Path based Algorithms(Term Paper)
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This document summarizes computational studies of two path-based traffic assignment algorithms: the disaggregate simplicial decomposition (DSD) algorithm and the gradient projection (GP) algorithm. The study used a large-scale real network in Chicago and five randomly generated networks. Results showed that the GP algorithm performed better than both versions of the DSD algorithm on all networks, finding solutions faster with fewer iterations. The GP algorithm was more efficient by maintaining a smaller set of active paths and avoiding expensive line searches through second derivative information. While DSD could find near-optimal solutions quickly, it took more time overall and maintained a larger set of paths in each iteration.
![Gradient projection (GP) algorithm: GP algorithm is based on the Goldstein–Levitin–Polyak GP
method formulated by Bertsekas for general nonlinear multi-commodity problems. The algorithm
operates directly in the space of path flows with moves are in the direction of the negative gradient,
scaled by the second derivative Hessian. The main idea is to eliminate the travel demand constraints
which is done by reformulating the path-flow variables in terms of the non-shortest path flows which
make the projection operation simpler.
• 𝑓𝐾𝑟𝑠
𝑟𝑠
= 𝑞 𝑟𝑠 − 𝑘 ∈ 𝐾𝑟𝑠,𝑘 =.𝑘𝑟𝑠
𝑓𝑘
𝑟𝑠
, ∀ r ∈ R, s ∈ S (10)
• Embedding constraints into the objective function, we obtain a new formulation with just the
nonnegativity constraints on the non-shortest path flows as the decision variables.
• min ˜ Z( ˜ f ) (11)
• subject to
• 𝑓𝑘
𝑟𝑠
≥ 0, ∀ k ∈ Krs, k _= .krs, ∀ r ∈ R, s ∈ S (12)
• where ˜Z is the transformed objective function expressed in terms of the non-shortest path flows ˜
f for all OD pairs. In each iteration, the scaled GP algorithm updates the path flows according to the
following iteration equations:
• 𝑓𝑘
𝑟𝑠
𝑛 + 1 = [𝑓𝑘
𝑟𝑠
𝑛 −
𝛼 𝑛
𝑠 𝑘
𝑟𝑠 𝑛
(𝑑 𝑘
𝑟𝑠
𝑛 − 𝑑 𝑘 𝑟𝑠(𝑛)
𝑟𝑠
)] ∀ k ∈ Krs, k _= .krs, ∀ r ∈ R, s ∈ S (13)
• 𝑓𝐾 𝑟𝑠(𝑛)
𝑟𝑠
𝑛 + 1 = 𝑞 𝑟𝑠 −, ∀ 𝑘 ∈ 𝐾𝑟𝑠,𝑘 =.𝑘𝑟𝑠
𝑓𝑘
𝑟𝑠
(𝑛 + 1), ∀ r ∈ R, s ∈ S (14)](https://image.slidesharecdn.com/project120040112-160416094958/85/Path-based-Algorithms-Term-Paper-5-320.jpg)
Path based Algorithms(Term Paper)
- 1. Computational study of state-of- the-art path-based traffic assignment algorithms Submitted by Pankaj Kumar 120040112 4th year undergraduate IIT Bombay
- 2. Content: • Introduction • Assignment Problem • Assignment Algorithms • Results • Conclusion
- 3. Introduction: Path-based algorithms, new area in solving traffic assignment problem, which has been made possible due to the dramatic advances in computing power. It has been found that path-based algorithms is a viable approach for traffic assignment problems with reasonably large network sizes. Path-formulated traffic assignment problem: Consider an urban traffic network represented as a graph G = (N,A), where N and A are the sets of nodes and links, respectively. Let R and S be subsets of N and qrs be the steady-state demand generated from origin r ∈ R to destination s ∈ S. The independent variables are a set of path flows 𝑓𝑘 𝑟𝑠 that satisfy the travel demand: • 𝑓𝑘 𝑟𝑠 = qrs, ∀ r ∈ R, s ∈ S , k∈Krs (1) where Krs is a set of cycle-free paths connecting r and s. Let xa be the traffic flow on link a ∈ A, then the total traffic flow on each link can be written as: xa = 𝑟∈𝑅 𝑠∈𝑆 𝑓𝑘 𝑟𝑠 for all k∈Krs (2) where • δrs= 1 if link a is on path k connecting r and s 0 otherwise To ensure meaningful solutions, all path flows must be non-negative: 𝑓𝑘 𝑟𝑠 ≥ 0, ∀ k ∈ Krs, r ∈ R, s ∈ S (3) The functional form is given by ta(xa) = tf(1+0.15( 𝑥𝑎 𝐶𝑎 )^4) (4) • ZUE = 0 𝑥𝑎 𝑡𝑎 (𝑤)𝑑𝑤 for all a∈A (5) • ZSO = 𝑡𝑎 (𝑥𝑎)𝑥𝑎 for all a∈A (6)
- 4. We are focusing on two particular algorithms: • Disaggregate simplicial decomposition (DSD) algorithm • Gradient projection (GP) algorithm Factors taken into consideration for sensitivity analysis are: • Network sizes • Congestion levels • Number of origin-destination (OD) pairs • Solution accuracy Disaggregate simplicial decomposition (DSD) algorithm: It belongs to the general class of simplicial decomposition algorithm, which states that any point in a bounded polyhedral set can be described by a convex combination of the extreme points. Its generalized solution iterates between a linear subproblem and a master problem to determine the next solution point. The subproblem solves a linear approximation of the objective function to obtain extreme points. • minZ(λ) = 𝑎∈𝐴 0 𝑡𝑎(𝑤) d𝑤 where upper limit is 𝑟=𝑅 𝑠∈𝑆 𝑞𝑟𝑠 𝑘€ˆ𝐾𝑟𝑠 𝜆𝑟𝑠𝛿𝑟𝑠 (7) • subject to • 𝑘∈𝐾𝑟𝑠 𝜆 𝑘 𝑟𝑠 = 1, ∀ r ∈ R, s ∈ S (8) • 𝜆 𝑘 𝑟𝑠 ≥ 0, ∀ k ∈ ˆKrs, r ∈ R, s ∈ S (9) • Path flows are readily available once 𝜆 𝑘 𝑟𝑠 are solved.
