Hybrid Optimization: The Ten Years of CPAIOR: 45 (Springer Optimization and Its Applications)

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Results on optimality and duality for a nonlinear scalar programming problem are presented, second and higher order duality results are given for a nonlinear scalar programming problem, and saddle point results are also presented. Invexity in multiobjective programming problems and Kuhn-Tucker optimality conditions are given for a multiobjecive programming problem, Wolfe and Mond-Weir type dual models are given for a multiobjective programming problem and usual duality results are presented in presence of invex functions.

Continuous-time multiobjective problems are also discussed. Quadratic and fractional programming problems are given for invex functions. Symmetric duality results are also given for scalar and vector cases. Hybrid Optimization focuses on the application of artificial intelligence and operations research techniques to constraint programming for solving combinatorial optimization problems. This book covers the most relevant topics investigated in the last ten years by leading experts in the field, and speculates about future directions for research.

BALCOR is an established biennial conference attended by a large number of faculty, researchers and students from the Balkan countries but also from other European and Mediterranean countries as well. Over the past decade, the BALCOR conference has facilitated the exchange of scientific and technical information on the subject of Operations Research and related fields such as Mathematical Programming, Game Theory, Multiple Criteria Decision Analysis, Information Systems, Data Mining and more, in order to promote international scientific cooperation. The carefully selected and refereed papers present important recent developments and modern applications and will serve as excellent reference for students, researchers and practitioners in these disciplines.

Scatter Search: Methodology and Implementations in C. Scatter Search SS -together with its generalized form called Path Relinking-constitutes the only evolutionary approach that embraces a collection of principles from Tabu Search TS , an approach popularly regarded to be divorced from evolutionary procedures. The TS perspective, which is responsible for introducing adaptive memory strategies into the metaheuristic literature at purposeful level beyond simple inheritance mechanisms , may at first seem to be at odds with population-based approaches.

Yet this perspective equips SS with a remarkably effective foundation for solving a wide range of practical problems. The successes documented by Scatter Search come not so much from the adoption of adaptive memory in the range of ways proposed in Tabu Search except where, as often happens, SS is advantageously coupled with TS , but from the use of strategic ideas initially proposed for exploiting adaptive memory, which blend harmoniously with the structure of Scatter Search.

From a historical perspective, the dedicated use of heuristic strategies both to guide the process of combining solutions and to enhance the quality of offspring has been heralded as a key innovation in evolutionary methods, giving rise to what are sometimes called "hybrid" or "memetic" evolutionary procedures. The underlying processes have been introduced into the mainstream of evolutionary methods such as genetic algorithms, for example by a series of gradual steps beginning in the late s.

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The theory and methods of multi-attribute decision making have been extensively applied in engineering, economics, management and military contexts, such as venture capital project evaluation, facility location, bidding, development ranking of industrial sectors and so on. An alternative way for defining large neighborhoods is to reduce Similarly to CP, see Section 3.

Among useful for searching large neighborhoods within a metaheuristic the more generally applicable ones are local branching constraints framework. If the MIP-solver finds resembles the Hamming distance, and thus, the neighborhood an improved solution, it becomes the new incumbent, a new large induced by the local branching constraint corresponds to the clas- neighborhood is defined around it, and the process is iterated.

Obvi- sical k-opt neighborhood. Parameter k controls the size of the ously, the selection of the variables that remain fixed and that neighborhood and its choice is critical. Frameworks that dynam- are subject to optimization, respectively, plays a crucial role: The ically adapt k are therefore common, e. Fischetti and Lodi [] also showed hood.

Too restricted neighborhoods — that is, subproblems — are how the local branching constraints can be generalized to non- unlikely to yield improved solutions, while too large neighborhoods binary integer variables. However, the major advantage of local might result in excessive running times for solving the subproblem branching constraints — namely that no variables must be explicitly by the MIP-solver. Therefore, a strategy for dynamically adapt- selected for fixing — also comes with a downside: Local branch- ing the number of free variables is sometimes used.

