L A METACOGNICIÓN DESDE UN ENFOQUE FLUIDO

For combinatorial optimization problems (COPs) that are NP-hard, no polynomial time algorithm exists, as- suming that

P

NP

. Therefore, complete methods might need exponential computational time in the worst- case.320 _{By using approximate methods, such as meta-heuristics, we sacrifice the guarantee of finding optimal} solutions for the sake of getting good solutions in a significantly reduced amount of time. Thus, the use of meta-heuristics has received more and more attention in the last 30 years. In the first two decades the applica- tions were confined to rather standard meta-heuristics. However, recent researches have shown that a skilled combination of a meta-heuristic with other optimization techniques, a so-called hybrid meta-heuristic, can provide a more efficient behavior and a higher flexibility when dealing with real-world and large-scale prob- lem321_.

 Classification of hybrid meta-heuristics

Nowadays we can observe a common agreement on the advantage of combining components from different search techniques. The tendency of designing hybrid techniques is widespread in the fields of OR and artificial intelligence (AI), mostly based on the no free lunch theorems322_.

319 _{For example, Wagner’s exact solution procedure for HLPs with 500 nodes, and Resende & Werneck solved the uncapacitated facility location prob-}

lem with hybrid multi-start heuristic under instance with 1000 nodes. See Wagner (2007), pp.391-401;.Resende/ Werneck (2006), pp.54-68.

320 _{See Blum/Roli (2008), p.1.} 321 _{See Blum/Roli (2008), p.1.}

We may distinguish hybrid heuristics in two dimensions, i.e. hybrid contents and hybrid level. A matrix is adopted to illustrate their relationships (see Fig. 4-1).

Figure 4-1: Classification of hybrid meta-heuristics

Dimension of hybrid contents

In terms of hybrid contents, we may distinguish between three categories: the first one combines meta- heuristics strategies; the second one combines meta-heuristics with certain algorithms specific for the problem; the third one combines meta-heuristics with other more general techniques coming from fields like OR and AI.323

A prominent example of the first category is the use of trajectory methods324 _{into population-based tech-} niques325_{. The reason becomes apparent by analyzing the respective strengths of trajectory methods and popu-} lation-based methods. The power of population-based methods lies in the capability of recombining solutions to obtain new ones. This enables the search process to perform a guided sampling of the search space and identify promising areas. Meanwhile, the strength of trajectory methods lies in the way they explore a promis- ing region in the search space. A promising area in the search space is searched by trajectory methods in a more structured way than by population-based methods so that the search is driven towards local optima or confined areas of the space in which many local optima are condensed326_{. In sum, population-based methods} are better in identifying promising areas in the search space, from which trajectory methods can quickly reach good local optima. Therefore, meta-heuristic hybrids can effectively combine the strengths of both population-

323 _{See Raidl (2006), p.4.}

324 _{Generally speaking, algorithms that work on a single solution are referred to as trajectory methods. They comprise all meta-heuristics that are based}

on local search, such as TS, iterated local search and variable neighborhood search. See Blum/Roli (2008), p.6.

325 _{Population-based meta-heuristics deal at each algorithm iteration with a set of solutions rather than with a single solution. From this set of solutions}

the population of the next iteration is produced by the application of certain operators.

326 _{See Chiarandini et al (2006), p. 118.}

Hybrid content Hybrid level

Collaborative combination of meta with problem-

specific algorithm

Integrative combination of meta with meta

Integrative combination of meta with problem-

specific algorithm Collaborative

combination of meta with meta

Collaborative combination of meta with general tech-

niques from OR/AI

Integrative combination of meta with general techniques from OR/AI

based methods and trajectory methods. Successful examples for the third category are hybrids of meta- heuristics with OR methods, such as linear programming327_{, branch & bound, tree-based search techniques}328_, dynamic programming and neutral networks. A recent literature review on this topic was made by Jourdan et al329_.

Dimension of hybrid level

In terms of hybrid level, we may distinguish between two categories: collaborative combinations and integra- tive combinations330_.

Collaborative combinations are based on the exchange of information about states, models, entire sub- problems, solutions or search space characteristics between several optimization techniques run sequentially (or in parallel). This kind of combination is more related to cooperative and parallel search and it in principle retain the individual identities of the original algorithms331_{. On the contrary, original algorithms in integrative} combinations strongly depend on each other. One technique is a subordinate or embedded component of the other technique. Thus, there is a distinguished master algorithm, and at least one integrated algorithms332_.  Hybrid principle

The motivation of hybridization of different algorithmic concepts is to obtain systems with better performance by exploiting and uniting advantages of the individual algorithm333_{. Hybridization is also a way to inject prob-} lem-specific knowledge according to No-Free-Lunch Theorem by Wolpert and Macready334_{. Of great im-} portance of hybridization is the dynamic balance between intensification and diversification, i.e. local exploita- tion and global exploration. They are two contrary but also complementary forces that largely determine the effectiveness of the algorithms335_{.In other words, local and intensive exploitation focuses on examining neigh-} bors of elite solutions, while global and extensive exploration is to encourage the search process to examine unvisited regions and to generate different solutions. Therefore, the hybrid meta-heuristics can, on the one side, quickly identify regions in search space with high quality solutions and, on the other side, not waste too much time in regions of search space which have already been explored or which do not provide high quality solutions.

327 _{Linear programming is often used either to solve a sub-problem or to provide dual information to a meta-heuristic in order to select the most prom-}

ising candidate solution or solution component. See e.g. Blum (2005), pp.1565-1591; Ibaraki/ Nakamura (2006), pp.13-27; Maniezzo (1999), pp.358-369.

328 _{See e.g. Focacci et al (2003), pp.369-403.} 329 _{See Jourdan et al (2009), pp.620-629.}

330 _{See e.g. Raidl (2006), p.3; Puchinger/ Raidl (2005), pp.41-53; Jourdan et al (2009), pp.620-629. Collaborative combination is also called parallel,}

cooperative or high level combination. Integrative combination is also called low level combination.

331 _{For interested readers, please refer to Alba (2005); Grainic/ Toulouse (2002), pp.247-249; Sondergeld/ Voß (1999), pp.297-312.} 332 _{See Talbi (2002), p.543; Puchinger/ Raidl (2005), p.42.}

333 _{See Puchinger/ Raidl (2005), p.42.} 334 _{See Wolpert / Macready (1997), p.68.} 335 _{See Yagiura/ Ibaraki (2001), pp.33-55.}

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