This paper presents the first convergence result for random search algorithms to a subset of the Pareto set of given maximum size k with bounds on the approximation quality. The core of the algorithm is a new selection criterion based on a hypothetical multilevel grid on the objective space. It is shown that, when using this criterion for accepting new search points, the sequence of solution archives converges with probability one to a subset of the Pareto set that epsilon-dominates the entire Pareto set. The obtained approximation quality epsilon is equal to the size of the grid cells on the finest level of resolution that allows an approximation with at most k points within the family of grids considered. While the convergence result is of general theoretical interest, the archiving algorithm might be of high practical value for any type iterative multiobjective optimization method, such as evolutionary algorithms or other metaheuristics, which all rely on the usage of a finite on-line memory to store the best solutions found so far as the current approximation of the Pareto set.
One option to control the approximation quality under size restrictions is to define a which maps each possible solution set to a real value that can then be used to decide on the inclusion of a new search point. Several algorithms have been proposed that implement this concept @cite_20 @cite_16 . In case that such a quality indicator fulfils certain monotonicity conditions, it can be used as a potential function in the convergence analysis. As shown in @cite_9 @cite_17 , this entails convergence to a subset of the Pareto set as a local optimum of the quality indicator, but it remained open how such a local optimum relates to a guarantee on the approximation quality @math . @cite_17 also analyzed an adaptive grid archiving method proposed in @cite_19 and proved that after finite time, even though the solution set itself might permanently oscillate, it will always represent an @math -approximation whose approximation quality depends on the granularity of the adaptive grid and on the number of allowed solutions. The results depend on the additional assumption that the grid boundaries converge after finite time, which is fulfilled in certain special cases.
@cite_9 @cite_19 @cite_16 @cite_20 @cite_17
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We introduce a simple evolution scheme for multiobjective optimization problems, called the Pareto Archived Evolution Strategy (PAES). We argue that PAES may represent the simplest possible nontrivial algorithm capable of generating diverse solutions in the Pareto optimal set. The algorithm, in its simplest form, is a (1 + 1) evolution strategy employing local search but using a reference archive of previously found solutions in order to identify the approximate dominance ranking of the current and candidate solution vectors. (1 + 1)-PAES is intended to be a baseline approach against which more involved methods may be compared. It may also serve well in some real-world applications when local search seems superior to or competitive with population-based methods. We introduce (1 + λ) and (μ | λ) variants of PAES as extensions to the basic algorithm. Six variants of PAES are compared to variants of the Niched Pareto Genetic Algorithm and the Nondominated Sorting Genetic Algorithm over a diverse suite of six test functions. Results are analyzed and presented using techniques that reduce the attainment surfaces generated from several optimization runs into a set of univariate distributions. This allows standard statistical analysis to be carried out for comparative purposes. Our results provide strong evidence that PAES performs consistently well on a range of multiobjective optimization tasks. Abstract The hypervolume measure (or S metric) is a frequently applied quality measure for comparing the results of evolutionary multiobjective optimisation algorithms (EMOA). The new idea is to aim explicitly for the maximisation of the dominated hypervolume within the optimisation process. A steady-state EMOA is proposed that features a selection operator based on the hypervolume measure combined with the concept of non-dominated sorting. The algorithm’s population evolves to a well-distributed set of solutions, thereby focussing on interesting regions of the Pareto front. The performance of the devised S metric selection EMOA ( SMS-EMOA ) is compared to state-of-the-art methods on two- and three-objective benchmark suites as well as on aeronautical real-world applications. This paper discusses how preference information of the decision maker can in general be integrated into multiobjective search. The main idea is to first define the optimization goal in terms of a binary performance measure (indicator) and then to directly use this measure in the selection process. To this end, we propose a general indicator-based evolutionary algorithm (IBEA) that can be combined with arbitrary indicators. In contrast to existing algorithms, IBEA can be adapted to the preferences of the user and moreover does not require any additional diversity preservation mechanism such as fitness sharing to be used. It is shown on several continuous and discrete benchmark problems that IBEA can substantially improve on the results generated by two popular algorithms, namely NSGA-II and SPEA2, with respect to different performance measures. Search algorithms for Pareto optimization are designed to obtain multiple solutions, each offering a different trade-off of the problem objectives. To make the different solutions available at the end of an algorithm run, procedures are needed for storing them, one by one, as they are found. In a simple case, this may be achieved by placing each point that is found into an "archive" which maintains only nondominated points and discards all others. However, even a set of mutually nondominated points is potentially very large, necessitating a bound on the archive's capacity. But with such a bound in place, it is no longer obvious which points should be maintained and which discarded; we would like the archive to maintain a representative and well-distributed subset of the points generated by the search algorithm, and also that this set converges. To achieve these objectives, we propose an adaptive archiving algorithm, suitable for use with any Pareto optimization algorithm, which has various useful properties as follows. It maintains an archive of bounded size, encourages an even distribution of points across the Pareto front, is computationally efficient, and we are able to prove a form of convergence. The method proposed here maintains evenness, efficiency, and cardinality, and provably converges under certain conditions but not all. Finally, the notions underlying our convergence proofs support a new way to rigorously define what is meant by "good spread of points" across a Pareto front, in the context of grid-based archiving schemes. This leads to proofs and conjectures applicable to archive sizing and grid sizing in any Pareto optimization algorithm maintaining a grid-based archive.