This paper proposes a decomposition method to compute a lower bound for unconstrained quadratic zero-one minimization. First, we show that any quadratic function can be expressed as a sum of particular quadratic functions whose minima can be computed by a simple branch and bound algorithm. Then, assuming some hypothesis, we prove that, among all possible decompositions, the best one can be found by a Lagrangian decomposition method. Moreover, we show that our algorithm gives at least the roof dual bound and should give better results in practice. Eventually, computational results and comparison with Pardalos and Rodgers' algorithm demonstrate the efficiency of our method for medium size problems (up to 100 variables).quadratic 0--1 programming, mixed integer programming, Lagrangean decomposition

We give a formal specification for a strategic network routing problem known as the convoy movement problem (CMP) and establish that the corresponding feasibility problem is NP-complete. We then introduce an integer programming (IP) model based on the concept of a time-space network and apply a Lagrangian relaxation to this model. We discuss how the dual function may be evaluated using a modified version of Dijkstra’s algorithm suitable to very large, implicitly defined graphs and show how heuristic solutions to the primal problem may be obtained. We present results for a number of instances of the CMP, most of which are based on real-world problems. The number of convoys in these instances varies between 15–25, and their movement time requires up to several thousand time units in networks ranging in size from a few dozen to several thousand vertices and edges. The most difficult instance tested involves 17 long convoys each taking four times the average link travel time to pass through a point in the network. This instance is solved within 3.3% of optimality in less than 3.5 hours of computing time on a Dell Precision 420 dual processor computer. Every other test instance is solved within 2% of the optimal value in less than 20 minutes of computing time

A new heuristic algorithm, PROBE_BA, which is based on the recently introduced metaheuristic paradigm population- reinforced optimization-based exploration (PROBE), is proposed for solving the Graph Partitioning Problem. The "exploration" part of PROBE_BA is implemented by using the differential-greedy algorithm of Battiti and Bertossi and a modification of the Kernighan-Lin algorithm at the heart of Bui and Moon's genetic algorithm BFS _GBA. Experiments are used to investigate properties of PROBE and show that PROBE_BA compares favorably with other solution methods based on genetic algorithms, randomized reactive tabu search, or more specialized multilevel partitioning techniques. In addition, PROBE_BA finds new best cut values for 10 of the 34 instances in Walshaw's graph partitioning archive

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