- 5. Gradient projection (GP) algorithm: GP algorithm is based on the Goldstein–Levitin–Polyak GP method formulated by Bertsekas for general nonlinear multi-commodity problems. The algorithm operates directly in the space of path flows with moves are in the direction of the negative gradient, scaled by the second derivative Hessian. The main idea is to eliminate the travel demand constraints which is done by reformulating the path-flow variables in terms of the non-shortest path flows which make the projection operation simpler. • 𝑓𝐾𝑟𝑠 𝑟𝑠 = 𝑞 𝑟𝑠 − 𝑘 ∈ 𝐾𝑟𝑠,𝑘 =.𝑘𝑟𝑠 𝑓𝑘 𝑟𝑠 , ∀ r ∈ R, s ∈ S (10) • Embedding constraints into the objective function, we obtain a new formulation with just the nonnegativity constraints on the non-shortest path flows as the decision variables. • min ˜ Z( ˜ f ) (11) • subject to • 𝑓𝑘 𝑟𝑠 ≥ 0, ∀ k ∈ Krs, k _= .krs, ∀ r ∈ R, s ∈ S (12) • where ˜Z is the transformed objective function expressed in terms of the non-shortest path flows ˜ f for all OD pairs. In each iteration, the scaled GP algorithm updates the path flows according to the following iteration equations: • 𝑓𝑘 𝑟𝑠 𝑛 + 1 = [𝑓𝑘 𝑟𝑠 𝑛 − 𝛼 𝑛 𝑠 𝑘 𝑟𝑠 𝑛 (𝑑 𝑘 𝑟𝑠 𝑛 − 𝑑 𝑘 𝑟𝑠(𝑛) 𝑟𝑠 )] ∀ k ∈ Krs, k _= .krs, ∀ r ∈ R, s ∈ S (13) • 𝑓𝐾 𝑟𝑠(𝑛) 𝑟𝑠 𝑛 + 1 = 𝑞 𝑟𝑠 −, ∀ 𝑘 ∈ 𝐾𝑟𝑠,𝑘 =.𝑘𝑟𝑠 𝑓𝑘 𝑟𝑠 (𝑛 + 1), ∀ r ∈ R, s ∈ S (14)
- 6. Experiment and Result: A real and large-scale network, known as the ADVANCE network in the Chicago area, is used for generating computational results from DSD and GP algorithms for empirically convincing that these algorithms are viable in real world. Other than this, 5 randomly generated networks are used to do sensitivity analysis of these algorithms. Table 1: Provides the statistics of these networks. Objective value is used as a common criterion to measure the performance of the algorithms. Table 2 reports the computational times and number of iterations for both algorithms based on the above termination criterion. DSD did not switch in any of the cases to the second-order operations. In Table 2, two versions of DSD, denoted by DSD1 for the first-order method and DSD2 for the second-order method, are provided and compared with GP.
- 7. Conclusion: Researchers have implemented two path-based algorithms (GP and DSD) for the traffic assignment problem and compared their efficiency using sensitivity analysis and Advance network. Results confirmed that GP performed better than DSD of both versions (DSD2 & DSD1) in all the cases. GP DSD Number of paths used in each iteration is less Number of path used in each iteration is high It uses one-at a time flow update strategy to avoid expensive line searches by keeping second derivative information DSD quickly obtain a near optimal solution by using first order reduced gradient method Less time taken to find a solution More time taken to find the ultimate solution GP avoids performing expensive line searches and remove unused paths by employing column dropping and maintain a much smaller path Main time spent on optimizing the switching strategy