Simultaneous Solution of Lagrangean Dual Problems Interleaved with Preprocessing for the Weight Constrained Shortest Path Problem

Ranga Muhandiramge(rangatartarus.uwa.edu.au)
Natashia Boland(natashiaunimelb.edu.au)

Abstract: Conventional Lagrangean preprocessing for the network Weight Constrained Shortest Path Problem (WCSPP calculates lower bounds on the cost of using each node and edge in a feasible path using a single optimal Lagrange multiplier for the relaxation of the WCSPP. These lower bounds are used in conjunction with an upper bound to eliminate nodes and edges. However, for each node and edge, a Lagrangean dual problem exists whose solution may differ from the relaxation of the full problem. Thus, using one Lagrange multiplier does not offer the best possible network reduction. Furthermore, eliminating nodes and edges from the network may change the Lagrangean dual solutions in the remaining reduced network, warranting an iterative solution and reduction procedure. We develop a method for solving the related Lagrangean dual problems for each edge simultaneously which is iterated with eliminating nodes and edges. We demonstrate the effectiveness of our method computationally: we test it against several others and show that it both reduces solve time and the number of intractable problems encountered. We use a modified version of the enumeration algorithm of Carlyle and Wood (2003) in the gap closing stage. We also make improvements to this algorithm and test them computationally.

Keywords: shortest paths; constrained shortest paths; enumeration algorithms; Lagrangean relaxation; preprocessing

Category 1: Network Optimization

Category 2: Other Topics (Dynamic Programming )

Category 3: Combinatorial Optimization (Other )

Citation: School of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway, Crawley, 6009, WA, Australia, November 2007

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Entry Submitted: 02/04/2008
Entry Accepted: 02/07/2008
Entry Last Modified: 02/04/2008

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