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A DUAL-RELAX PENALTY FUNCTION APPROACH FOR SOLVING nonlinear bilevel programming WITH LINEAR LOWER LEVEL PROBLEM

Acta Mathematica Scientia 2011年第2期31卷 652-660页

作者：万仲平王广民吕一兵 School of Mathematics and Statistics Wuhan University School of Economics and Management China University of Geosciences Institute of Information and Mathematics Yangtze University

The penalty function method, presented many years ago, is an important nu- merical method for the mathematical programming problems. In this article, we propose a dual-relax penalty function approach, which is significantly different from penalty func- tion approach existing for solving the bilevel programming, to solve the nonlinear bilevel programming with linear lower level problem. Our algorithm will redound to the error analysis for computing an approximate solution to the bilevel programming. The error estimate is obtained among the optimal objective function value of the dual-relax penalty problem and of the original bilevel programming problem. An example is illustrated to show the feasibility of the proposed approach.

关键词： nonlinear bilevel programming penalty function approach dual-relax strategy

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An evolutionary algorithm for solving nonlinear bilevel programming based on a new constraint-handling scheme

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IEEE TRANSACTIONS ON SYSTEMS MAN AND CYBERNETICS PART C-APPLICATIONS AND REVIEWS 2005年第2期35卷 221-232页

作者： Wang, YP Jiao, YC Li, H Xidian Univ Fac Sci Xian 710071 Peoples R China Xidian Univ Inst Antennas & Electomagnet Scattering Xian 710071 Peoples R China

In this paper, a special nonlinear bilevel programming problem (nonlinear BLPP) is transformed into an equivalent single objective nonlinear programming problem. To solve the equivalent problem effectively, we first construct a specific optimization problem with two objectives. By solving the specific problem, we can decrease the leader's objective value, identify the quality of any feasible solution from infeasible solutions and the quality of two feasible solutions for the equivalent single objective optimization problem, force the infeasible solutions moving toward the feasible region, and improve the feasible solutions gradually. We then propose a new constraint-handling scheme and a specific-design crossover operator. The new constraint-handling scheme can make the individuals satisfy all linear constraints exactly and the nonlinear constraints approximately. The crossover operator can generate high quality potential offspring. Based on the constraint-handling scheme and the crossover operator, we propose a new evolutionary algorithm and prove its global convergence. A distinguishing feature of the algorithm is that it can be used to handle nonlinear BLPPs with nondifferentiable leader's objective functions. Finally, simulations on 31 benchmark problems, 12 of which have nondifferentiable leader's objective functions, are made and the results demonstrate the effectiveness of the proposed algorithm.

关键词： constraint handling evolutionary algorithm (EA) global optimization nonlinear bilevel programming

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A neural network approach for solving nonlinear bilevel programming problem

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COMPUTERS & MATHEMATICS WITH APPLICATIONS 2008年第12期55卷 2823-2829页

作者： Lv, Yibing Hu, Tiesong Wang, Guangmin Wan, Zhongping Wuhan Univ State Key Lab Water Resources & Hydropower Engn S Wuhan 430072 Peoples R China Wuhan Univ Sch Math & Stat Wuhan 430072 Peoples R China

A neural network model is presented for solving nonlinear bilevel programming problem, which is a NP-hard problem. The proposed neural network is proved to be Lyapunov stable and capable of generating approximal optimal solution to the nonlinear bilevel programming problem. The asymptotic properties of the neural network are analyzed and the condition for asymptotic stability, solution feasibility and solution optimality are derived. The transient behavior of the neural network is Simulated and the validity of the network is verified with numerical examples. (c) 2007 Elsevier Ltd. All rights reserved.

关键词： nonlinear bilevel programming neural network asymptotic stability optimal solution

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An augmented Lagrangian multiplier method based on a CHKS smoothing function for solving nonlinear bilevel programming problems

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KNOWLEDGE-BASED SYSTEMS 2014年 55卷 9-14页