Furthermore, ing constraints are dense, i. For variant. After performing tabu search, a large neighborhood is defined missioned cars along a production line, that is, a permutation. More formally, solutions, is built. Due to the relatively high running time of the MIP-solver, the large Prandtstetter and Raidl describe a generalized variable neighbor- neighborhood search is not iterated here, but only applied once as a hood search that makes use of eight different types of neighborhood final refinement phase.

Experimental results document the positive structures. Besides the more straight-forward simple move and impact of this approach. These cars are then released from their described by Oncan et al. Large neighborhood search by means of MIP solvers is nowadays a relatively frequent approach which is promising in many cases. Here, the strategy for selecting the next tree node to be processed is modified in such 4. Literature overview a way that the search is focused on the neighborhoods of promis- ing incumbent solutions in order to quickly identify high-quality The hybridization of metaheuristics with tree search techniques solutions.

Danna et al. Instead be processed next is chosen to be the one in which the branching of trying to mention all the articles that have appeared in this field, variable is allowed to take the value it has in an incumbent solu- we focus on a representative selection of works, different to the tion. Guided dives are repeatedly applied at regular intervals during ones mentioned already above, that is, different to Beam-ACO and the whole optimization process.

Again, this concept is included in MIP-based large neighborhood search. The work by Nagar et al. Beam search purges its queue of may be seen as a multi-stage approach. On the other side, the memetic algorithm is guided by tree. The second stage consists in the execution of the evolution- injecting information about promising regions of the search space ary algorithm. Hereby, each generated partial solution is mapped identified by beam search into the population.

In [] the authors operators are applied for changing the partial solution. One search methods for finding improved solutions do not necessar- example is the application of a tabu search algorithm to the job ily have to be defined based on a single incumbent solution only. Applegate et al. However, the search tree of nested partitioning cor- set of promising solutions is derived by a series of runs of the responds to an explicit search space partitioning, rather than an chained Lin-Kernighan iterated local search.

The sets of edges of implicit one obtained by variable-value assignments. The obtained all these solutions are merged and the TSP finally solved to opti- sub-spaces are usually evaluated by a metaheuristic. In [], mality on this resulting reduced graph. Solution merging further ACO is applied for this purpose, whereas in [] local search is is sometimes used as a replacement for naive crossover oper- used. Hybridizing metaheuristics with problem relaxation lems. Eremeev [] studies the computational complexity of producing a best possible offspring from two parents for binary Enhancing metaheuristics with information gained from prob- representations from a theoretical point of view.

He concludes lem relaxation has turned into a quite popular hybridization that the optimal recombination problem is polynomially solv- approach in recent years. When weight partitioning problem, and linear Boolean programming removing constraints, they may either be dropped, or they may, for problems with at most two variables per inequality. In case the relaxed problem can be efficiently solved, the hope ger programming with three or more variables per inequality, is that the structure of an optimal solution to the relaxed problem the knapsack problem, set covering, the p-median problem, and together with its objective function value may facilitate somehow others.

However, the literature bound. This is because the optimal solution value of a relaxed prob- also offers examples where metaheuristics are used for guiding lem can be regarded as a bound for the optimal solution value of the the search process of tree search. For mixed integer program- original problem, and hence, it can be used for pruning the search ming, Rothberg [] suggests a tight integration of an evolutionary tree. The resulting relaxation, which is a intervals, and MIP-based optimal merging is done by first fixing all linear program LP , can then be solved to optimality by efficient variables that are common in a set of selected elite solutions.

Muta- methods such as the well-known simplex algorithm. In the first one a search algorithm remaining problem. Experimental results indicate that this hybrid is guided by Lagrangian relaxation. Example 1: hybrid metaheuristics based on Lagrangian nicely working example of a Lagrangian metaheuristic applied to relaxation the generalized assignment problem can be found in []. Algorthm 5 Lagrangian Metaheuristic In [,], Boschetti et al. However, most of these approaches have not been 6: until termination conditions are satisfied 7: output: the best feasible solution obtained for problem P developed from a metaheuristic perspective.

