Variable fixing method by weighted average for the continuous quadratic knapsack problem

<p style='text-indent:20px;'>We analyze the method of solving the separable convex continuous quadratic knapsack problem by weighted average from the viewpoint of variable fixing. It is shown that this method, considered as a variant of the variable fixing algorithms, and Kiwiel's variable fixing method generate the same iterates. We further improve the algorithm based on the analysis regarding the semismooth Newton method. Computational results are given and comparisons are made among the state-of-the-art algorithms. Experiments show that our algorithm has significantly good performance; it behaves very much like an <inline-formula><tex-math id="M1">\begin{document}$ O(n) $\end{document}</tex-math></inline-formula> algorithm with a very small constant.</p>

Benders Decomposition for the Profit Maximizing Capacitated Hub Location Problem with Multiple Demand Classes

This paper models the profit maximizing capacitated hub location problem with multiple demand classes to determine an optimal hub network structure that allocates available capacities of hubs to satisfy demand for commodities from different market segments. A strong deterministic formulation of the problem is presented, and a Benders reformulation is described to optimally solve large-size instances of the problem. A new two-phase methodology is developed to decompose the Benders subproblem, and two effective separation routines are derived to strengthen the Benders optimality cuts. The algorithm is enhanced by the integration of improved variable-fixing techniques. The deterministic model is further extended by considering uncertainty associated with the demand to develop a two-stage stochastic program. To solve the stochastic version, a Monte Carlo simulation–based algorithm is developed that integrates a sample average approximation scheme with the proposed Benders decomposition algorithm. Novel acceleration techniques are presented to improve the convergence of the algorithm proposed for the stochastic version. The efficiency and robustness of the algorithms are evaluated through extensive computational experiments. Computational results show that large-scale instances with up to 500 nodes and three demand classes can be solved to optimality, and that the proposed separation routines generate cuts that provide significant speedups compared with using Pareto-optimal cuts. The developed two-phase methodology for solving the Benders subproblem as well as the variable-fixing and acceleration techniques can be used to solve other discrete location and network design problems.

Variable Fixing for Two-Arc Sequences in Branch-Price-and-Cut Algorithms on Path-Based Models

Variable fixing by reduced costs is a popular technique for accelerating the solution process of mixed-integer linear programs. For vehicle-routing problems solved by branch-price-and-cut algorithms, it is possible to fix to zero the variables associated with all routes containing at least one arc from a subset of arcs determined according to the dual solution of a linear relaxation. This is equivalent to removing these arcs from the network used to generate the routes. In this paper, we extend this technique to routes containing sequences of two arcs. Such sequences or their arcs cannot be removed directly from the network because routes traversing only one arc of a sequence might still be allowed. For some of the most common vehicle-routing problems, we show how this issue can be overcome by modifying the route-generation labeling algorithm in order to remove indirectly these sequences from the network. The proposed acceleration strategy is tested on benchmark instances of the vehicle-routing problem with time windows (VRPTW) and four variants of the electric VRPTW. The computational results show that it yields a significant speedup, especially for the most difficult instances.

A Matheuristic for Joint Optimal Power and Scheduling Assignment in DVB-T2 Networks

Because of the introduction and spread of the second generation of the Digital Video Broadcasting—Terrestrial standard (DVB-T2), already active television broadcasters and new broadcasters that have entered in the market will be required to (re)design their networks. This is generating a new interest for effective and efficient DVB optimization software tools. In this work, we propose a strengthened binary linear programming model for representing the optimal DVB design problem, including power and scheduling configuration, and propose a new matheuristic for its solution. The matheuristic combines a genetic algorithm, adopted to efficiently explore the solution space of power emissions of DVB stations, with relaxation-guided variable fixing and exact large neighborhood searches formulated as integer linear programming (ILP) problems solved exactly. Computational tests on realistic instances show that the new matheuristic performs much better than a state-of-the-art optimization solver, identifying solutions associated with much higher user coverage.