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Minimizing Cubic and Homogeneous Polynomials over integers in the Plane

MATHEMATICS OF OPERATIONS RESEARCH 2016年第2期41卷 511-530页

作者： Del Pia, Alberto Hildebrand, Robert Weismantel, Robert Zemmer, Kevin Univ Wisconsin Dept Ind & Syst Engn Madison WI 53706 USA Univ Wisconsin Wisconsin Inst Discovery Madison WI 53706 USA ETH Inst Operat Res CH-8092 Zurich Switzerland

We complete the complexity classification by degree of minimizing a polynomial over the integer points in a polyhedron in a real vector space of dimension two. Previous work shows that optimizing a quadratic polynomial over the integer points in a polyhedral region in a real vector space of dimension two can be done in polynomial time, whereas optimizing a quartic polynomial in the same type of region is NP-hard. We close the gap by showing that this problem can be solved in polynomial time for cubic polynomials. Furthermore, we show that the problem of minimizing a homogeneous polynomial of any fixed degree over the integer points in a bounded polyhedron in a real vector space of dimension two is solvable in polynomial time. We show that this holds for polynomials that can be translated into homogeneous polynomials, even when the translation vector is unknown. We demonstrate that such problems in the unbounded case can have smallest optimal solutions of exponential size in the size of the input, thus requiring a compact representation of solutions for a general polynomial time algorithm for the unbounded case.

关键词： nonlinear integer programming polynomial optimization

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An exact solution method for unconstrained quadratic 0-1 programming: a geometric approach

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JOURNAL OF GLOBAL OPTIMIZATION 2012年第4期52卷 797-829页

作者： Li, D. Sun, X. L. Liu, C. L. Chinese Univ Hong Kong Dept Syst Engn & Engn Management Shatin Hong Kong Peoples R China Fudan Univ Sch Management Dept Management Sci Shanghai 200433 Peoples R China Shanghai Univ Finance & Econ Dept Appl Math Shanghai 200433 Peoples R China

We explore in this paper certain rich geometric properties hidden behind quadratic 0-1 programming. Especially, we derive new lower bounding methods and variable fixation techniques for quadratic 0-1 optimization problems by investigating geometric features of the ellipse contour of a (perturbed) convex quadratic function. These findings further lead to some new optimality conditions for quadratic 0-1 programming. Integrating these novel solution schemes into a proposed solution algorithm of a branch-and-bound type, we obtain promising preliminary computational results.

关键词： Quadratic 0-1 programming nonlinear integer programming Optimality condition Lower bounds Variable fixation Branch-and-bound method

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Design of manufacturing cells in the presence of alternative cell locations and material transporters

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JOURNAL OF THE OPERATIONAL RESEARCH SOCIETY 2003年第10期54卷 1059-1075页

作者： Logendran, R Karim, Y Oregon State Univ Dept Ind & Mfg Engn Corvallis OR 97331 USA

This paper addresses two significant issues in the design of cellular manufacturing (CM) systems: (i) the availability of alternative locations for a cell, and (ii) the use of alternative routes to move part loads between cells when the capacity of the material transporter (MT) employed is limited. In addition, several other important factors in the design of CM systems including machine capacity limitations, batches of part demands, non-consecutive operations of parts, and maximum number of machines assigned to a cell are considered. A nonlinear programming model, comprised of binary and general integer variables, is formulated for the research problem. A higher-level heuristic solution algorithm based upon a concept known as 'tabu search' is presented for solving industry-size problems. Six different versions of the heuristic are developed to investigate the impact of long-term memory and the use of fixed versus variable tabu-list sizes. Explicit method-based techniques are developed to convert the original nonlinear programming model into an equivalent mixed (binary)-integer linear programming model in order to test the efficacy of the proposed solution technique for solving small problem instances. The solutions obtained from the heuristics have average deviation of less than 3% of the optimal solutions, and require less than a minute in comparison with optimizing methods that needed 1-10h of computation time. A carefully designed statistical experiment is used to compare the performance of the heuristics by solving three different problem structures, ranging from four to 30 parts, and three to nine locations. The experiment shows that as the problem size increases, the tabu-search-based heuristic using fixed tabu list size and long-term memory based on minimal frequency strategy is preferred over the other heuristics.