Therefore, the authors of [,] see a large potential for enhancing this type of algo- When to use this technique? The potential advantages of hybrid rithm with algorithmic components from the metaheuristics field. First, due Lagranging relaxation. For this purpose we start by shortly dis- to the fact that both lower and upper bounds are improved dur- cussing the main ideas of Lagrangian relaxation. Consider the ing the search process, quality conditions may be derived for the following general mixed integer program P: obtained solutions.

The availability of a constantly improv- subject to: ing lower bound also allows the potential pruning of the search space. Wilbaut and Hanafi [] are matrices. Moreover, zP is the optimal solution value of problem present several iterative relaxation based heuristics to solve 0—1 P. A Lagrangian relaxation is obtained by moving some of the con- mixed integer programming problems. They combine LP as well as straints — for example, constraints 2 — in the following way to the MIP relaxations into a powerful set of heuristics. Finally a cut is added to the problem excluding the already visited search space.

These weights are also called the Lagrangian multipliers. In MIPA, first an obtain the best relaxation possible, it is necessary to find the weight LP relaxation of the problem is solved yielding a solution xLP. In the next iteration integrality is enforced on those but simple way. This procedure can also be used in the follow- binary variables with non-integral values in xMIPR of the previous ing way for solving the original problem P.

At each iteration, the iteration. This process is repeated as in LPA. This may be done by means of simple heuristics, from one iteration to the next. In practice, however, the number of or alternatively by means of metaheuristic concepts. This way of binary variables in the MIP relaxation will be limited as will be the tackling a problem, which is sketched in Algorithm 5, was labelled number of iterations.

Lagrangian Metaheuristic in [,]. The authors of these works also provide example implementations for combinatorial problems such as the single source capacitated facility location problem.

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In IRH, both the LP In contrast to the above mentioned approaches, Chu and Beasley relaxation as well as the MIP relaxation obtained by enforcing inte- [] present an evolutionary algorithm for the MKP that exploits grality on those binary variables that have non-integral values in the dual variable values, coming as a by-product of solving LP the LP relaxation are solved at each iteration.

The lower bounds relaxations.

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Angel, E. Dras, Kernelization as heuristic structure for the vertex cover [75] J. The availability of a constantly improv- subject to: ing lower bound also allows the potential pruning of the search space. Smith, A tutorial for competent memetic algorithms: model, vehicle routing problems, in: M. This book includes contributions by experts from different but related areas of research including constraint programming, decision theory, operations research, SAT, artificial intelligence, as well as others.

On the basis of the dual variable values they calculate obtained by both relaxations are compared and the better one is pseudo-utility ratios for the variables. Interestingly, these pseudo- retained. Furthermore, as in the previous algorithms, the solutions utility ratios tend to give good indications of the likeliness of the obtained by both relaxations are used to define two reduced prob- corresponding items to be included in an optimal solution.

Cuts obtained by the solutions of both relaxations are then Vasquez and Hao [,], which was also applied to the MKP. This is done in a first are working with separate problems to be solved, therefore keep- phase. Afterwards, in a second phase, tabu search is used to search ing the obtained cuts separated. Only the current best solution and around the optimal solutions to these relaxed problems.

Hereby, the lower bound are shared between this interleaved version of LPA tabu search is enforced to search within a certain distance to the and MIPA. In a theoretical article by the volume algorithm. The original graph is reduced by cutting [] Glover proposes different ways of using cuts obtained from edges, meaningful initial solutions are generated, and the objective relaxations in metaheuristic algorithms. Among others, the cuts function is modified by considering reduced costs.

Glover shows how such inequalities can be genetic algorithms is described in Pirkwieser et al. Moreover, a combination of a Lagrangian relaxation approach search. The main advantage to be expected Leitner and Raidl [] for a real-world fiber optic network design from combining relaxations with metaheuristics is the global prob- problem. A different use of Lagrangian relaxation is proposed in Tamura This allows to lead metaheuristics and local search towards promis- et al. Given ing regions of the search space and possibly obtain higher quality the MIP formulation of the problem, the domain of the variables is solutions requiring less run-time.

