Evolutionary Multi-objective Optimization

Multi-objective optimization problems are often difficult to solve as the task is not to compute a single optimal solution but a set of solutions representing the different trade-offs with respect to the given objective functions. The number of these trade-offs can be exponential with regard to the problem size, which implies that not all trade-offs can be computed efficiently. In this case, one is interested in good approximations of the Pareto front consisting of a not too large set of Pareto-optimal solutions. It has been observed empirically that multi-objective evolutionary algorithms (MOEAs) are able to obtain good approximations for a wide range of multi-objective optimization problems. Our aim of is to contribute to the theoretical understanding of MOEAs in particular with respect to their approximation behavior.

Additional Objectives

Investigators: Tobias Friedrich, Nils Hebbinghaus, Christian Klein, and Frank Neumann in cooperation with Dimo Brockhoff (ETH Zurich) and Eckart Zitzler (ETH Zurich)

Most studies on evolutionary multi-objective optimization investigate problems where the number of considered objectives is low, i.e., between two and four, while studies with many objectives are rare, cf. [2]. The reason is that a large number of objectives leads to further difficulties with respect to decision making, visualization, and computation. Nevertheless,

from a practical point of view it is desirable with most applications to include as many objectives as possible without the need to specify preferences among the different criteria.

An open question in this context is how the inclusion of additional objectives affects the search efficiency of an evolutionary algorithm to generate the set of Pareto optimal solutions.

In [1], we examined how adding objectives to a given optimization problem affects the computational effort required to generate the set of Pareto optimal solutions. Experimental studies show that additional objectives may change the running time behavior of an algo-rithm drastically. Often it is assumed that more objectives make a problem harder as the number of different trade-offs may increase with the problem dimension. We show that addi-tional objectives, however, may be both beneficial and obstructive depending on the chosen objective. Our results are obtained by rigorous running time analyses that show the different effects of adding objectives to a well-known plateau-function. Additional experiments show that the theoretically shown behavior can be observed for problems with more than one objective.

In [3], we pointed out a different obstacle when using multi-objective models for single-objective optimization problems. To the best of our knowledge, there is so far no rigorous analysis of a problem on which the multi-objective approach is slower by more than a factor bounded by the population size compared to the respective single-objective one. We showed that a multi-objective model may lead to a totally inefficient optimization process (in com-parison to a single-objective one) even if the population size is always small. This effect is first illustrated for simple plateau functions and later pointed out for a multi-objective model of the SetCover problem.

References

• [1] D. Brockhoff, T. Friedrich, N. Hebbinghaus, C. Klein, F. Neumann, and E. Zitzler. On the effects of adding objectives to plateau functions. IEEE Transactions on Evolutionary Computation, 2009.

To appear.

[2] C. A. Coello Coello, G. B. Lamont, and D. A. Van Veldhuizen. Evolutionary Algorithms for Solving Multi-Objective Problems (Genetic and Evolutionary Computation). Kluwer Academic Publishers, New York, USA, 2002.

• [3] T. Friedrich, N. Hebbinghaus, and F. Neumann. Plateaus can be harder in multi-objective op-timization. In IEEE Congress on Evolutionary Computation 2007, Singapore, Singapore, 2007, pp. 2622–2629. IEEE.

Fairness in Evolutionary Multi-objective Optimization

Investigators: Tobias Friedrich and Frank Neumann in cooperation with Christian Horoba (TU Dortmund)

Many multi-objective evolutionary algorithms give priority to regions in the decision or objective space that have been rarely explored. This leads to the use of fairness in evolu-tionary multi-objective optimization. The idea behind using fairness is that the number of descendants generated by individuals with certain properties should be balanced. In [1], we investigated the model of fairness introduced in [2]. The algorithms that are subject to our analyses count the number of descendants that have been generated by the individuals in

the population. The first idea is to count the number of descendants with respect to the decision space, i. e., a separate counter is dedicated to each decision vector. The descendants are generated by individuals that have not produced many descendants in order to discover new regions of the decision space. This prevents individuals that have achieved less progress towards other non-dominated decision vectors from producing additional descendants. The other idea we examined is the usage of a counter with respect to the objective space. This implies that many decision vectors potentially depend on the same counter. We compared the runtime behavior of these two variants and pointed out the differences of these two fairness mechanisms by rigorous analyses.

References

• [1] T. Friedrich, C. Horoba, and F. Neumann. Runtime analyses for using fairness in evolutionary multi-objective optimization. In G. Rudolph, T. Jansen, S. M. Lucas, C. Poloni, and N. Beume, eds., Parallel Problem Solving from Nature (PPSN X), Dortmund, Germany, 2008, LNCS 5199, pp. 671–680. Springer.

[2] M. Laumanns, L. Thiele, and E. Zitzler. Running time analysis of multiobjective evolution-ary algorithms on pseudo-boolean functions. IEEE Transactions on Evolutionevolution-ary Computation, 8(2):170–182, 2004.

