Un-evaluated Solutions May Be Valuable in Expensive Optimization

TL;DR

This work targets expensive optimization problems where evaluating a single solution is resource-intensive. It introduces USEA, a framework that augments surrogate-assisted evolutionary algorithms by incorporating high-quality unevaluated solutions predicted by surrogate models into the next generation, thereby increasing population disparity and improving reproduction efficiency. Across GA, DE, and EDA operators, USEA (notably USEA-EDA) demonstrates strong performance gains and notable runtime advantages over mainstream SAEAs and Bayesian optimization methods, while maintaining solution quality under limited function evaluations. The approach offers practical impact by enabling faster convergence on real-world costly optimization tasks and opens avenues for extending the strategy to multi-objective settings.

Abstract

Expensive optimization problems (EOPs) are prevalent in real-world applications, where the evaluation of a single solution requires a significant amount of resources. In our study of surrogate-assisted evolutionary algorithms (SAEAs) in EOPs, we discovered an intriguing phenomenon. Because only a limited number of solutions are evaluated in each iteration, relying solely on these evaluated solutions for evolution can lead to reduced disparity in successive populations. This, in turn, hampers the reproduction operators' ability to generate superior solutions, thereby reducing the algorithm's convergence speed. To address this issue, we propose a strategic approach that incorporates high-quality, un-evaluated solutions predicted by surrogate models during the selection phase. This approach aims to improve the distribution of evaluated solutions, thereby generating a superior next generation of solutions. This work details specific implementations of this concept across various reproduction operators and validates its effectiveness using multiple surrogate models. Experimental results demonstrate that the proposed strategy significantly enhances the performance of surrogate-assisted evolutionary algorithms. Compared to mainstream SAEAs and Bayesian optimization algorithms, our approach incorporating the un-evaluated solution strategy shows a marked improvement.

Figures (10)

• Figure 1: Visualization of the offspring distributions generated by GA, DE, and EDA operators with and without incorporating un-evaluated solutions in one-dimensional function optimization. The orange $\triangleleft$ represent the evaluated population $\mathcal{P}_e$, and the green $\triangleright$ represent the un-evaluated population $\mathcal{P}_u$. The histograms illustrate the offspring distributions generated by the operators.
• Figure 2: The basic framework of the proposed algorithm. Black arrows connect the fundamental SAEA framework, red arrows indicate our proposed unevaluated solution strategy. Blue circles represent evaluated solutions, green triangles denote offspring, and orange triangles signify solutions assessed by the surrogate model. Darker colors indicate higher quality.
• Figure 3: Illustration of the variable width histogram modeling for the $\mathbf{x}_i$ with $K=5$. The first and last bins are assigned very small probabilities due to the absence of individuals falling into them, while bins 2 to 4 are of equal width and model based on the number of individuals within each interval. Here, the blue circles represent the distribution of evaluated solutions ($\mathcal{P}_e$) in the i-th dimension, and the orange triangles indicate the distribution of un-evaluated solutions ($\mathcal{P}_u$) in the i-th dimension.
• Figure 4: Illustration of the surrogate-assisted selection process. The offspring solutions are predicted by the surrogate model, and the top 1 individual is selected as $\mathcal{O}^*$. The top $N/2$ individuals are selected as un-evaluated solutions $\mathcal{P}_u$.
• Figure 5: Performance improvement of the GA operator with different $\beta_1$ and $\beta_2$ values of un-evaluated solutions on the LZG test suite.