关键词： cellular manufacturing nonlinear integer programming mathematical modeling optimization heuristics

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The Chvatal-Gomory Closure of a Strictly Convex Body

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MATHEMATICS OF OPERATIONS RESEARCH 2011年第2期36卷 227-239页

作者： Dadush, Daniel Dey, Santanu S. Vielma, Juan Pablo Georgia Inst Technol H Milton Stewart Sch Ind & Syst Engn Atlanta GA 30332 USA IBM Corp Thomas J Watson Res Ctr Business Analyt & Math Sci Dept Yorktown Hts NY 10598 USA Univ Pittsburgh Dept Ind Engn Pittsburgh PA 15261 USA

In this paper, we prove that the Chvatal-Gomory closure of a set obtained as an intersection of a strictly convex body and a rational polyhedron is a polyhedron. Thus, we generalize a result of Schrijver [Schrijver, A. 1980. On cutting planes. Ann. Discrete Math. 9 291-296], which shows that the Chvatal-Gomory closure of a rational polyhedron is a polyhedron.

关键词： nonlinear integer programming cutting planes Chvatal-Gomory closure

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Feasible roadmap for CCS retrofit of coal-based power plants to reduce Chinese carbon emissions by 2050

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APPLIED ENERGY 2020年第0期259卷 114112-000页

作者： Guo, Jian-Xin Huang, Chen Chinese Acad Sci Inst Sci & Dev Beijing 100190 Peoples R China Univ Chinese Acad Sci Sch Publ Policy & Management Beijing 100049 Peoples R China

In the medium to long term, China must conduct deep emission reduction actions to transform the country's economy and meet the conditions of the Paris Agreement. As an advanced emission reduction technology, carbon capture and storage (CCS) is undoubtedly an important means of achieving this goal. China must consider how to retrofit existing thermal power plants to install CCS technology. Thus, the expected future roadmap for power plants with CO2 capture is of significant interest. To achieve this aim, we propose a new CCS project investment model, which helps design the CCS development roadmap for achieving the emission targets for 2050. Reductions in the cost of technologies as a result of learning-by-doing is considered to enrich the model to describe the reality more sensibly. Through some mathematical skills, we transform the original continuous problem into a nonlinear integer programming problem. By solving the model, we are trying to answer questions about when to adopt CCS technology and the cost. The results reveal that early large-scale CCS demonstrations are not necessary. The peak investment period of CCS is around 2035. Operating costs account for 80% of the overall cost of CCS, and thus, policymakers must fully motivate the development and investment of operating technologies, especially capture technologies. The results also show that when the scale of CCS technology is promoted, flexible installation levels can be considered to improve efficiency and reduce costs.

关键词： Abatement strategy CCS development roadmap Learning by doing nonlinear integer programming

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Reliability optimisation of flexible manufacturing systems with spare tooling

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INTERNATIONAL JOURNAL OF ADVANCED MANUFACTURING TECHNOLOGY 2001年第1期17卷 69-77页

作者： Altumi, AA Philipose, AM Taboun, SM Univ Windsor Windsor ON N9B 3P4 Canada

Tool reliability plays an important role in the performance and justification of flexible manufacturing systems (FMSs). Failure of a single tool can cause downtimes over the entire system. This would cause due dates to be missed and can result in inferior products. Therefore, in order to justify the large capital investment associated with FMSs, the system must perform in a reliable manner to give an acceptable or required rate of return on the investment. In order to arrive at this objective, FMS reliability must be studied at the planning and design stages, tool failures pose a major obstacle to achieving this objective. In this paper, a mathematical model has been developed to determine the spare tooling requirement for the tooling system in an FMS, so that a desired system reliability is achieved and the cost is minimised. The influence of tool sharing on cost, reliability, spares requirement, and tool magazine capacity of the FMS are analysed. The tools and tool transporter are subject to general failure distributions.

关键词： flexible manufacturing systems nonlinear integer programming reliability optimisation spare requirements tool sharing

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Improving the performance of weighted Lagrange-multiplier methods for nonlinear constrained optimization

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INFORMATION SCIENCES 2000年第1-4期124卷 241-272页

作者： Wah, BW Wang, T Shang, Y Wu, Z Univ Missouri Dept Comp Sci & Comp Engn Columbia MO 65211 USA Univ Illinois Dept Elect & Comp Engn Urbana IL 61801 USA Univ Illinois Coordinated Sci Lab Urbana IL 61801 USA

nonlinear constrained optimization problems in discrete and continuous spaces are an important class of problems studied extensively in artificial intelligence and operations research. These problems can be solved by a Lagrange-multiplier method in continuous space and by an extended discrete Lagrange-multiplier method indiscrete space. When constraints are satisfied, these methods rely on gradient descents in the objective space to find high-quality solutions. On the other hand, when constraints are violated, these methods rely on gradient ascents in the Lagrange-multiplier space in order to increase the penalties on unsatisfied constraints and to force the constraints into satisfaction. The balance between gradient descents and gradient ascents depends on the relative weights between the objective function and the constraints, which indirectly control the convergence speed and solution quality of the Lagrangian method. To improve convergence speed without degrading solution quality, we propose an algorithm to dynamically control the relative weights between the objective and the constraints. Starting from an initial weight, the algorithm automatically adjusts the weights based on the behavior of the search progress. With this strategy, we are able to eliminate divergence, reduce oscillation, and speed up convergence. We show improved convergence behavior of our proposed algorithm on both nonlinear continuous and discrete problems. (C) 2000 Elsevier Science Inc. All rights reserved.