作者： Jiang, Yan Li, Xuyong Huang, Chongchao Wu, Xianing Chinese Acad Sci State Key Lab Urban & Reg Ecol Res Ctr Ecoenvironm Sci Beijing 100085 Peoples R China Wuhan Univ Sch Math & Stat Wuhan 430072 Peoples R China Sinohydro Resources Ltd Beijing 100048 Peoples R China

bilevel programming techniques deal with decision processes involving two decision makers with a hierarchical structure. In this paper, an augmented Lagrangian multiplier method is proposed to solve nonlinear bilevel programming (NBLP) problems. An NBLP problem is first transformed into a single level problem with complementary constraints by replacing the lower level problem with its Karush-Kuhn-Tucker optimality condition, which is sequentially smoothed by a Chen-Harker-Kanzow-Smale (CHKS) smoothing function. An augmented Lagrangian multiplier method is then applied to solve the smoothed nonlinear program to obtain an approximate optimal solution of the NBLP problem. The asymptotic properties of the augmented Lagrangian multiplier method are analyzed and the condition for solution optimality is derived. Numerical results showing viability of the approach are reported. (C) 2013 Elsevier B.V. All rights reserved.

关键词： nonlinear bilevel programming Karush-Kuhn-Tucker condition Complementary constraints Smoothing function Augmented Lagrangian multiplier method

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Application of particle swarm optimization based on CHKS smoothing function for solving nonlinear bilevel programming problem

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APPLIED MATHEMATICS AND COMPUTATION 2013年第9期219卷 4332-4339页

作者： Jiang, Yan Li, Xuyong Huang, Chongchao Wu, Xianing Chinese Acad Sci Ecoenvironm Sci Res Ctr State Key Lab Urban & Reg Ecol Beijing 100085 Peoples R China Wuhan Univ Sch Math & Stat Wuhan 430072 Peoples R China Sinohydro Corp Ltd Beijing 100044 Peoples R China

Particle Swarm Optimization (PSO) is a new optimization technique originating from artificial life and evolutionary computation. It completes optimization through following the personal best solution of each particle and the global best value of the whole swarm. PSO can be used to solve nonlinear programming problems for global optimal solutions efficiently, so a novel approach based on particle swarm optimization is proposed to solve nonlinear bilevel programming problem (NBLP). In the proposed approach, applying Karush-Kuhn-Tucker (KKT) condition to the lower level problem, we transform the NBLP into a regular nonlinear programming with complementary constraints, which is sequentially smoothed by Chen-Harker-Kanzow-Smale (CHKS) smoothing function. The PSO approach is then applied to solve the smoothed nonlinear programming for getting the approximate optimal solution of the NBLP problem. Simulations on 5 benchmark problems and practical example about watershed water trading decision-making problem are made and the results demonstrate the effectiveness of the proposed method for solving NBLP. (C) 2012 Elsevier Inc. All rights reserved.

关键词： nonlinear bilevel programming Karush-Kuhn-Tucker condition Smoothing method Particle swarm optimization

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nonlinear bilevel programming approach for decentralized supply chain using a hybrid state transition algorithm

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KNOWLEDGE-BASED SYSTEMS 2022年 240卷

作者： Zhou, Xiaojun Tian, Jituo Wang, Zeyu Yang, Chunhua Huang, Tingwen Xu, Xuesong Cent South Univ Sch Automat Changsha 410083 Peoples R China Texas A&M Univ Qatar Doha 23874 Qatar Hunan Univ Technol & Business Sch Frontier Crossover Studies Changsha 410205 Peoples R China Peng Cheng Lab Shenzhen 518000 Peoples R China

This paper investigates a decentralized supply chain that is composed of one manufacturer and multiple distributors. The manufacturer produces goods and wholesales them to multiple distributors and then the distributors sell products to various markets. The entire production period of the manufacturer is divided into several intervals. The decision-making problem of the entire decentralized supply chain is presented as a two-echelon coordination game network, in which each decision-maker can influence decision-making of other levels. A Stackelberg game framework is proposed to coordinate the decision-making process. And then two nonlinear bi-level programming (BLP) models are developed to find the optimal equilibrium decision scheme by switching the leader and follower roles between the manufacturer and the distributors. The models consider the manufacturer's budget constraints in each interval and the market demands are affected by distributors' selling price and advertising strategies. According to the hierarchy and complexity of bi-level programming problem (BLPP), a nested bi-level method based on hybrid state transition algorithm is proposed to address the BLP models, and mapping approximation strategy is utilized to improve computational efficiency. Finally, the numerical experiments are performed to demonstrate the superiority of the proposed method in terms of accuracy and computational efficiency. (c) 2022 Elsevier B.V. All rights reserved.

