Multilevel Optimization: Algorithms and Applications

Transcription

1 Multilevel Optimization: Algorithms and Applications Edited by Athanasios Migdalas Division of Optimization, Department of Mathematics, Linköping Institute of Technology, Linköping, Sweden Panos M. Pardalos Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL, U.S.A. and Peter Värbrand Division of Optimization, Department of Mathematics, Linköping Institute of Technology, Linköping, Sweden KLUWER ACADEMIC PUBLISHERS DORDRECHT / BOSTON / LONDON

2 CONTENTS PREFACE CONGESTED O-D TRIP DEMAND ADJUSTMENT PROBLEM: BILEVEL FORMULATION AND OPTIMALITY CONDITIONS Yang Chen and Michael Florian 1 1 Introduction 2 2 Literature Review 3 3 Model Analysis 9 4 Necessary Optimality Conditions of the DAP 12 5 Conclusions 18 DETERMINING TAX CREDITS FOR CONVERTING NONFOOD CROPS TO BIOFUELS: AN APPLICATION OF BILEVEL Jonathan F. Bard, John Plummer and Jean Claude Sourie 23 1 Introduction 24 2 Mathematical Model 25 3 Description of Algorithms 29 4 Computational Results 36 5 Discussion 41

3 viii MULTILEVEL OPTIMIZATION METHODS IN MECHANICS P.D. Panagiotopoulos, E.S. Mistakidis, G.E. Stavroulakis and O.K. Panagouli 51 1 Introduction 51 2 Presentation of the Multilevel Decomposition Methods 54 3 Large Cable Structures 59 4 Large Elastoplastic Structures 61 5 Validation and Improvements of Simplified Models 63 6 Extension to other Problems. Decomposition Algorithms for Nonconvex Minimization Problems 67 7 A Multilevel Method for the Approximation of a Nonconvex Minimum Problem by Convex ones 69 8 Multilevel Decomposition into two Convex Problems 76 9 Structures with Fractal Interfaces 82 OPTIMAL STRUCTURAL DESIGN IN NONSMOOTH MECHANICS Georgios E. Stavroulakis and Harald Günzel 91 1 Introduction 92 2 Parametric Nonsmooth Structural Analysis Problems 95 3 Optimal Design Problems Mathematical Analysis and Algorithms Discussion 112 REFERENCES 112 OPTIMIZING THE OPERATIONS OF AN ALUMINIUM SMELTER USING NON-LINEAR BI-LEVEL Miles G. Nicholls Introduction The Mathematical Model of the Aluminium Smelter The Solution Algorithm The Mathematical Model Representing the Multi-period Operations of the Aluminium Smelter Concluding Remarks 137

4 IX REFERENCES 137 COMPLEXITY ISSUES IN BILEVEL LINEAR Xiaotie Deng Introduction Difficulty in Approximation A Special Case Solvable in Polynomial Time Regret Ratio in Decision Analysis Future Directions 159 REFERENCES 160 THE COMPUTATIONAL COMPLEXITY OF MULTI-LEVEL BOTTLENECK PROBLEMS Tibor Dudäs, Bettina Klinz and Gerhard J. Woeginger Introduction Problem Statement and Previous Complexity Results Hardness Proof for Multi-Level Bottleneck Programs Hardness Proof for Multi-Level Linear Programs The Complexity of Bi-Level Programs Discussion 177 REFERENCES 177 ON THE LINEAR MAXMIN AND RELATED PROBLEMS Charles Audet, Pierre Hansen, Brigitte Jaumard and Gilles Savard Introduction Reformulations Tools for Resolution Solving the Linear Maxmin Problem 195

5 X 9 PIECEWISE SEQUENTIAL QUADRATIC FOR MATHEMATICAL PROGRAMS WITH NONLINEAR COMPLEMENTARITY CONSTRAINTS Zhi-Quan Luo, Jong-Shi Pang, Daniel Ralph Introduction Application to Optimal Design of Mechanical Structures The Piecewise Smooth Approach to NCP-MP The PSQP Method for NCP-MPEC Computational Testing of PSQP 222 REFERENCES A NEW BRANCH AND BOUND METHOD FOR BILEVEL LINEAR PROGRAMS Hoang Tuy and Saied Ghannadan Introduction The Equivalent Reverse Convex Program Solution Method Implementation Issues Illustrative Example 241 IIA PENALTY METHOD FOR LINEAR BILEVEL PROBLEMS Mahyar A. Amouzegar, Khosrow Moshirvaziri Introduction Linear Bilevel Programming Problem The Method Globalization of the Solution Numerical Examples Concluding Remarks AN IMPLICIT FUNCTION APPROACH TO BILEVEL PROBLEMS Stephan Dempe Introduction Lipschitz Continuity of Optimal Solutions 275

6 XI 3 Application of the Bündle Method Non-uniquely Solvable Lower Level Problems Nonconvex Lower Level Problems and Coupling Constraints in the Upper Level Problem BILEVEL LINEAR, MULTIOBJECTIVE, AND MONOTONIC REVERSE CONVEX Hoang Tuy Introduction Optimization over the Efncient Set Bilevel Linear Programming Basic Properties of (FMRP) Different D.C. Approaches to {FMRP) EXISTENCE OF SOLUTIONS TO GENERALIZED BILEVEL PROBLEM Maria Beatrice Lignola and Jacqueline Morgan Introduction Notations and Preliminaries Parametric Implicit Variational Problem Existence Results for Generalized Bilevel Problems Final Remarks APPLICATION OF TOPOLOGICAL DEGREE THEORY TO COMPLEMENTARITY PROBLEMS Vladimir A. Bulavsky, George Isac and Vyacheslav V. Kalashnikov Problem Specification and Topological Degree Theory General Complementarity Problem Sufficient Conditions for Solution Existence Standard Complementarity Problem Implicit Complementarity Problem General Order Complementarity Problem 357 REFERENCES 358

7 Xll 16 OPTIMALITY AND DUALITY IN PARAMETRIC CONVEX LEXICOGRAPHIC C. A. Floudas and S. Zlobec 1 Introduction 2 Orientation 3 Continuity 4 Global Optimality 5 Local Optimality 6 Duality 7 Bilevel Zermelo's Problems INDEX 381