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Containment of a single polygon using mathematical programming

EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 1996年第2期92卷 368-386页

作者： Grinde, RB Cavalier, TM PENN STATE UNIV DEPT IND & MANAGEMENT SYST ENGNUNIVERSITY PKPA 16802

This paper investigates the polygon containment problem, to determine whether a given polygon can be translated and/or rotated to fit inside a fixed polygon. A primary application is the stock cutting problem, when pieces have irregular shapes and can be rotated. The case in which both polygons are convex is treated. An optimization-based approach is used, in contrast to previous computational geometry methods. Optimality conditions of the math program are found and are used to develop an efficient algorithm which finds the best placement and to draw relationships between the two approaches. The problem becomes a parametric linear program with a nonlinear parameter corresponding to the orientation of the moving figure. All alternate-optimal placements with at least three contacts can be found. In addition, the algorithm provides an analytic description of the optimal objective as a function of the orientation of the moving polygon.

关键词： polygon containment computational geometry stock cutting pattern layout nesting parametric programming algorithm

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DUALITY OF REGULARIZATIONS AND HULLFUNCTIONS FOR MATHEMATICAL-programming PROBLEMS

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OR SPEKTRUM 1994年第3期16卷 187-191页

作者： SCHADE, L 1. Erich-Schmidt-Weg 14 D-37083 G?ttingen Germany

For the study of mathematical programming problems and solution methods the duality theory forms a powerful tool. There are also some concepts of regularization and stabilization of a given problem for a better behavior in practical solution procedures. The aim of this paper is the investigation of duality aspects of such regularizations and the forming of hullfunctions on the other hand. Applications for handling of so-called ill-posed problems (Eremin) using some parametrizations of the original problem will emphasize the importance for practical numerical methods, especially. This results will inspire some applications to solution methods for parametric and multicriteria optimization.

关键词： NONLINEAR programming DUALITY SOLUTION METHODS parametric programming MULTICRITERIA OPTIMIZATION ILL-POSED PROBLEMS

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STABILITY IN DISTRIBUTION PROBLEMS OF NONLINEAR STOCHASTIC-programming

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ASIA-PACIFIC JOURNAL OF OPERATIONAL RESEARCH 1991年第2期8卷 128-134页

作者： WANG, JD NANJING UNIV DEPT MATHNANJINGPEOPLES R CHINA

In this paper concerned is the stability property of the optimum of nonlinear distribution problems with respect to the random vector. We give conditions, under which the optimum of the problem with approximate random vectors will converge to the optimum of the original problem in distribution sense, L1-sense or in the sense of mean-convergence respectively. Based upon this stability an approximation method is suggested.

关键词： STABILITY DISTRIBUTION PROBLEM NONLINEAR STOCHASTIC programming parametric programming

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Generalized linear multiplicative and fractional programming

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Annals of Operations Research 1990年第1期25卷 147-161页

作者： Konno, Hiroshi Kuno, Takahito Institute of Human and Social Sciences Tokyo Institute of Technology Tokyo 152 2-12-1 Oh-Okayama Meguro-ku Japan Department of Social Engineering Tokyo Institute of Technology Tokyo 152 2-12-1 Oh-Okayama Meguro-ku Japan

This paper addresses itself to the algorithm for minimizing the sum of a convex function and a product of two linear functions over a polytope. It is shown that this nonconvex minimization problem can be solved by solving a sequence of convex programming problems. The basic idea of this algorithm is to embed the original problem into a problem in higher dimension and apply a parametric programming (path following) approach. Also it is shown that the same idea can be applied to a generalized linear fractional programming problem whose objective function is the sum of a convex function and a linear fractional function. © 1990 J.C. Baltzer A.G. Scientific Publishing Company.

关键词： generalized linear fractional function generalized linear multiplicative function lobal minimization Nonconvex minimization parametric programming

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A bilevel programming framework for enterprise-wide process networks under uncertainty

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COMPUTERS & CHEMICAL ENGINEERING 2004年第6-7期28卷 1121-1129页

作者： Ryu, JH Dua, V Pistikopoulos, EN Univ London Imperial Coll Sci & Technol Dept Chem Engn Ctr Proc Syst Engn London SW7 2AZ England

Enterprise-wide supply chain planning problems naturally exhibit a multi-level decision network structure, where for example, one level may correspond to a local plant control/scheduling/planning problem and another level to a corresponding plant-wide planning/network problem. Such a multi-level decision network structure can be mathematically represented by using multi-level programming principles. In this paper, we specifically address bilevel decision-making problems under uncertainty in the context of enterprise-wide supply chain optimization with one level corresponding to a plant planning problem, while the other to a distribution network problem. We first describe how such problems can be modelled as bilevel programming problems and then we present an effective solution strategy based on parametric programming techniques. An attractive feature of the proposed strategy is the fact that it transforms the bilevel problem into a family of single parametric optimization problems, which can be solved to global optimality. A numerical example is presented to illustrate the proposed framework. (C) 2003 Elsevier Ltd. All rights reserved.

