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A reducibility method for the weak linear bilevel programming problems and a case study in principal-agent

INFORMATION SCIENCES 2018年 454卷 46-58页

作者： Zheng, Yue Zhang, Guangquan Zhang, Zhen Lu, Jie Huaibei Normal Univ Sch Management Huaibei 235000 Peoples R China Univ Technol Sydney Ctr Artificial Intelligence CAI Decis Syst & E Serv Intelligence Lab Sydney NSW 2007 Australia Dalian Univ Technol Inst Syst Engn Dalian 116024 Peoples R China

A weak linear bilevel programming (WLBP) problem often models problems involving hierarchy structure in expert and intelligent systems under the pessimistic point. In the paper, we deal with such a problem. Using the duality theory of linear programming, the WLBP problem is first equivalently transformed into a jointly constrained bilinear programming problem. Then, we show that the resolution of the jointly constrained bilinear programming problem is equivalent to the resolution of a disjoint bilinear programming problem under appropriate assumptions. This may give a possibility to solve the WLBP problem via a single-level disjoint bilinear programming problem. Furthermore, some examples illustrate the solution process and feasibility of the proposed method. Finally, the WLBP problem models a principal-agent problem under the pessimistic point that is also compared with a principal-agent problem under the optimistic point. Crown Copyright (C) 2018 Published by Elsevier Inc. All rights reserved.

关键词： Bilevel programming Duality theory bilinear programming Principal-agent

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On the construction of convex and concave envelope formulas for bilinear and fractional functions on quadrilaterals

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2004年第1期27卷 5-22页

作者： Benson, HP Univ Florida Warrington Coll Business Adm Gainesville FL 32611 USA

Convex and concave envelopes play important roles in various types of optimization problems. In this article, we present a result that gives general guidelines for constructing convex and concave envelopes of functions of two variables on bounded quadrilaterals. We show how one can use this result to construct convex and concave envelopes of bilinear and fractional functions on rectangles, parallelograms and trapezoids. Applications of these results to global optimization are indicated.

关键词： convex envelope bilinear programming fractional programming global optimization

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Transmission Expansion Planning Considering a High Share of Wind Power to Maximize Available Transfer Capability

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IEEE ACCESS 2023年 11卷 23136-23145页

作者： Alshamrani, Ahmad M. El-Meligy, Mohammed A. Sharaf, Mohamed Abdel Fattah Saif, Wigdan Aref Mohammed Awwad, Emad Mahrous King Saud Univ Coll Sci Stat & Operat Res Dept Riyadh 11451 Saudi Arabia King Saud Univ Adv Mfg Inst Riyadh 11421 Saudi Arabia King Saud Univ Coll Engn Ind Engn Dept Riyadh 11421 Saudi Arabia King Saud Univ Coll Engn Elect Engn Dept Riyadh 11421 Saudi Arabia

To have an effective transmission network and support free trading, the open and non-discriminatory access to transmission services for all market participants under all conditions is of significant concern. To this end, a sufficient value of available transfer capability (ATC) is required, which can significantly affect the electricity market efficiency. This paper constructs a methodology for reinforcing an existing transmission network considering wind power investment to enhance ATC. In this regard, a bi-level structure is adopted whose upper level is the joint transmission expansion planning (TEP) and wind power investment subject to the technical power grid limitations. After solving the upper level, a security-constrained economic dispatch (SC-ED) is formulated to acquire the optimal generation scheduling. Then, the lower level is designed to calculate the ATC. To solve this problem, the SC-ED is first replaced by its primal and dual feasibility constraints and strong duality equality. Then, these constraints are added to the upper level. The lower level is also replaced by its corresponding constraints. Consequently, a bilinear single-level optimization problem is extracted, which is further solved by employing Konno's cutting plane algorithm. The experimental results on the IEEE 24-bus RTS show the proposed method's effectiveness. It is indicated that the ATC can be improved by 27.2 % using the formulated model at the expense of more investment cost resulting in less load shedding and less wind curtailment.

