strategy that involves the solution of a multiparametric linear programming sub-problem at leaf nodes and appropriate comparison procedures to update the tree. McCormick relaxation procedures are employed to overcome the presence of bilinear terms in the model. The algorithm generates an envelope of parametric profiles, containing the optimal solution of the mp-MILP problem. The parameter space is partitioned into polyhedral convex critical regions. Two examples are presented to illustrate the steps of the proposed algorithm.

关键词： Multiparametric programming mixed integer linear programming Branch and bound

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Cutting plane algorithms for the inverse mixed integer linear programming problem

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OPERATIONS RESEARCH LETTERS 2009年第2期37卷 114-116页

作者： Wang, Lizhi Iowa State Univ Ames IA 50011 USA

We present cutting plane algorithms for the inverse mixed integer linear programming problem (InvMILP), which is to minimally perturb the objective function of a mixed integer linear program in order to make a given f... 详细信息

关键词： Inverse optimization mixed integer linear programming

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A mixed integer linear programming based approach for optimal placement of different types of automation devices in distribution networks

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ELECTRIC POWER SYSTEMS RESEARCH 2017年第Jul.期148卷 136-146页

作者： Popovic, Z. Brbaklic, B. Knezevic, S. Univ Novi Sad Fac Tech Sci Fruskogorska 11 Novi Sad 21000 Serbia Schneider Elect DMS NS Narodnog Fronta 25a Novi Sad 21000 Serbia

This paper presents a mixed integer linear programming (MILP) based approach for determining the optimal number, type and location of automation devices to be installed in the network by considering different types of devices simultaneously (remotely controlled circuit breakers/reclosers, sectionalizing switches, remotely supervised fault passage indicators). Simultaneously, it determines the new (optimal) locations of the automation devices that already exist in the network. In determining the most effective network automation scenario, the proposed approach takes into account the outage cost of consumers/producers due to momentary, short-term, and long-term interruptions, the commonly used network reliability indices (SAIFI, SAIDI, MAIFI, and ASIDI) as well as the cost of automation devices and the cost of crews. It provides the best network automation scenario in distribution systems if the network reliability indices are used for measuring the distribution system reliability, if cost of interruptions is defined to all consumers/producers, and if both aforementioned approaches (criteria) are used. (C) 2017 Elsevier B.V. All rights reserved.

关键词： Distribution network Network automation Interruption costs Reliability indices mixed integer linear programming

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A robust mixed integer linear programming framework for underground cut-and-fill mining production scheduling

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INTERNATIONAL JOURNAL OF MINING RECLAMATION AND ENVIRONMENT 2020年第6期34卷 397-414页

作者： Huang, Shuwei Li, Guoqing Ben-Awuah, Eugene Afum, Bright Oppong Hu, Nailian Univ Sci & Technol Beijing Sch Civil & Resource Engn Beijing Peoples R China Laurentian Univ Min Optimizat Lab MOL Bharti Sch Engn Sudbury ON Canada

A review of general optimization studies that have been proposed for underground mining shows that previous works lack flexibility, operability and practicality in relation to cut-and-fill mining production scheduling. This paper presents a robust mixed integer linear programming (MILP) formulation for underground cut-and-fill mining. The objective function of the model is to maximize the net present value (NPV) of the operation while meeting all mining and processing operational and technical constraints. The MILP model features stope development and extraction sequencing constraints, mining and processing tonnage fluctuation constraints, and extraction duration and active levels control constraints. These features make the model more practical and expandable. The MILP model is verified and validated with two case studies from an existing mine and the results are compared with the actual mining strategy. The comparison shows a 9% to 17% improved NPV in both case studies resulting from mining higher grades and processing less tonnes thereby generating a better cash flow.

关键词： mixed integer linear programming production scheduling cut-and-fill underground mining backfilling

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A mixed integer linear programming framework for optimising the extraction strategy of open pit - underground mining options and transitions

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INTERNATIONAL JOURNAL OF MINING RECLAMATION AND ENVIRONMENT 2020年第10期34卷 700-724页

作者： Afum, Bright Oppong Ben-Awuah, Eugene Askari-Nasab, Hooman Laurentian Univ Bharti Sch Engn MOL Sudbury ON Canada Univ Alberta Sch Min & Petr Engn Dept Civil & Environm Engn MOL Edmonton AB Canada

The decision to exploit a mineral deposit that can be extracted by open pit and/or underground mining involves consideration of the following options: (a) independent open pit extraction;(b) independent underground extraction;(c) simultaneous open pit and underground (OPUG) extraction;(d) sequential OPUG extraction;and (e) combinations of (c) and (d). This paper investigates the extraction strategy for deposits using mixed integer linear programming (MILP) optimisation framework to maximise the net present value and determine the schedules for mining, processing, underground capital and operational developments, and 3D crown pillar position. The MILP framework is implemented for a gold deposit. The results showed a combined sequential and simultaneous open pit and underground (OPUG) mining with crown pillar as the optimal extraction option generating NPV that is 11% and 13% better than independent OP or UG mining, respectively.

关键词： mixed integer linear programming open pit and underground mine planning mining options mine optimisation transition depth crown pillar

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