Just like in the case of the first split into sub-domains, which are then indexed. Moreover, the orig- example of this section, this hybrid technique should only be con- inal domains are replaced by the indices of the sub-domains. Then, sidered if solving the corresponding problem relaxation is not a GA is applied to this reduced problem version and the fitness of computationally expensive. Literature overview the corresponding solution. When the GA terminates, an exhaus- Metaheuristics that are guided by problem relaxation can be tive search of the region identified as the most promising one is found quite frequently in the literature.

In the following we present carried out. TSP where an optimal solution to the minimum spanning tree MST A straightforward way to make use of an optimal solution to the relaxation is used for biasing the search of the artificial ants towards LP relaxation of a problem at hand is to directly derive a heuristic edges that form part of the minimum spanning tree.

The proposed integer solution which is feasible for the original problem. For example, Raidl and edges in common with an optimal MST solution. Feltl [] present a hybrid genetic algorithm GA for the gener- alized assignment problem. In their GA, the initial population is 6.

Hybridizing metaheuristics with dynamic programming obtained by a randomized rounding procedure for generating fea- sible integer solutions from the LP relaxation. As these solutions are Dynamic programming DP [] is an algorithmic scheme for often infeasible, randomized repair and improvement operators are optimization that solves a combinatorial problem as follows.

First, applied as well. Then a solution to Optimal solutions to LP relaxations may also be exploited for the given problem is obtained by combining the solutions to already guiding local search or for repairing infeasible candidate solutions. A crucial point of DP is that the solutions case for hybrid algorithms, is considered.

The items are sorted to already solved subproblems are stored. This has the advantage according to increasing LP-values of their corresponding variables. Basically, an optimization problem must exhibit two from the knapsack until all constraints are fulfilled. In this case, a problem is said to show optimal by choosing the minimum of the tardiness values computed as a substructure. The space of subproblems should be relatively small. Typically, computed recursively by a DP algorithm that runs in O n3 and the total number of distinct subproblems is polynomial in the requires O n space.

A best-improvement local search based on the dynasearch neighborhood has, on average, a better performance than a 6. Example 1: iterated dynasearch best-improvement local search using the 2-exchange or the 3-exchange neighborhoods. In other words, the average total Iterated dynasearch is a hybrid metaheuristic that uses DP as tardiness of the local optimum returned in the case of the a neighborhood exploration strategy inside iterated local search dynasearch neighborhood is lower.

Furthermore, this local search [4]. The rationale behind this integration is the same as for LNS, as can be taken as the inner local search component for an described in Sections 3. In some cases, DP can make it iterated local search ILS algorithm [31], as illustrated in Algo- possible to completely explore a neighborhood of exponential size rithm 6.

The algorithm iteratively perturbs the current solution in polynomial time and space. In this paragraph, we will illustrate s to provide an initial solution for a best improvement local the principles of iterated dynasearch with respect to its application search. Further contributions to this work can be found criterion. More formally, for each of the n jobs, a processing neighborhood is DP, it is obvious that a precondition for the appli- time pj , a positive weight wj and a due date dj are given.

Jobs cability of this hybrid method is the availability of an efficient DP are available at time zero and must be processed one at a time approach for solving the sub-problem corresponding to neighbor- without interruption. Once a job ordering is provided, for each hood exploration. Example 2: corridor method based on dynamic programming A natural neighborhood structure can be defined in terms of job permutations.

Any permutation of n objects can be obtained by the The so-called corridor method [] is a hybrid metaheuris- repeated application of swaps. Each swap consists in exchanging tic inspired by DP. It has its origins in attempts to deal with the two objects. The resulting neighborhood is called the 2-exchange curse of dimensionality [] in large-scale DP applications.

Con- neighborhood. In general, the k-exchange neighborhood, defined ceptually, the idea is to optimize the objective function over a by sequences of swaps involving k objects, has an O nk size. The best solution found in this cor- considered. This process is repeated until the new incumbent. At this point the procedure erated by a series of independent swaps. However, the independence that kind were devised in the context of reservoir control and of moves makes it possible to define a recursive enumeration algo- operation problems see, for example, [].