Diversity Mechanisms for Evolutionary Multi-objective Optimization

Investigator: Frank Neumann in cooperation with Christian Horoba (TU Dortmund) When using evolutionary algorithms for multi-objective optimization to approximate a large Pareto front, specific diversity mechanisms are applied to spread the individuals of the pop-ulation over the whole Pareto front. In [1], we studied the concept of -dominance [3] and investigated its impact with respect to the runtime behavior. In the mentioned approach diversity is ensured by partitioning the objective space into boxes of appropriate size. The applied evolutionary algorithm is allowed to keep at most one individual of each box in its population. A usual scenario is to divide the objective space into boxes such that their number is logarithmic with respect to the number of objective vectors. We compared a vari-ation of a simple evolutionary algorithm for multi-objective optimizvari-ation using the concept of -dominance with the original algorithm. To point out situations where this concept leads provably to a better optimization process, we presented a class of instances with an exponen-tial number of non-dominated feasible objective vectors. We showed that using the concept of -dominance a good approximation of the Pareto front is constructed efficiently while the approach not using this concept can not achieve this goal in expected polynomial time. Later on, we presented instances where the concept of -dominance prevents the algorithm from constructing good approximations of the Pareto front. For the efficient optimization of these instances it is essential that the population contains more than one individual per box to construct other individuals that are needed for a good approximation of the Pareto front.

In contrast to this, we proved that the approach without using the diversity mechanism constructs the whole Pareto front in expected polynomial time.

Another popular approach to diversify the population of an evolutionary multi-objective algorithm is to use the density estimator. This diversity mechanism is used in a well-known

evolutionary algorithm for multi-objective optimization called SPEA2 [4]. For each individ-ual in the population the distances to all other individindivid-uals are computed. Based on these distances individuals are preferred for the next generation that do not belong to crowded re-gions of the objective space. In [2], we considered evolutionary algorithms for multi-objective optimization using this mechanism and examined when it is provably helpful to achieve a good approximation of the Pareto optimal set. Thereby, we also related it to the -dominance approach and pointed out in which situations one mechanism favors over the other.

References

• [1] C. Horoba and F. Neumann. Benefits and drawbacks for the use of epsilon-dominance in evolu-tionary multi-objective optimization. In C. Ryan and M. Keijzer, eds., Genetic and Evoluevolu-tionary Computation Conference 2008, Atlanta, USA, 2008, pp. 641–680. ACM Press.

• [2] C. Horoba and F. Neumann. Additive approximations of pareto-optimal sets by evolutionary multi-objective algorithms. In Foundations of Genetic Algorithms 2009, Orlando, USA, 2009.

ACM. To appear.

[3] M. Laumanns, L. Thiele, K. Deb, and E. Zitzler. Combining convergence and diversity in evolu-tionary multiobjective optimization. Evoluevolu-tionary Computation, 10(3):263–282, 2003.

[4] E. Zitzler, M. Laumanns, and L. Thiele. SPEA2: Improving the strength Pareto evolutionary algorithm for multiobjective optimization. In Proceedings of the EUROGEN, 2002, pp. 95–100.

CIMNE.

Computation of the Hypervolume

Investigators: Karl Bringmann and Tobias Friedrich

How to compare Pareto sets lies at the heart of research in multi-objective optimization.

A measure that has been the subject of much recent study in evolutionary multi-objective optimization is the “hypervolume indicator” (HYP). It measures the volume of the domi-nated portion of the objective space and is of exceptional interest as it possesses the highly desirable feature of strict Pareto compliance [8]. We have shown in [3] that not only the the hypervolume indicator is #P-hard, but also most measures of unions of high-dimensional geometric objects. For rectangular boxes this is known as Klee’s measure problem. [3] also presents an efficient FPRAS (fully polynomial-time randomized approximation scheme) for computing the volume of the unions of objects where one can (a) test whether a given point lies inside the object, (b) sample a point uniformly, and (c) calculate the volume of the object in polynomial time.

Most hypervolume indicator based optimization algorithms like SIBEA [7], SMS-EMOA [1, 5] or MO-CMA-ES [6] remove the solution with the smallest contribution to the dominated hypervolume from the population. This is usually iterated λ times until the size of the population no longer exceeds a fixed size µ. We show in [4] that this greedy selection scheme can perform arbitrarily bad and present the first hypervolume algorithm which calculates directly the contribution of every set of λ solutions. Given a population of size n = µ + λ, our algorithm can calculate a set of λ ≥ 1 solutions with minimal d-dimensional hypervolume contribution in time On^d/2log n + n^λ for d > 2. This improves all previously published algorithms by a factor of order n^min{λ,d/2} for d > 3.

