On multiobjective selection for multimodal optimization

Abstract

Multiobjective selection operators are a popular and straightforward tool for preserving diversity in evolutionary optimization algorithms. One application area where diversity is essential is multimodal optimization with its goal of finding a diverse set of either globally or locally optimal solutions of a single-objective problem. We therefore investigate multiobjective selection methods that identify good quality and diverse solutions from a larger set of candidates. Simultaneously, unary quality indicators from multiobjective optimization also turn out to be useful for multimodal optimization. We focus on experimentally detecting the best selection operators and indicators in two different contexts, namely a one-time subset selection and an iterative application in optimization. Experimental results indicate that certain design decisions generally have advantageous tendencies regarding run time and quality. One such positive example is using a concept of nearest better neighbors instead of the common nearest-neighbor distances.

Appendix: Quality indicators

Throughout the section, ${\mathcal {P}} = \{{\varvec{x}}_1, \ldots , {\varvec{x}}_\mu \}$, $\mu < \infty$, denotes the approximation set that is to be assessed.

1.1 Indicators without problem knowledge

Solow–Polasky diversity (SPD) Solow and Polasky [41] developed an indicator to measure a population’s biological diversity and showed that it has superior theoretical properties compared to SD and other indicators. Ulrich et al. [47] discovered its applicability to multiobjective optimization. They also verified the inferiority of SD experimentally by directly optimizing the indicator values. To compute this indicator for ${\mathcal {P}}$, it is necessary to build a $\mu \times \mu$ correlation matrix ${\mathbf {C}}$ with entries $c_{ij} = \exp (-\theta d({\varvec{x}}_i, {\varvec{x}}_j))$. The indicator value is then the scalar resulting from ${\text {SPD}}({\mathcal {P}}) := {\varvec{e}}^\top {\mathbf {C}}^{-1}{\varvec{e}}$, where ${\varvec{e}} = (1, \ldots , 1)^\top$. As the matrix inversion requires time $O(\mu ^3)$, the indicator is only applicable to relatively small sets. It also requires a user-defined parameter $\theta$, which depends on the size of the search space. We chose $\theta = 1/n$ throughout this paper.

Sum of distances (SD) The sum of distances ${\text {SD}}({\mathcal {P}}) := \sqrt{\sum _{i=1}^\mu \sum _{j=i+1}^\mu d({\varvec{x}}_i, {\varvec{x}}_j)}$ is criticized by [25, 41, 47] as being inappropriate for a diversity measure, because it only rewards the spread, but not the diversity of a population. The figure should therefore not be used. However, if it is used, we suggest to take the square root of the sum, to obtain indicator values of reasonable magnitude.

Sum of distances to nearest neighbor (SDNN) As [47] showed that SD has some severe deficiencies, we also consider the sum of distances to the nearest neighbor ${\text {SDNN}}({\mathcal {P}}) := \sum _{i=1}^{\mu } d_{{\mathrm {nn}}}({\varvec{x}}_i, {\mathcal {P}})$. In contrast to SD, SDNN penalizes the clustering of solutions, because only the nearest neighbor is considered. Emmerich et al. [14] mention the arithmetic mean gap $\frac{1}{\mu }{\text {SDNN}}({\mathcal {P}})$ and two other similar variants. We avoid the averaging here to reward larger sets. However, it is still possible to construct situations where adding a new point to the set decreases the indicator value.

Average objective value (AOV) The sample mean of objective values is ${\text {AOV}}({\mathcal {P}}) := \frac{1}{\mu } \sum _{i=1}^\mu f({\varvec{x}}_i)$.

1.2 Indicators requiring knowledge of the optima

Peak ratio (PR) Ursem [49] defined the number of found optima $\ell = |\{{\varvec{z}} \in {\mathcal {Q}} \mid d_{{\mathrm {nn}}}({\varvec{z}}, {\mathcal {P}}) \le \epsilon \}|$ divided by the total number of optima as peak ratio ${\text {PR}}({\mathcal {P}}) := \ell /m$. The indicator requires some constant $\epsilon$ to be defined by the user, to decide if an optimum has been approximated appropriately.