关键词： Available transfer capability bi-level optimization bilinear programming Konno's algorithm transmission expansion planning wind power investment

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Convexification method for bilinear transmission expansion problem

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INTERNATIONAL TRANSACTIONS ON ELECTRICAL ENERGY SYSTEMS 2014年第5期24卷 638-652页

作者： Cedeno, Enrique B. Arora, Sant Univ Minnesota Ind & Syst Engn Dept Minneapolis MN 55455 USA

This research presents a method for convexifying the non-linear constraints and the objective function for the Transmission Network Expansion Planning Problem (TNEP). The TNEP seeks to identify the best set of transmission capacity additions to meet a future electric power demand. The TNEP is a non-convex Mixed Integer Non-linear programming problem for which a variety of primal solution methods have been designed. In this paper, the TNEP is formulated as a bilinear programming problem subject to bilinear constraints. This allows resolving the non-linearities through convexification of the non-linear objective function and the constraints for a relaxation of the TNEP in which integrality requirements are removed. The final solution for the original TNEP problem is obtained by using a branch and bound strategy. It was found by using a two-node test case that the convexification method presented in this paper in conjunction with a brand and bound strategy identifies the optimal solution. Successfully solving a small-scale test problem through convexification presents a promising scenario to solve larger scale problems. This primal-dual method allows identifying the duality gap which is a limitation of primal methods. Further multidisciplinary research is needed to extend the method presented here to larger cases and real life networks. Copyright (c) 2013 John Wiley & Sons, Ltd.

关键词： bilinear programming non-linear programming convexification transmission planning

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An effective approximation algorithm for joint lot-sizing and pricing problem

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INTERNATIONAL JOURNAL OF ADVANCED MANUFACTURING TECHNOLOGY 2013年第9-12期65卷 1429-1437页

作者： Askarpoor, Hamid Reza Davoudpour, Hamid Amirkabir Univ Technol Dept Ind Engn Tehran Iran

Emphasis on effective demand management is becoming increasingly recognized as an important factor in operations performance as well as the supply management. One of the known leverages to demand management is pricing. In joint pricing and production planning problem, manufacturer or supplier faces a price-sensitive demand and it should maximize its profit by making decision on the production policy among the finite time horizon while predicting the demand via determination of the prices of the products it sells. In this paper, a single-item joint pricing and production planning with concave revenue function has been discussed. By a piecewise linear approximation and reduction of the resulting problem to a bilinear one, an effective heuristic procedure has been proposed to solve it optimally. The efficiency of the procedure has been shown by numerical experiments.

关键词： Lot-sizing Pricing Demand management bilinear programming Approximation algorithm

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Feature minimization within decision trees

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 1998年第2期10卷 111-126页

作者： Bredensteiner, EJ Bennett, KP Rensselaer Polytech Inst Dept Math Sci Knowledge Discovery & Data Min Grp Troy NY 12180 USA

Decision trees for classification can be constructed using mathematical programming. Within decision tree algorithms, the feature minimization problem is to construct accurate decisions using as few features or attributes within each decision as possible. Feature minimization is an important aspect of data mining since it helps identify what attributes are important and helps produce accurate and interpretable decision trees. In feature minimization with bounded accuracy, we minimize the number of features using a given misclassification error tolerance. This problem can be formulated as a parametric bilinear program and is shown to be NP-complete. A parametric Frank-Wolfe method is used to solve the bilinear subproblems. The resulting minimization algorithm produces more compact, accurate, and interpretable trees. This procedure can be applied to many different error functions. Formulations and results for two error functions are given. One method, FM_RLP-P, dramatically reduced the number of features of one dataset from 147 to 2 while maintaining an 83.6% testing accuracy. Computational results compare favorably with the standard univariate decision tree method, C4.5, as well as with linear programming methods of tree construction.

关键词： data mining machine learning feature minimization decision trees bilinear programming

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Absolute Value Constraint Based Method for Interval Optimization to SCED Model

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IEEE TRANSACTIONS ON POWER SYSTEMS 2014年第2期29卷 980-981页

作者： Ding, Tao Bo, Rui Gu, Wei Guo, Qinglai Sun, Hongbin Tsinghua Univ Dept Elect Engn Beijing 100084 Peoples R China Midwest Independent Transmiss Syst Operator Midwe St Paul MN 55108 USA Southeast Univ Sch Elect Engn Nanjing Jiangsu Peoples R China

This letter proposes an absolute value based method to solve the interval linear optimization problem with application to security constrained economic dispatch (SCED). To avoid expensive computation associated with solving combinatorial linear programming (LP) problems for interval upper bound, firstly a bilinear programming model is formulated using duality theory. The equality constraints are then converted to absolute value constraints. Lastly, the absolute value operator is eliminated through introducing two nonnegative slack variables and complementary slackness condition. The resulting new bilinear programming model can be effectively solved by the branch and bound method with linear relaxation technique to obtain the global optimal solution. Numerical results demonstrate the effectiveness of the proposed method in improving solution and computation time.

