A Comparison-Relationship-Surrogate Evolutionary Algorithm for Multi-Objective Optimization

TL;DR

This work tackles expensive multi-objective optimization under a tight evaluation budget by introducing a comparison-relationship surrogate that learns pairwise objective comparisons between candidates. The CRSEA algorithm integrates a symmetry-aware neural network that predicts comparisons and uses it to drive NSGA-II–style search, with procedures to enforce transitivity and properly handle equalities. Empirical results show CRSEA achieves strong convergence on several WFG biobjective problems and demonstrates practical value on a real-world accelerator physics problem, though performance on some DTLZ problems indicates a need for diversity-preserving mechanisms. The study highlights future directions in diversity maintenance and constraint incorporation to broaden applicability and robustness of comparison-based surrogates in expensive multi-objective optimization.

Abstract

Evolutionary algorithms often struggle to find well converged (e.g small inverted generational distance on test problems) solutions to multi-objective optimization problems on a limited budget of function evaluations (here, a few hundred). The family of surrogate-assisted evolutionary algorithms (SAEAs) offers a potential solution to this shortcoming through the use of data driven models which augment evaluations of the objective functions. A surrogate model which has shown promise in single-objective optimization is to predict the "comparison relationship" between pairs of solutions (i.e. who's objective function is smaller). In this paper, we investigate the performance of this model on multi-objective optimization problems. First, we propose a new algorithm "CRSEA" which uses the comparison-relationship model. Numerical experiments are then performed with the DTLZ and WFG test suites plus a real-world problem from the field of accelerator physics. We find that CRSEA finds better converged solutions than the tested SAEAs on many of the medium-scale, biobjective problems chosen from the WFG suite suggesting the comparison-relationship surrogate as a promising tool for improving the efficiency of multi-objective optimization algorithms.

Figures (6)

• Figure 1: A diagram of the how the CRSEA algorithm (Alg. \ref{['alg:crsea']}) works. An initial population is generated and the comparison-relationship model trained. Then, the comparison-relationship model is used in the place of the objective functions in an evolutionary algorithm for some number of generations ($w_{\text{max}}$). Several ($\mu$) randomly chosen solutions from the resulting candidate solutions are evaluated and used to retrain the model. The process is repeated until the budget of function evaluations ($\text{FE}_{max}$) is exhausted.
• Figure 2: A schematic of the comparison surrogate model. Decision variables of the two solutions being compared are fed into feature extraction networks made of densely connected layers with shared weights. Additional features in the form of the comparisons $x_{1,i} < x_{2,i}$ are generated. All features are combined into a symmetric vector such that a permutation of the two input solutions will cause the feature vector to reverse itself ($f_i\to f_{n-i+1}$ for a vector $f$ with $n$ elements). These features are fed into comparison networks (one for each objective) which use special symmetric dense layers to ensure the correct behavior under permutation of the inputs.
• Figure 3: An example of the comparison model trained on data from the WFG8 ($n=16$) problem. (a) loss function on the train and test set during training; (b) the accuracy during training; (c) and (d) predicted comparisons for both objectives on never-before-seen samples where solutions are indexed in sorted order for the objective being compared (i.e. predictions from a perfect classifier would show up as a solid yellow upper triangle); (e) the receiver operating characteristic curve for each model output as well as domination predictions.
• Figure 4: Mean value of the IGD metric and 95% prediction interval for non-dominated solutions to DTLZ5 ($m=3$, $n=10$).
• Figure 5: Results from the experiments performed in Tab. \ref{['tab:high-dimension-results']} for the WFG7 (n=16) test problem : (a) Mean value of the IGD metric and 95% prediction interval for non-dominated solutions; (b) Solutions after 300 function evaluations from the optimization with the median value of the IGD metric for each algorithm. The analytical Pareto front (labeled "PF") is plotted in black.