UPTEC F 10 045 Examensarbete 30 hp Juli 2010 Towards the Solution of Large-Scale and Stochastic Traffic Network Design Problems Fredrik Hellman Abstract Towards the Solution of Large-Scale and Stochastic Traffic Network Design Problems Fredrik Hellman Teknisk- naturvetenskaplig fakultet UTH-enheten This thesis investigates the second-best toll pricing and capacity expansion problems when stated as mathematical programs with equilibrium constraints (MPEC). Three Besöksadress: main questions are rised: First, whether conventional descent methods give Ångströmlaboratoriet Lägerhyddsvägen 1 sufficiently good solutions, or whether global solution methods are to prefer. Second, Hus 4, Plan 0 how the performance of the considered solution methods scale withnetwork size. Third, how a discretized stochastic mathematical program with equilibrium Postadress: constraints (SMPEC) formulation of a stochastic network design problem can be Box 536 751 21 Uppsala practically solved. An attempt to answer these questions is done through a series of numerical experiments. Telefon: 018 – 471 30 03 The traffic system is modeled using the Wardrop’s principle for user behavior, Telefax: separable cost functions of BPR- and TU71-type. Also elastic demand is considered 018 – 471 30 00 for some problem instances. Hemsida: Two already developed method approaches are considered: implicit programming and http://www.teknat.uu.se/student a cutting constraint algorithm. For the implicit programming approach, several methods—both local and global—are applied and for the traffic assignment problem an implementation of the disaggregate simplicial decomposition (DSD) method is used. Regarding the first question concerning local and global methods, our results don’t give a clear answer. The results from numerical experiments of both approaches on networks of different sizes shows that the implicit programming approach has potential to solve large-scale problems, while the cutting constraint algorithm scales worse with network size. Also for the stochastic extension of the network design problem, the numerical experiments indicate that implicit programming is a good approach to the problem. Further, a number of theorems providing sufficient conditions for strong regularity of the traffic assignment solution mapping for OD connectors and BPR cost functions are given. Handledare: Michael Patriksson Ämnesgranskare: Per Lötstedt Examinator: Tomas Nyberg ISSN: 1401-5757, UPTEC F 10 045 Acknowledgements I would like to express my appreciation and gratitude to my supervisor, Prof. Michael Patriksson for giving me the opportunity to write this thesis and for improving the final report by proofreading and through inspiring conversations. I also want to acknowledge Dr. Christoffer Cromvik for helping me in the initial phase, Prof. Clas Rydergren and Joakim Ekström at Linköping University for pro- viding me with data on the Stockholm network, and Dr. Napsu Karmitsa at University of Turku for providing the source code for LMBM-B. I want to thank my new friends at the Department of Mathematical Sciences at Chalmers for this year and all the nice lunch and coffee breaks. My gratitude also goes to my family and friends for their support during this process, which lasted longer than planned. I am very grateful to my beloved Hanna for giving me great support all the way, despite us being literally parted by an ocean. i Contents Acronyms iv Glossary of Notation v 1 Introduction 1 2 Contribution 1 3 Solution Approaches 1 4 Traffic Assignment Problem 2 4.1 Model Description . 2 4.2 As Solution to an Optimization Problem . 4 4.3 System Optimal Traffic Assignment . 7 4.4 Convergence Measures . 7 5 Network Data: Graph and Functions 8 5.1 Cost Functions . 8 5.2 Demand Functions . 8 5.3 Centroids and OD Connectors . 9 6 Network Design Problem 10 6.1 Problem Definition . 10 6.2 Mathematical Programming Models . 11 6.3 Stability and Subgradients of Traffic Assignment Solution . 13 6.4 Computing Subgradients of Traffic Assignment Solutions . 15 6.5 Uniqueness and Strong Regularity in Practice . 17 6.6 Existence of Solutions to MPEC . 21 6.7 Stochastic MPEC . 22 6.8 Toll Pricing Problem (TP) . 25 6.9 Capacity Expansion Problem (CEP) . 26 7 Approach I: Implicit Programming 28 7.1 Solving the Traffic Assignment Problem . 28 7.2 Solving the Sensitivity Analysis Problem . 30 7.3 Solving the Stochastic Extension . 31 7.4 Solving the Network Design Problem Locally . 32 7.4.1 SDBH . 32 7.4.2 SNOPT . 34 7.4.3 LMBM-B . 34 7.5 Solving the Network Design Problem Globally . 34 7.5.1 Branch and Bound . 35 7.5.2 NFFM . 36 7.5.3 EGO . 38 7.5.4 DIRECT . 40 7.6 Implementation and Usage Details . 40 7.6.1 DSDTAP . 40 7.6.2 SDBH . 41 7.6.3 SNOPT . 42 7.6.4 LMBM-B . 42 7.6.5 NFFM . 42 7.6.6 EGO . 43 7.6.7 DIRECT . 43 8 Approach II: Cutting Constraint Algorithm 44 ii 9 Numerical Experiments 46 9.1 Problem Set . 