One PLOT to Show Them All: Visualization of Efficient Sets in Multi-Objective Landscapes

Visualization techniques for the decision space of continuous multi-objective optimization problems (MOPs) are rather scarce in research. For long, all techniques focused on global optimality and even for the few available landscape visualizations, e.g., cost landscapes, globality is the main criterion. In contrast, the recently proposed gradient field heatmaps (GFHs) emphasize the location and attraction basins of local efficient sets, but ignore the relation of sets in terms of solution quality. In this paper, we propose a new and hybrid visualization technique, which combines the advantages of both approaches in order to represent local and global optimality together within a single visualization. Therefore, we build on the GFH approach but apply a new technique for approximating the location of locally efficient points and using the divergence of the multi-objective gradient vector field as a robust second-order condition. Then, the relative dominance relationship of the determined locally efficient points is used to visualize the complete landscape of the MOP. Augmented by information on the basins of attraction, this Plot of Landscapes with Optimal Trade-offs (PLOT) becomes one of the most informative multi-objective landscape visualization techniques available.Comment: This version has been accepted for publication at the 16th International Conference on Parallel Problem Solving from Nature (PPSN XVI

Multi-objective Optimization by Uncrowded Hypervolume Gradient Ascent

Evolutionary algorithms (EAs) are the preferred method for solving black-box multi-objective optimization problems, but when gradients of the objective functions are available, it is not straightforward to exploit these efficiently. By contrast, gradient-based optimization is well-established for single-objective optimization. A single-objective reformulation of the multi-objective problem could therefore offer a solution. Of particular interest to this end is the recently introduced uncrowded hypervolume (UHV) indicator, which takes into account dominated solutions. In this work, we show that the gradient of the UHV can often be computed, which allows for a direct application of gradient ascent algorithms. We compare this new approach with two EAs for UHV optimization as well as with one gradient-based algorithm for optimizing the well-established hypervolume. On several bi-objective benchmarks, we find that gradient-based algorithms outperform the tested EAs by obtaining a better hypervolume with fewer evaluations whenever exact gradients of the multiple objective functions are available and in case of small evaluation budgets. For larger budgets, however, EAs perform similarly or better. We further find that, when finite differences are used to approximate the gradients of the multiple objectives, our new gradient-based algorithm is still competitive with EAs in most considered benchmarks. Implementations are available at https://github.com/scmaree/uncrowded-hypervolume.Comment: T.M.D. and S.C.M. contributed equally. The final authenticated version is available in the conference proceedings of Parallel Problem Solving from Nature - PPSN XVI. Changes in new version: removed statement about Pareto compliance in abstract; added related work; corrected minor mistake

Multiple-Gradient Descent Algorithm (MGDA) for Pareto-Front Identification

International audienceThis article compounds and extends several publications in which aMultiple-Gradient Descent Algorithm (MGDA), has been proposed and tested forthe treatment of multi-objective differentiable optimization. Originally introducedin [8], the method has been tested and reformulated in [9]. Its efficacy to identifythe Pareto front [4] has been demonstrated in [22], in comparison with an evolutionarystrategy. Recently, a variant, MGDA-II, has been proposed in which the descentdirection is calculated by a direct procedure [10] based on a Gram-Schmidtorthogonalization process (GSP) with special normalization. This algorithm wastested in the context of a simulation by domain partitioning, as a technique to matchthe different interface components concurrently [11]. The experimentation revealedthe importance of scaling, and a slightly modified normalization procedure wasproposed (”MGDA-IIb”). Two novel variants have been proposed since. The first,MGDA-III, realizes two enhancements. Firstly, the GSP is conducted incompletelywhenever a test reveals that the current estimate of the direction of search is adequatealso w.r.t. the gradients not yet taken into account; this improvement simplifiesthe identification of the search direction when the gradients point roughly in thesame direction, and makes the directional derivative common to several objectivefunctionslarger. Secondly, the order in which the different gradients are consideredin the GSP is defined in a unique way devised to favor an incomplete GSP. In thesecond variant, MGDA-IV, the question of scaling is addressed when the Hessiansare known. A variant is also proposed in which the Hessians are estimated by theBroyden-Fletcher-Goldfarb-Shanno (BFGS) formula. Lastly, a solution is proposedto adjust the step-size optimally in the descent step