DMulti-MADS: mesh adaptive direct multisearch for bound-constrained blackbox multiobjective optimization

Abstract

The context of this research is multiobjective optimization where conflicting objectives are present. In this work, these objectives are only available as the outputs of a blackbox for which no derivative information is available. This work proposes a new extension of the mesh adaptive direct search (MADS) algorithm to multiobjective derivative-free optimization with bound constraints. This method does not aggregate objectives and keeps a list of non dominated points which converges to a (local) Pareto set as long as the algorithm unfolds. As in the single-objective optimization MADS algorithm, this method is built around a search step and a poll step. Under classical direct search assumptions, it is proved that the so-called DMulti-MADS algorithm generates multiple subsequences of iterates which converge to a set of local Pareto stationary points. Finally, computational experiments suggest that this approach is competitive compared to the state-of-the-art algorithms for multiobjective blackbox optimization.

Appendices

Appendix A A study of the influence of the integer parameter $w^+$ on the performance of the DMulti-MADS algorithm

This appendix presents some experiments which led to the choice of the value of the integer parameter $w^+$. For readability, only the DMulti-MADS variants without opportunistic polling and the spread strategy are considered. For the set of $w^+$ integer values presented here, similar observations can be done as the ones presented in Subsection 6.3 concerning the influence of opportunistic polling and the spread strategy on the concrete performance of the DMulti-MADS algorithm. Thus, the variants compared here are the better ones for each considered $w^+$ value. They use the same settings as the ones described at the beginning of Subsection 6.3.

Fig. 15

Data profiles obtained on $10$ replications from $100$ multiobjective optimization problems from [22] for DMulti-MADS variants with strict success strategy without opportunistic polling and with spread strategy for tolerance $\varepsilon _\tau \in \{10^{-2}, 5 \times 10^{-2}, 10^{-1}\}$

Figure 15 presents data profiles of the DMulti-MADS strict success strategy variants for several integer values of $w^+$. Fixing $w^+$ to $0$ means that only points with maximum frame size parameter in the current Pareto front approximation can be selected as current poll centers. When $w^+$ is high (for example $w^+ \in \{10,14,20\}$), the poll selection of the algorithm is similar to the one of the DMS algorithm: all points of the current incumbent list can be chosen as poll centers. From Fig. 15, one can note that allowing only points with maximum frame size parameters to be selected as poll centers grandly decreases the performance of DMulti-MADS for all considered tolerances. However, removing all restrictions on the choice of the current incumbent as long as it belongs to the current incumbent list is not the most performant variant (i.e. for example with $w^+ \in \{10,14,20\}$). Indeed, for the lowest tolerance (see Fig. 15a), the data profiles reveal than strict strategy variants with high value of $w^+ \in \{10,14,20\}$ solve slightly less problems compared to the choice of $w^+ =5$. For higher tolerances, a value $w^+ \in \{3,5\}$ is preferable, as shown in Fig. 15b, c.

Fig. 16

Data profiles obtained on $10$ replications from $100$ multiobjective optimization problems from [22] for DMulti-MADS variants with DMS strategy with tolerance $\varepsilon _\tau \in \{10^{-2}, 5 \times 10^{-2}, 10^{-1}\}$

One can make similar observations when comparing DMulti-MADS variants with DMS success strategy for different $w^+$ values, as shown on Fig. 16. For the DMS success strategy, a $w^+$ integer value comprised between $3$ and $5$ implies a better performance.

Appendix B Comparing DMulti-MADS with other algorithms: performance profiles

This appendix reports comparison results between the two best DMulti-MADS variants and the other multiobjective solvers BiMADS, DMS, MOIF and NSGA-II in term of the purity metric, spread metrics $\Gamma$ and $\Delta$ metrics as described and used in [22]. All settings for running the solvers are the same as the ones described in Subsection 6.4. Specifically the maximal allowed number of evaluations for each problem and each solver is fixed to $30,000$. Algorithms are compared in pairs (see [22, 27] for an explanation).

Fig. 17

Purity performance profiles using NOMAD (BiMADS), DMS, DMulti-MADS, MOIF and NSGA-II obtained on $100$ multiobjective optimization problems ($69$ with $m=2$, $29$ with $m=3$ and $2$ with $m=4$) from [22] with $50$ different runs of NSGA-II

When looking at Fig. 17, one can observe that the two variants of DMulti-Mads are more efficient in term of purity than DMS and MOIF. On the contrary, it is less efficient than BiMADS (with and without model) and NSGA-II (worst and best versions) in terms of purity metric. This can be explained by the fact that BiMADS generates more points in the Pareto front reference, due to its scalarization approach when DMulti-MADS generates points that are close to the Pareto front reference, but not part of it. Concerning NSGA-II, a closer look at the runs shows that all deterministic solvers can stop before the exhaustion of the whole budget of evaluations (because the solver reaches a threshold), which can prevent them from exploring potential interesting areas in the objective space. NSGA-II always exploits its full budget of evaluations, which allows to generate more points in the Pareto front reference, and consequently to have a better purity metric.

Fig. 18

$\Delta$ spread performance profiles using NOMAD (BiMADS), DMS, DMulti-MADS, MOIF and NSGA-II obtained on $100$ multiobjective optimization problems ($69$ with $m=2$, $29$ with $m=3$ and $2$ with $m=4$) from [22] with $50$ different runs of NSGA-II

In terms of $\Delta$ spread metric results reported in Fig. 18, one can see that the two DMulti-MADS variants are slightly less performant than the other algorithms. For DMS and MOIF, the use of coordinate directions seems to play a role in the distribution of their generated points for this set of problems. However, the use of dense directions enables DMulti-MADS to find new non-dominated points contrary to DMS and MOIF as shown in Fig. 17.

Fig. 19

$\Gamma$ spread performance profiles using NOMAD (BiMADS), DMS, DMulti-MADS with DMS strategy, MOIF and NSGA-II obtained on $100$ multiobjective optimization problems ($69$ with $m=2$, $29$ with $m=3$ and $2$ with $m=4$) from [22] with $50$ different runs of NSGA-II

In terms of $\Gamma$ spread metric, both variants of DMulti-MADS generate less dense Pareto front approximations than BiMADS and NSGA-II, as shown in Fig. 19. The DMulti-MADS variant with DMS strategy performs better when compared to MOIF and DMS than the DMulti-MADS variant with strict success strategy. Thus, for an important budget of evaluations, the DMulti-MADS variant with DMS strategy generates denser Pareto front approximations.