关键词： Absolute value constraint bilinear programming complementary slackness condition interval optimization security constrained economic dispatch (SCED)

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Computing upper and lower bounds in interval decision trees

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2007年第2期181卷 808-816页

作者： Danielson, Mats Ekenberg, Love Stockholm Univ Royal Inst Technol Dept Comp Sci & Syst SE-16440 Kista Sweden Mid Sweden Univ FSCN SE-85170 Sundsvall Sweden

This article presents algorithms for computing optima in decision trees with imprecise probabilities and utilities. In tree models involving uncertainty expressed as intervals and/or relations, it is necessary for the evaluation to compute the upper and lower bounds of the expected values. Already in its simplest form, computing a maximum of expectancies leads to quadratic programming (QP) problems. Unfortunately, standard optimization methods based on QP (and BLP - bilinear programming) are too slow for the evaluation of decision trees in computer tools with interactive response times. Needless to say, the problems with computational complexity are even more emphasized in multi-linear programming (MLP) problems arising from multi-level decision trees. Since standard techniques are not particularly useful for these purposes, other, non-standard algorithms must be used. The algorithms presented here enable user interaction in decision tools and are equally applicable to all multi-linear programming problems sharing the same structure as a decision tree. (c) 2006 Elsevier B.V. All rights reserved.

关键词： decision analysis decision trees bilinear programming multi-linear programming

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Hybrid extreme point tabu search

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 1998年第2-3期106卷 676-688页

作者： Blue, JA Bennett, KP Rensselaer Polytech Inst Dept Math Sci Troy NY 12180 USA

We develop a new hybrid tabu search method for optimizing a continuous differentiable function over the extreme points of a polyhedron. The method combines extreme point tabu search (EPTS) with traditional descent algorithms based on linear programming. The tabu search algorithm utilizes both recency-based and frequency-based memory and oscillates between local improvement and diversification phases. The hybrid algorithm iterates between using a descent algorithm to find a local minimum and tabu search to improve locally and then move to a new area of the search space. The descent algorithm acts as a form of intensification within the tabu search. This algorithm can be used on many important classes of problems in global optimization including bilinear programming, multilinear programming, multiplicative programming, concave minimization, and complementarity problems. The algorithm is applied to two practical problems: the quasistatic multi-rigid-body contact problem (QCP) in robotics and the global tree optimization (GTO) problem in machine learning. We perform computational experiments on problems of significant real world dimensions: the smallest machine learning problem tested has on the order of 10(485) extreme points. The results show our method obtains high quality solutions both by comparison with the embedded descent algorithm and by comparison to a version of tabu search that does not make use of descent. (C) 1998 Elsevier Science B.V. All rights reserved.

关键词： tabu search global optimization bilinear programming machine learning classification

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Piecewise MILP under- and overestimators for global optimization of bilinear programs

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AICHE JOURNAL 2008年第4期54卷 991-1008页

作者： Wicaksono, Danan Suryo Karimi, I. A. Natl Univ Singapore Dept Chem & Biomol Engn Singapore 117576 Singapore

Many practical problems of interest in chemical engineering and other fields can be formulated as bilinear programs (BLPs). For such problems, a local nonlinear programming solver often provides a suboptimal solution or even fails to locate a feasible one. Numerous global optimization algorithms devised for bilinear programs rely on linear programming (LP) relaxation, which is often weak, and, thus, slows down the convergence rate of the global optimization algorithm. All interesting recent development is the idea of using an ab initio partitioning of the. search domain to improve the relaxation quality, which results in a relaxation problem that is a mixed-integer linear program (MILP) rather than LP, called as piecewise MILP relaxation. However, much work is in order to fully exploit the potential of such approach. Several novel formulations are developed for piecewise MILP under- and overestimators for BLPs via three systematic approaches, and two segmentation schemes. As is demonstrated and evaluated the superiority of the novel models is shown, using a variety of examples. In addition, metrics are defined to measure the effectiveness of piecewise MILP relaxation within a two-level-relaxation framework, and several theoretical results are presented, as well as valuable insights into the properties of such relaxations, which may prove useful in developing global optimization algorithms. (c) 2008 American Institute of Chemical Engineers.

关键词： global optimization bilinear programming mixed-integer linear programming piecewise under- and overestimators process network synthesis

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