simulation of a cooling process in metallurgy. Tabu search [74, 75] generally accepts the best solution in the neighborhood of a current solution but restricts this neigh-borhood by forbidding certain moves leading to just visited solutions. Iterated local search [129] randomly perturbs reached local optimal solutions in order to continue the search and escape the current bounds of attraction.
Nature inspired Metaheuristics
Another important strategy which is frequently used to develop metaheuristics is to mimic successful behavior from nature. As opposed to the methods discussed above, the resulting approaches often operate on a whole set of solutions, called population, rather than on single solutions only. An important class of nature inspired methods are evolutionary algorithms [10] which imitate the theory of evolution as described by Darwin [47] and Mendel [139]. Here, new candidate solutions are created by selecting parental solutions and inheriting their properties on a random basis. In analogy to mutation in nature, these offsprings are further randomly modified to a small degree. Concrete variants of this concept include evolution strategies [159], evolutionary programming [66], and genetic algorithms [92].
A further important variant of evolutionary algorithms which additionally incorpo-rates local search to further improve promising candidate solutions is called memetic algorithms [141].
Another important stream of nature inspired metaheuristics are methods that try to simulate cooperative behavior of individuals instead of their competition as described above. Prominent methods in this area include the diverse variants of ant colony optimization [38, 51] and the simulation of swarm intelligence such as particle swarm optimization [105].
More complete and thorough overviews on above mentioned and further metaheuris-tics can be found in the books of Dreo et al. [52] or Glover and Kochenberger [76].
2.3 Hybrid Methods
Each of the methods presented in the previous sections have their individual advan-tages and drawbacks. While (meta-) heuristics are normally able to generate feasible solutions relatively fast, usually no information on the gap to an optimal solution is available. On the contrary, exact methods are in principle able to compute proven
Chapter 2 Methodologies
optimal solutions. However, due to their dramatical runtime and resource usage increase if the problem sizes get larger, they might terminate yielding a huge opti-mality gap or even no feasible solution at all after their maximum allowed runtime.
Thus, many successful hybrid methods trying to avoid the drawbacks of individual methods have been proposed recently, see e.g. [25, 133, 158].
Hybrid methods can be classified into collaborative combinations where algorithms exchange information but are not part of each other and integrative combinations, where one method is used as a subordinate of another. In the collaborative approach one can further distinguish between sequential, intertwined, and parallel approaches depending on the execution order of the different methods [151, 156, 157].
Another important distinction can be made between approaches combining different metaheuristic methods on the one hand and approaches combining exact methods and metaheuristics on the other hand. In particular, combinations of metaheuristics and mathematical programming techniques turned out to often be very fruitful [133];
such hybrids are also frequently called matheuristics.
Combinations of different metaheuristics have been successfully used in many ap-plications. Prominent examples are memetic algorithms [141] or the integration of local search based methods in population based metaheuristics such as EAs and ACO approaches. See e.g. [25] for an overview on and several successful applications of hybrid metaheuristics.
Using exact methods as subordinates of metaheuristic methods often allows for find-ing extremely good solutions. In very large scale neighborhood search (VLSN) [5], subproblems of appropriate size are frequently solved by exact methods. Successful applications of this concept include dynasearch [39, 40] where dynamic program-ming is used to explore large neighborhoods or the embedding of IP techniques within VNS approaches [96, 148].
Other successful approaches are based on using metaheuristics within IP based meth-ods. Those include the metaheuristic generation of cutting planes [80] as well as the use of metaheuristics for solving the pricing subproblem in column generation meth-ods, see e.g. [145, 150, 152].
Finally, another frequently used combination is the use of metaheuristics to obtain tighter upper bounds within Lagrangian relaxation based methods, see e.g. [116, 146].
An extensive survey on combinations of integer programming techniques with meta-heuristics can also be found in [154].
Chapter 3
The b max -Survivable Network Design Problem
3.1 Introduction
Thebmax-Survivable Network Design Problem (bmax-SNDP) is a real-world commu-nication network design problem which arises for instance in the expansion of fiber optic networks. Recently, fiber-to-the-home has become economically feasible for individual households in urban areas. However, covering larger districts with such networks requires enormous financial resources from an operators point of view.
Since customers are usually not willing to pay significantly more than for exist-ing lower bandwidth connections, good algorithms for findexist-ing cost-efficient network layouts are crucial.
bmax-SNDP considers the problem of augmenting an existing network infrastructure by additional links and switches in order to connect additional customer nodes.
Here, we distinguish between standard (type-1) customer nodes for which a single link connection suffices and type-2 customer nodes representing business customers which require a more reliable connection, ensuring connectivity even when a single link or routing node fails. Since offering full redundancy to each type-2 customer often is too expensive and does not pay off from an economic point of view, we consider a problem variant where the redundancy condition for type-2 customers is relaxed in the sense that a connection is allowed via a final non-redundant branch line that does not exceed a certain lengthbmax. Thus, we restrict the length of the
Chapter 3 The bmax-Survivable Network Design Problem
non-redundant part of a connection taking a compromise between reliability and construction costs.
In this chapter, we formally introducebmax-SNDP in Section 3.2 and review previous and related work in Section 3.3. Afterwards, we present two new mixed integer programming approaches for solving bmax-SNDP to proven optimality based on an exponential number of so-called connection variables which can be solved by branch-and-price. As one main contribution within this section, we show how to significantly speed up the solution of the linear relaxation of these models by using alternative dual-optimal solutions in the pricing subproblem. Theoretical comparisons of the corresponding polyhedra of those two as well to previously existing formulations are given in Section 3.7.
Section 3.8 details a Lagrangian decomposition (LD) approach, while Section 3.9 introduces neighborhood structures for bmax-SNDP. In Section 3.10, we present a heuristic for generating feasible solutions to bmax-SNDP as well as two metaheuris-tic approaches based on variable neighborhood search (VNS) and greedy randomized adaptive search (GRASP), respectively. The details of hybrid approaches combin-ing Lagrangian decomposition with variable neighborhood descent (VND) are given in Section 3.11. Test instances for benchmarking our various approaches and com-putational results are discussed in Sections 3.12 and 3.13, before we finally draw conclusions and outline potential future work in Section 3.14.
The approaches presented in this chapter have been previously published in [116, 122, 120, 123].