Catálogo de publicaciones - libros

The problem of computing a good approximation set of the Pareto front of a multiobjective optimization problem can be recasted as the maximization of its S-metric value, which measures the dominated hypervolume. In this way, the S-metric has recently been applied in a variety of metaheuristics. In this work, a novel high-precision method for computing approximation sets of a Pareto front with maximal S-Metric is proposed as a high-level relay hybrid of an evolutionary algorithm and a gradient method, both guided by the S-metric. First, an evolutionary multiobjective optimizer moves the initial population close to the Pareto front. The gradient-based method takes this population as its starting point for computing a local maximal approximation set with respect to the S-metric. Thereby, the population is moved according to the gradient of the S-metric.

This paper introduces expressions for computing the gradient of a set of points with respect to its S-metric on basis of the gradients of the objective functions. It discusses singularities where the gradient is vanishing or differentiability is one sided. To circumvent the problem of vanishing gradient components of the S-metric for dominated points in the population a penalty approach is introduced.

In order to test the new hybrid algorithm, we compute the precise maximizer of the S-metric for a generalized Schaffer problem and show, empirically, that the relay hybrid strategy linearly converges to the precise optimum. In addition we provide first case studies of the hybrid method on complicated benchmark problems.

The connected facility location (ConFL) problem generalizes the facility location problem and the Steiner tree problem in graphs. Given a graph  = (,), a set of customers , a set of potential facility locations (including a root ), and a set of Steiner nodes in the graph  = (,), a solution (,) of ConFL represents a set of open facilities , such that each customer is assigned to an open facility and the open facilities are connected to the root via a Steiner Tree . The total cost of the solution (,) is the sum of the cost for opening the facilities, the cost of assigning customers to the open facilities and the cost of the Steiner tree that interconnects the facilities.

We show how to combine a variable neighborhood search method with a reactive tabu-search, in order to find sub-optimal solutions for large scale instances. We also propose a branch-and-cut approach for solving the ConFL to provable optimality. In our computational study, we test the quality of the proposed hybrid strategy by comparing its values to lower and upper bounds obtained within a branch-and-cut framework.

For the NP-hard (OCST) problem a cost minimizing spanning tree has to be found, where the cost depends on the communication volume between each pair of nodes routed over the tree. We present a memetic algorithm (MA) for this problem and focus our discussion on the evaluation of recombination operators for the OCST. The proposed algorithm outperforms evolutionary algorithms (EA) for known benchmark instances and outperforms state-of-the-art solvers for non-Euclidean instances.

We discuss a general approach to hybridize traditional construction heuristics for combinatorial optimization problems with numerical based evolutionary algorithms. Therefore, we show how to augment a construction heuristic with real-valued parameters, called . An evolutionary algorithm for numerical optimization uses this enhanced heuristic to find assignments for these control values, which in turn enable the latter to find high quality solutions for the original combinatorial problem. Additionally to the actual optimization task, we thereby experimentally analyze the heuristic’s substeps.

Furthermore, after finding a good assignment for a specific instance set, we can use it for similar yet different problem instances, without the need of an additional time-consuming run of the evolutionary algorithm. This concept is of particular interest in the context of computing efficient bounds within Branch-and-Cut algorithms. We apply our approach to a real-world problem in network optimization, and present a study on its effectiveness.