However, the cor- rithm based on DP such that the resulting exploration is polynomial ridor method is not restricted to the use of DP. In the case of in time and space. Two cases must be sen complete method can explore them in pseudo- polynomial considered: time. The pseudo-code of a general corridor method is shown in Algorithm 7. This is done by DP. In [], the authors propose a hybrid In a way, the corridor method is similar to large neighborhood method combining adaptive memory, sparse DP, and reduc- search LNS see Sections 3.

However, while — at least tion techniques to reduce and explore the search space. First, a the early — LNS approaches were developed with the aim of mak- bi-partition of the variables is generated, which leads to the identi- ing local search based metaheuristics more efficient, the corridor fication of small core problems with at most 15 variables.

guipaiviastaner.ga/map7.php These method was designed with the aim of supporting complete tech- small problems are solved using the forward phase of DP. The niques such as dynamic programming in a heuristic way when space defined by the remaining variables is explored using tabu applied to large scale problems. In fact, the neighborhoods used in search. Hereby, partial solutions are completed using the informa- local search based metaheuristic and in early LNS methods see, for tion stored during the forward phase of DP.

The authors indicate example, [] are generally move-based. This refers to the fact that that their approach can be seen as a global intensification mecha- the neighborhood around the incumbent solution is usually gener- nism, since at each iteration, the move evaluations involve solving ated on a topological basis of moves, that is, relatively small changes a reduced problem implicitly.

On the contrary, neighbor- The application of DP to subproblems is also proposed in hoods used in the context of the corridor method are method-based, [], where the authors introduce and tackle a multi-drug can- which means that these neighborhoods are designed in order to ful- cer chemotherapy model to simulate the possible response of the fill the needs and the requirements of the complete optimization tumor cells under drug administration.

The objective is to mini- technique used to explore the neighborhood. A so-called adaptive When to use this technique? The corridor method should be con- elitist GA is combined with a local search technique called iter- sidered as an option in cases in which efficient complete methods ative dynamic programming.

This local search technique works by are known for solving sub-problems of the original problem at subdividing the problem into subproblems, and optimizing the sub- hand. If this is given, the algorithm designer is required to specify problems separately by DP. Although not yet widely in use, the cor- approach presented in [], where the authors tackle the multi- ridor method has been successfully applied, for example, to a blocks ple sequence alignment problem.

One of the main approaches for relocation problem [], to a DNA sequencing problem [], and multiple sequence alignment uses DP to align sequences as follows.

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First, two of the sequences are optimally aligned. Then, the outcome is aligned with a third sequence. This process is repeated until all sequences have been considered. In this article, Juang and Su pro- 6. Literature overview pose the application of particle swarm optimization for improving the alignment result at each step of the afore-mentioned iterative Apart from the examples outlined above, a few other hybrids process.

In this section A recent heuristic version of DP labelled bounded dynamic pro- we discuss a representative sample of them. In [], for example, gramming was proposed in [] for the simple assembly line Blum and Blesa present the use of a DP algorithm in two different balancing problem. Hereby, the number of states is heuristically metaheuristics for the k-cardinality tree KCT problem. The general reduced at each level. In this way, the authors were able to find idea of their approaches is not limited to the KCT problem and can, optimal solutions in a reduced amount of computation time.

Basically, the idea Finally, in [] DP is purely used as a solution decoder in the is to let the metaheuristic generate objects that are bigger than context of the rectangle packing problem with general spatial costs, solutions. Ideally, these objects contain an exponential number of which consists in packing given rectangles without overlap in the solutions to the problem under consideration. DP is then used to plane so that the maximum cost of the rectangles is minimized.

Another example is the article by Hu and Raidl [], where DP is used in the context of an evolutionary algorithm for obtaining 7. Discussion and conclusions the best solution that can be generated from an incomplete solu- tion. The problem considered in this article is the generalized TSP in In this article we have provided a survey on the hybridization of which a clustered graph is given and a shortest tour visiting exactly metaheuristics with other techniques for optimization. We divided one node from each cluster is required. Hu and Raidl study a repre- this growing research area into five different lines of hybridization.