关键词： supply chain planning bilevel programming parametric programming uncertainty

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Linear programming and Resource Allocation Modeling 1

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丛书名： []

2018年

作者： Michael J. Panik

ISBN: (数字)9781119509455

ISBN: (纸本)9781119509448

Guides in the application of linear programming to firm decision making, with the goal of giving decision-makers a better understanding of methods at their disposal Useful as a main resource or as a supplement in an economics or management science course, this comprehensive book addresses the deficiencies of other texts when it comes to covering linear programming theory--especially where data envelopment analysis (DEA) is concerned--and provides the foundation for the development of DEA. Linear programming and Resource Allocation Modeling begins by introducing primal and dual problems via an optimum product mix problem, and reviews the rudiments of vector and matrix operations. It then goes on to cover: the canonical and standard forms of a linear programming problem; the computational aspects of linear programming; variations of the standard simplex theme; duality theory; single- and multiple- process production functions; sensitivity analysis of the optimal solution; structural changes; and parametric programming. The primal and dual problems are then reformulated and re-examined in the context of Lagrangian saddle points, and a host of duality and complementary slackness theorems are offered. The book also covers primal and dual quadratic programs, the complementary pivot method, primal and dual linear fractional functional programs, and (matrix) game theory solutions via linear programming, and data envelopment analysis (DEA). This book: Appeals to those wishing to solve linear optimization problems in areas such as economics, business administration and management, agriculture and energy, strategic planning, public decision making, and health care Fills the need for a linear programming applications component in a management science or economics course Provides a complete treatment of linear programming as applied to activity selection and usage Contains many detailed example problems as well as textual and graphical explanations Linear programming and Resourc

关键词： linear programming linear programming applications DEA modeling linear programming fundamentals network flows linear models mathematical programming economics linear programming for economics applied mathematics statistics applied statistics data envelopment analysis tools for productivity comparisons solving resource allocation problems parametric programming DEA for finance DEA for business DEA for strategic planning DEA for decision making guide to DEA guide to linear programming linear programming for business linear programming for finance linear programming for decision making linear programming and management computer programming Michael J. Panik Linear programming and Resource Allocation Modeling

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A bilevel programming framework for enterprise-wide process networks under uncertainty

A bilevel programming framework for enterprise-wide process ...

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International Conference on Foundations of Computer-Aided Process Operations

作者： Ryu, JH Dua, V Pistikopoulos, EN Univ London Imperial Coll Sci & Technol Dept Chem Engn Ctr Proc Syst Engn London SW7 2AZ England

关键词： supply chain planning bilevel programming parametric programming uncertainty

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Pivot rules for linear programming: A survey on recent theoretical developments

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Annals of Operations Research 1993年第1期46-47卷 203-233页

作者： Terlaky, Tamás Zhang, Shuzhong Faculty of Technical Mathematics and Computer Science Delft University of Technology Delft 2600 GA P.O. Box 5031 Netherlands Department of Econometrics University of Groningen Groningen 9700 AV P.O. Box 800 Netherlands

The purpose of this paper is to discuss the various pivot rules of the simplex method and its variants that have been developed in the last two decades, starting from the appearance of the minimal index rule of Bland. We are mainly concerned with finiteness properties of simplex type pivot rules. Well known classical results concerning the simplex method are not considered in this survey, but the connection between the new pivot methods and the classical ones, if there is any, is discussed. In this paper we discuss three classes of recently developed pivot rules for linear programming. The first and largest class is the class of essentially combinatorial pivot rules including minimal index type rules and recursive rules. These rules only use labeling and signs of the variables. The second class contains those pivot rules which can actually be considered as variants or generalizations or specializations of Lemke's method, and so they are closely related to parametric programming. The last class has the common feature that the rules all have close connections to certain interior point methods. Finally, we mention some open problems for future research. © 1993 J.C. Baltzer AG, Science Publishers.

关键词： cycling Linear programming minimal index rule parametric programming pivot rules recursion simplex method

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Multiparametric linear fractional functionals programming

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Trabajos de Investigacion Operativa 1989年第1期4卷 3-10页

作者： Chada, S.S. Universidad de Wisconsin Eau Claire Eau-Claire 54701 Wisconsin United States

In this paper a multiparametric linear fractional functionals program, with parameters appearing only in the objective function, is generated. The optimum solution of this parametric program is supposed to satisfy the constraints as equations only. It is also shown that the set of parameters forms a convex polyhedron. © 1989 SEIO.

关键词： AMS Classification: 90C31 (parametric programming) linear fractional programming optimization parametric programming

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Stochastic programming with Incomplete Information: A Survey of Results on Postoptimization and Sensitivity Analysis

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Optimization 1987年第4期18卷 507-532页

作者： Dupačová, Jitka Charles University Department of Statistics Prague Czech Republic

The possibility of successful applications of stochastic programming decision models has been limited by the assumed complete knowledge of the distribution F of the random parameters as well as by the limited scope of the existing numerical procedures. We shall give a survey of selected methods wich can be used to deal with the incomplete knowledge of the distribution F, namely to study robustness of the optimal solution and the optimal value of the objective function relative to small changes of the underlying distribution and to get error bounds in approximation schemes. © 1987, Taylor & Francis Group, LLC. All rights reserved.

关键词： Asymptotic properties Contamination Incomplete information Minimax approach parametric programming Robustness Stability Stochastic programming

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