Peak distance (PD) This indicator simply calculates the average distance ${\text {PD}}({\mathcal {P}}) := \frac{1}{m}\sum _{i=1}^{m} d_{{\mathrm {nn}}}({\varvec{z}}_i, {\mathcal {P}})$ of a member of the reference set ${\mathcal {Q}}$ to the nearest individual in ${\mathcal {P}}$. A first version of this indicator (without the averaging) was presented by [42] as “distance accuracy”. With the 1 / m part, peak distance is analogous to the indicator inverted generational distance [7], which is computed in the objective space of multiobjective problems.

Peak inaccuracy (PI) Thomsen [43] proposed the basic variant of the indicator ${\text {PI}}({\mathcal {P}}) := \frac{1}{m}\sum _{i=1}^{m} |f({\varvec{z}}_i) - f({\text {nn}}({\varvec{z}}_i, {\mathcal {P}}))|$ under the name “peak accuracy”. To be consistent with PR and PD, we also add the 1 / m term here. It is also relabeled to peak inaccuracy, because speaking of accuracy is a bit misleading as the indicator must be minimized.

Averaged Hausdorff distance (AHD) This indicator can be seen as an extension of PD due to its relation to the inverted generational distance. It was defined by [36] as

$$\begin{aligned} \Delta _p({\mathcal {P}},{\mathcal {Q}})&= \max \left\{ \left( \textstyle \frac{1}{m}\sum _{i=1}^{m} d_{{\mathrm {nn}}}({\varvec{z}}_i, {\mathcal {P}})^p\right) ^{1/p}, \left( \textstyle \frac{1}{\mu }\sum _{i=1}^{\mu } d_{{\mathrm {nn}}}({\varvec{x}}_i, {\mathcal {Q}})^p\right) ^{1/p} \right\} \!. \end{aligned}$$

The definition contains a parameter p that controls the influence of outliers on the indicator value (the more influence the higher p is). For $1 \le p < \infty$, AHD has the property of being a semi-metric [36]. We chose $p = 1$ throughout this paper, analogously to [14]. The practical effect of the indicator is that it rewards the approximation of the optima (as PD does), but as well penalizes any unnecessary points in remote locations.

1.3 Indicators requiring knowledge of the basins

For the implementation of indicators in this section, we assume the existence of a function

$$\begin{aligned} b({\varvec{x}}, {\varvec{z}}) = {\left\{ \begin{array}{ll} 1 &{} {\text {if }}{\varvec{x}} \in {\text {basin}}({\varvec{z}}),\\ 0 &{} {\text {else}}. \end{array}\right. } \end{aligned}$$

Basin ratio (BR) The number of covered basins is calculated as

$$\begin{aligned} \textstyle \ell = \sum _{i=1}^m \min \{1, \sum _{j=1}^\mu b({\varvec{x}}_j, {\varvec{z}}_i)\}\,. \end{aligned}$$

The basin ratio is then ${\text {BR}}({\mathcal {P}}) := \ell /m$, analogous to PR. This indicator can only assume $m+1$ distinct values, and in lower dimensions it should be quite easy to obtain a perfect score by a simple random sampling of the search space. It makes sense especially when not all of the existing optima are relevant. Then, its use can be justified by the common assumption in global optimization that the actual optima can be found relatively easily with a hill climber, once there is a start point in each respective basin [45].

Basin inaccuracy (BI) This combination of BR and PI was proposed by [32]. It is defined as

$$\begin{aligned} {\text {BI}}({\mathcal {P}}) := \frac{1}{m}\sum _{i=1}^{m} {\left\{ \begin{array}{ll} \min \left\{ |f({\varvec{z}}_i) - f({\varvec{x}})| \mid {\varvec{x}} \in {\mathcal {P}} \wedge b({\varvec{x}}, {\varvec{z}}_i) \right\} &{} \exists {\varvec{x}} \in {\text {basin}}({\varvec{z}}_i)\,, \\ f_{\max } &{} {\text {else}}, \end{array}\right. } \end{aligned}$$

where $f_{\max }$ denotes the difference between the global optimum and the worst possible objective value. For each optimum, the indicator calculates the minimal difference in objective values between the optimum and all solutions that are located in it’s basin. If no solution is present in the basin, a penalty value is assumed for it. Finally, all the values are averaged. The rationale behind this indicator is to enforce a good basin coverage, while simultaneously measuring the deviation of objective values $f({\varvec{x}})$ from $f({\varvec{z}}_i)$.