In addition, a literature review has been ited. A DP procedure is then used to derive a corresponding optimal provided for each research line. We hope that this work will serve selection of particular nodes from each cluster. However, we would recommend that, before starting technique and DP for the application to a dynamic facility lay- to develop a hybrid metaheuristic, researchers carefully consider out problem with unequal sizes of departments, which may even whether a hybrid metaheuristic technique is the appropriate solver change from one period to the next.

A number of T evolutionary method for the problem at hand. The following questions should be algorithms is run in parallel, one for each of T periods. What is the optimization goal? If man-power and computation time are critical, hybrid meta- heuristics are, in general, not advisable. Only when very good References solutions are needed which cannot be obtained by any complete method in a feasible time frame, the development of a hybrid [1] C.

Reeves Ed. Is there still room to improve over the results of existing [2] F. Glover, G. Kochenberger Eds. In Academic Publishers, Blum, A. Roli, Metaheuristics in combinatorial optimization: overview already very well for the problem instances that are to be and conceptual comparison, ACM Computing Surveys 35 3 — Or, alternatively, the problem instances under consider- [4] H.

Hoos, T. In these cases it does not make [5] L. Perron, M. Trick Eds. Which type of hybrid metaheuristic might work well for my [6] W. Hooker Eds. The process of designing and implement- [7] A. Lodi, M. Milano, P. Toth Eds.

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Blesa Aguilera, C. Blum, C. Cotta, A. Gallardo, A. Roli, algorithm engineering and statistics. For the development of M. Sampels Eds. Blum, L. Roli, M. Sampels, A. Schaerf Eds. Maniezzo, P. Hansen, S. Voss Eds. For the extraction of useful guidelines for the development of [11] P.

Hansen, V. Maniezzo, M. Fischetti, T. Unfortunately, the used research methodology is often tional Workshop on Model Based Metaheuristics, Vienna, Austria, Maniezzo, T. Blum, M. Blesa Aguilera, A. In our opinion, the research community Studies in Computational Intelligence, Springer-Verlag, Berlin, Germany, should make an effort to move towards a sound scientific method- Cotta, A study of hybridisation techniques and their application to the ology consisting of theoretical models for describing properties design of evolutionary algorithms, AI Communications 11 3—4 of hybrid metaheuristics and using an experimental methodol- — In fact, among the key points [16] I.

Dumitrescu, T. Raidl, et al. Raidl, A unified view on hybrid metaheuristics, in: F. Almeida, M. Blum, J. Moreno Vega, M. Berlin, Germany, , pp. Raidl, J. Puchinger, C. Blum, Metaheuristic hybrids, in: M. Gendreau, J. Besides the already cited papers and book, we mention the well Potvin Eds.

Further- Berlin, Germany, , pp. Puchinger, G. Raidl, A. Silc one of its issues, such as parameter tuning or the statistical assess- Conference on Bio-Inspired Optimization Methods and their Applications, ment of results, can be found in [,—]. Jozef Stefan Institute, Ljubljana, Slovenia, , pp.

We are convinced that research on hybrid metaheuristics is still [20] C. Roli, Hybrid metaheuristics, in: M. In the years to come, most publications on meta- Springer-Verlag, Berlin, Germany, , pp. We hope that [21] M. Ehrgott, X. Gandibleux, Hybrid Metaheuristics for Multi-objective Com- this work contributes to give some more structure and guidance to binatorial Optimization, Vol. Michalewicz, P. Siarry, Special issue on adaptation of discrete metaheuris- tics to continuous optimization, European Journal of Operational Research — Acknowledgements [23] K.

Price, R. Storn, J. Molina, M. Lozano, C. Chen, Y. Lu, G. Cotta, E. Talbi, E. Parallel — Hybrid Metaheuristics.

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Boettcher, A. Hoboken, New Jersey, Shi, Y. Liang, H. Lee, C. Individual chapters A tantalizing problem that cuts The rapid progress of computer technologies, including new parallel In particular, many design and operational problems give rise to nonlinear and mixed-integer One of the motivating factors was the Constrained linear optimization models were soon adopted in numerous In many of these problems it is necessary to compute the global optimum or a good approximation of a multivariable function. The variables that define the