The #P-hardness result of [3] for calculation of the hypervolume does not rule out that the hypervolume contribution is hard as well. In [2] it is shown that this problem is #P-hard to solve exactly and NP-hard to approximate by a factor of 2^d1−ε for any ε > 0. It is also shown that even finding the solution with contribution at most (1 + ε) times the minimal contribution of any solution is NP-hard. Though this dashes the hope for a provable efficient approximation algorithm, [2] also presents a very fast approximation algorithm for this prob-lem. We prove that for arbitrarily given ε, δ > 0 it calculates a solution with contribution at most (1 + ε) times the minimal contribution with probability at least (1 − δ). The algorithm solves very large problem instances which are intractable for all previous algorithms (e.g., 10000 solutions in 100 dimensions) within a few seconds.

References

[1] N. Beume, B. Naujoks, and M. Emmerich. SMS-EMOA: Multiobjective selection based on dom-inated hypervolume. European Journal of Operational Research, 181(3):1653–1669, 2007.

• [2] K. Bringman and T. Friedrich. Approximating the least hypervolume contributor: NP-hard in general, but fast in practice. In Proceedings of the 5th International Conference on Evolutionary Multi-Criterion Optimization (EMO 2009), Nantes, France, 2009. ACM.

• [3] K. Bringmann and T. Friedrich. Approximating the volume of unions and intersections of high-dimensional geometric objects. In S.-H. Hong, H. Nagamochi, and T. Fukunaga, eds., Proceedings of the 19th International Symposium on Algorithms and Computation (ISAAC 2008), Gold Coast, Australia, 2008, LNCS 5369, pp. 436–447. Springer.

• [4] K. Bringmann and T. Friedrich. Don’t be greedy when calculating hypervolume contributions.

In T. Jansen, I. Garibay, W. R. Paul, and A. S. Wu, eds., Proceedings of the 10th International Workshop on Foundations of Genetic Algorithms (FOGA 2009), Orlando, USA, 2009. ACM.

[5] M. Emmerich, N. Beume, and B. Naujoks. An EMO algorithm using the hypervolume measure as selection criterion. In Proc. Third International Conference on Evolutionary Multi-Criterion Optimization (EMO ’05), 2005, pp. 62–76.

[6] C. Igel, N. Hansen, and S. Roth. Covariance matrix adaptation for multi-objective optimization.

Evol. Comput., 15(1):1–28, 2007.

[7] E. Zitzler, D. Brockhoff, and L. Thiele. The hypervolume indicator revisited: On the design of Pareto-compliant indicators via weighted integration. In Proc. Fourth International Conference on Evolutionary Multi-Criterion Optimization (EMO ’07), 2007, LNCS 4403, pp. 862–876. Springer.

[8] E. Zitzler, L. Thiele, M. Laumanns, C. M. Fonseca, and V. G. da Fonseca. Performance assessment of multiobjective optimizers: an analysis and review. IEEE Trans. Evolutionary Computation, 7(2):117–132, 2003.

Hypervolume-based Algorithms

Investigators: Tobias Friedrich and Frank Neumann in cooperation with Dimo Brockhoff (ETH Zurich) and Christian Horoba (TU Dortmund)

Indicator-based methods to tackle multi-objective problems have become popular recently, mainly because they allow to incorporate user preferences into the search explicitly. Multi-objective Evolutionary Algorithms (MOEAs) using the hypervolume indicator in particular showed better performance than classical MOEAs in experimental comparisons. In [1], the

use of indicator-based MOEAs is investigated for the first time from a theoretical point of view. We carried out runtime analyses for an evolutionary algorithm with a (µ + 1)-selection scheme based on the hypervolume indicator as it is used in most of the recently proposed MOEAs. Our analyses point out two important aspects of the search process. First, we examined how such algorithms can approach the Pareto front. Later on, we pointed out how they can achieve a good approximation for an exponentially large Pareto front.

In [2], we examined the hypervolume-based approach with respect to the achieved multi-plicative approximation ratio for a given multi-objective problem and related it to a set of µ points on the Pareto front that achieves the best possible approximation ratio. For the class of linear functions and a class of convex functions, we proved that the hypervolume gives the best possible approximation ratio. In addition, we examined Pareto fronts of different shapes by numerical calculations and showed where and when the approximation computed by the hypervolume is different to an optimal one.

References

• [1] D. Brockhoff, T. Friedrich, and F. Neumann. Analyzing hypervolume indicator based algorithms.

In G. Rudolph, T. Jansen, S. M. Lucas, C. Poloni, and N. Beume, eds., Parallel Problem Solving from Nature (PPSN X), Dortmund, Germany, 2008, LNCS 5199, pp. 651–660. Springer.

• [2] T. Friedrich, C. Horoba, and F. Neumann. Multiplicative approximations and the hypervolume indicator. In G. Raidl and F. Rothlauf, eds., Genetic and Evolutionary Computation Conference 2009, Montreal, Canada, 2009. ACM. To appear.