关键词： Decentralized supply chain nonlinear bilevel programming State transition algorithm Mapping approximation

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A hybrid differential evolution algorithm for solving nonlinear bilevel programming with linear constraints

A hybrid differential evolution algorithm for solving nonlin...

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5th IEEE International Conference on Cognitive Informatics (ICCI 2006)

作者： Zhu, Xiaobo Yu, Qian Wang, Xianjia Wuhan Univ Inst Syst Engn Wuhan 430072 Peoples R China

ISBN: (纸本)9781424404759

A hybrid differential evolution method was proposed for the general (nonlinear) bilevel programming (GBP) with linear constraint functions by combining the Differential Evolution algorithm (DE) with the interior point algorithm. The differential evolution algorithm was designed to operate in the up-level problems of GBP and the interior point algorithm was employed to solve the lower-level problems. With testing the solvability of the lower-level problem by the interior point algorithm, the infeasible individuals can be screened out and the difficulty of using penalty functions to deal with the constraints is avoided Finally, six numerical experiments with the offline-performance analyses were et given to show the effectiveness of the algorithm.

关键词： differential evolution (DE) nonlinear bilevel programming interior point algorithm offline-performance

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Combining TOPSIS and Particle Swarm Optimization for a Class of nonlinear bilevel programming Problems 10

Combining TOPSIS and Particle Swarm Optimization for a Class...

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10th International Conference on Computational Intelligence and Security CIS 2014

作者： Jia, Liping Zou, Guocheng Fan, Lei Leshan Normal Univ Coll Math & Informat Sci Leshan 614000 Peoples R China Xidian Univ Sch Software Inst Software Engn Xian 710071 Peoples R China

ISBN: (纸本)9781479974344

bilevel programming has a nested structure and widely used in many real-life problems. This paper addresses a kind of nonlinear bilevel programming, the lower level problem is transformed into a series of constraints for the upper level problem by KKT condition. In the transformed problem, the constraints are taken as an objective for the upper level, thus the nonlinear bilevel programming becomes an equivalent two-objective unconstrained optimization problem. To solve the equivalent problem, TOPSISP and particle swarm optimization are combined and then TOPSISPSO is proposed, computational results for two benchmarks are presented to show the effectiveness of the algorithm.

关键词： nonlinear bilevel programming PSO TOPSIS KKT condition

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An Ameliorated Teaching-learning Based Optimization Algorithm for nonlinear bilevel programming 12

An Ameliorated Teaching-learning Based Optimization Algorith...

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12th International Conference on Computational Intelligence and Security (CIS)

作者： Jia, Liping Li, Zhonghua Leshan Normal Univ Coll Math & Informat Sci Leshan 614000 Peoples R China Leshan Normal Univ Coll Comp Sci Leshan 614000 Peoples R China

ISBN: (纸本)9781509048403

bilevel programming has a hierarchical leaderfollower structure in which the leader acts as the upper level, and the follower acts as the lower level, respectively. Both of them wish to optimize its objective. In this paper, a heuristic method called teaching-learning based optimization algorithm is proposed for solving nonlinear bilevel programming. The detailed procedure is presented, and the performance of the proposed algorithm is validated by two numerical results.

关键词： nonlinear bilevel programming ameliorated teaching-learning based optimization algorithm KKT condition

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Exact Penalty Method for the nonlinear bilevel programming Problem

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Wuhan University Journal of Natural Sciences 2010年第6期15卷 471-475页

作者： PAN Qingfei AN Zhonghua QI Hui Department of Physics and Mechanics-Electronic Engineering Sanming University Sanming 365004 Fujian China Department of Mathematics and Measure Economics Hu-bei University of Education Wuhan 430205 Hubei China Department of Mathematics and Computer SanmingUniversity Sanming 365004 Fujian China

In this paper,following the method of replacing the lower level problem with its Kuhn-Tucker optimality condition,we transform the nonlinear bilevel programming problem into a normal nonlinear programming problem with the complementary slackness constraint ***,we get the penalized problem of the normal nonlinear programming problem by appending the complementary slackness condition to the upper level objective with a *** prove that this penalty function is exact and the penalized problem and the nonlinear bilevel programming problem have the same global optimal solution ***,we propose an algorithm for the nonlinear bilevel programming *** numerical results show that the algorithm is feasible and efficient.

关键词： convex-quadratic programming nonlinear bilevel programming Kuhn-Tucker optimality condition penalty function method optimal solution

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