关键词： adaptive weights Lagrange-multiplier method multiplierless filter-bank design nonlinear constrained optimization nonlinear integer programming nonlinear programming QMF filter-bank design

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A mixed integer nonlinear optimization model for gas sale company

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OPTIMIZATION LETTERS 2007年第1期1卷 61-69页

作者： Allevi, E. Bertocchi, M. I. Vespucci, M. T. Innorta, M. Bergamo Univ Dept Math Stat Comp Sci & Applicat I-24127 Bergamo Italy Univ Brescia Dept Quantitat Methods I-25122 Brescia Italy Bergamo Univ Dept Engn Management I-24044 Dalmine Italy

In this paper the authors propose an optimisation model, called OMoGaS (Optimisation Modelling for Gas Seller), to assist companies dealing with gas retail commercialisation. The model takes into account the limits on price imposed by law on small consumers as well as the gas company policies in order to explore the commercial consequences of different policies. The GAMS framework is used for the optimisation of the defined MINLP model where the profit function is based on the number of contracts with the final consumers, on the tipology of consumers and on the cost supported to meet the final demand while the constraints include information on a maximum daily gas consumption, on yearly maximum and minimum comsumption in order to avoid penalties and on consumption profiles. A case study is presented.

关键词： Natural gas market Gas sales company Tariff components nonlinear integer programming

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An Ant Colony Optimization algorithm for partner selection in Virtual Enterprises

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INTERNATIONAL JOURNAL OF MATERIALS & PRODUCT TECHNOLOGY 2009年第3期34卷 227-240页

作者： Cheng, Fangqi Ye, Feifan Yang, Jianguo Ningbo Univ Fac Engn Ningbo 315211 Zhejiang Peoples R China Zhejiang Business Technol Inst Dept Mechatron Ningbo Zhejiang Peoples R China Shanghai Jiao Tong Univ Sch Mech Engn Shanghai 200030 Peoples R China

Partner selection problem seeks to find a best combination of enterprises by optimising a nonlinear objective over the given constraints. In this paper the partner selection problem is modelled as a nonlinear integer programming problem and an Ant Colony Optimization (ACO) algorithm embedded project scheduling is presented for solving the problem with the lead time, subproject cost and risk factor constraints in Virtual Enterprises (VE). Genetic Algorithm (GA) and enumeration algorithm are introduced for comparison to check the effectiveness of the ACO algorithm. A case study is implemented to verify the feasibility of the proposed approach and the computational results are satisfactory.

关键词： VE virtual enterprise partner selection nonlinear integer programming ACO ant colony optimization algorithm

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A superior linearization method for signomial discrete functions in solving three-dimensional open-dimension rectangular packing problems

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ENGINEERING OPTIMIZATION 2017年第5期49卷 746-761页

作者： Lin, Ming-Hua Tsai, Jung-Fa Chang, Shu-Chuan Shih Chien Univ Dept Informat Technol & Management Taipei Taiwan Natl Taipei Univ Technol Dept Business Management Taipei Taiwan St Johns Univ Dept Informat Management New Taipei Taiwan Natl Taipei Univ Technol Coll Management Taipei Taiwan

This article studies the three-dimensional open-dimension rectangular packing problem (3D-ODRPP) in which a set of given rectangular boxes is packed into a large container of minimal volume. This problem is usually formulated as a mixed-integer nonlinear programming problem with a signomial term in the objective. Existing exact methods experience difficulty in solving large-scale problems within a reasonable amount of time. This study reformulates the original problem as a mixed-integer linear programming problem by a novel method that reduces the number of constraints in linearizing the signomial term with discrete variables. In addition, the range reduction method is used to tighten variable bounds for further reducing the number of variables and constraints in problem transformation. Numerical experiments are presented to demonstrate that the computational efficiency of the proposed method is superior to existing methods in obtaining the global optimal solution of the 3D-ODRPP.

关键词： Three-dimensional open-dimension rectangular packing problem global optimization nonlinear integer programming linearization signomial term

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