47 9.1.1 Harker and Friesz CEP (HF CEP) . 47 9.1.2 Sioux Falls Fixed Demand CEP (SFF CEP) . 48 9.1.3 Sioux Falls Elastic Demand TP (SFE TP) . 48 9.1.4 Small Stockholm Elastic Demand TP (STHLM TP) . 49 9.1.5 Anaheim Fixed Demand CEP (ANA CEP) . 50 9.1.6 Barcelona Fixed Demand (BARC) . 51 9.1.7 Summary of Problems . 51 9.2 Precision of Objective Function . 51 9.3 Evaluation of Rules for Defining Used Routes . 53 9.4 NFFM on Six-Hump Camel Function . 54 9.5 DSDTAP on a Trivial Elastic Demand Problem . 57 9.6 Time Complexity in Number of Scenarios of LMBM-B and CCA for Stochastic Extension 58 9.7 Sioux Falls with Elastic Demand First-Best Toll Pricing Problem (SFE FB TP) . 58 9.8 Harker and Friesz Capacity Expansion Problem (HF CEP) . 60 9.8.1 Local and Global Optimization . 60 9.8.2 Investigation of Failure of NFFM on HF CEP . 61 9.8.3 Global Optimization on HF 2 CEP . 63 9.8.4 Stochastic Optimization . 64 9.9 Sioux Falls with Fixed Demands Capacity Expansion Problem (SFF CEP) . 67 9.10 Stockholm Toll Pricing Problem with Cordon J2 (STHLM J2 TP) . 68 9.10.1 Local and Global Optimization . 68 9.10.2 Stochastic Optimization . 69 9.11 Anaheim Capacity Expansion Problem (ANA CEP) . 71 9.12 Barcelona with Fixed Demand (BARC) . 73 10 Discussion 73 11 Conclusions 75 A Network data 79 A.1 Anaheim . 79 iii Acronyms BFGS Broyden-Fletcher-Goldfarb-Shanno (Hessian approximation method) BPR Bureau of Public Roads (cost function) CCA Cutting Constraint Algorithm (Approach II) CEP Capacity Expansion Problem (special Network Design Problem) CVAR Conditional Value at Risk (objective function for stochastic optimization) DIRECT Dividing Rectangles (method for global optimization) DSD Disaggregate Simplicial Decomposition (method for solving the traffic assignment problem) DSDTAP DSD Traffic Assignment Problem (implementation of DSD) EGO Efficient Global Optimization (method for global optimization LMBM-B Limited Memory Bundle Method Bounded (method for local optimzation) MFCQ Mangasarian-Fromovitz Constraint Qualification (constraing qualification) MNL Multinomial Logit model (model for elastic demand) MPCC Mathematical Program with Complementarity Constraints (optimization model class) MPEC Mathematical Program with Equilibrium Constraints (optimization model class) NDP Network Design Problem (optimzation model) NLP Nonlinear Programming (optimization model class) OBA Origin-Based Algorithm for Traffic Assignment Problem (method for solving the traffic assignment problem) QP Quadratic Programming (optimization model class) RMP Restriced Master Problem (optimization model part of CCA) SAA Sample Approximation Method (discretization method) SCEP Stochastic CEP (special stochastic Network Design Problem) SDBH Steepest-Descent-BFGS-Hybrid (method and implementation for local optimization) SMPEC Stochastic MPEC (optimization model class) SNOPT Software for Large-Scale Nonlinear Optimization (implementation of local optimiza- tion method) SQP Sequential Quadratic Programming (method for local optimization) SQOPT Software for Large-Scale Linear and Quadratic Programming) STP Stochastic TP (special stochastic Network Design Problem) TAP Traffic Assignment Problem (optimization model) TAPAS Traffic Assignment by Paired Alternative Segments (method for solving the traffic assignment problem) TP Toll Pricing Problem (special Network Design Problem) TU71 Unknown (cost functions) VOT Value of Time (parameter) iv Glossary of Notation Network Model Definitions can be found in the section specified by last column. G = (A; N ) Traffic network graph G with links A and nodes N 4.1 a Link 4.1 C; k Set of origin-destination pairs and member k 4.1 Gk = (Ak; NK ) Traffic network graph understood by travelers on OD pair k 4.1 R Set of all routes 4.1 Rk Set of routes for OD pair k 4.1 r Route 4.1 M = (mij) Incidence matrix for G 4.1 Mk Incidence matrix for Gk 4.1 d = (dk) Demand variable 4.1 D = (Dk) Demand function 4.1 π = (πk) Travel cost variable 4.1 t = (ta) Link cost function 4.1 v = (va) Link flow variable 4.1 h = (hkr) = (hr) Route flow variable 4.1 δ = (δkra) = (δra) Route-link incidence value 4.1 Λ Route-link incidence matrix 4.1 Γ Route-OD pair incidence matrix 4.1 z Traffic assignment objective function 4.2 y = (v; d) Composed traffic assignment solution 4.2 V Set of feasible cycle-free flow solutions y 4.2 C Gradient of z w.r.t y, i.e, C = ryz 4.2 SS Social surplus function 4.3 UC User cost function 4.3 SC Social cost function 4.3 RDG Relative duality gap of traffic assignment solution 4.4 Network Data Definitions can be found in the section specified by last column. b Parameters for BPR cost function 5.1 c Parameters for TU71 cost function 5.1 0 Tk;Ak;Kk; π ; α Parameters for MNL demand function 5.2 max min dk ; πk Maximum demand and corresponding minimum cost 5.2 Network Design Problem Definitions can be found in the section specified by last column.