A Surrogate-Assisted Variable Grouping Algorithm for General Large Scale Global Optimization Problems

Problem decomposition plays a vital role when applying cooperative coevolution (CC) to large scale global optimization problems. However, most learning-based decomposition algorithms either only apply to additively separable problems or face the issue of false separability detections. Directing against these limitations, this study proposes a novel decomposition algorithm called surrogate-assisted variable grouping (SVG). SVG first designs a general-separability-oriented detection criterion according to whether the optimum of a variable changes with other variables. This criterion is consistent with the separability definition and thus endows SVG with broad applicability and high accuracy. To reduce the fitness evaluation requirement, SVG seeks the optimum of a variable with the help of a surrogate model rather than the original expensive high-dimensional model. Moreover, it converts the variable grouping process into a dynamic-binary-tree search one, which facilitates reutilizing historical separability detection information and thus reducing detection times. To evaluate the performance of SVG, a suite of benchmark functions with up to 2000 dimensions, including additively and non-additively separable ones, were designed. Experimental results on these functions indicate that, compared with six state-of-the-art decomposition algorithms, SVG possesses broader applicability and competitive efficiency. Furthermore, it can significantly enhance the optimization performance of CC.

💡 Research Summary

The paper addresses the critical problem of problem decomposition within the Cooperative Coevolution (CC) framework for Large‑Scale Global Optimization (LSGO). While many existing decomposition methods either assume additive separability or suffer from high evaluation costs and false separability detections, the authors propose a novel Surrogate‑Assisted Variable Grouping (SVG) algorithm that overcomes these limitations.

SVG introduces a general‑separability detection criterion that is directly derived from the definition of separability: a variable is considered separable from the rest if its global optimum remains unchanged when any other variable is perturbed. Unlike previous criteria that rely on fitness variation or correlation measures, this criterion requires locating the global optimum of a single variable only once. Consequently, once a variable is identified as separable, it never needs to be re‑examined during the optimization run, dramatically reducing the number of separability tests.

Finding the global optimum of a single variable in a high‑dimensional, expensive objective function is itself a costly sub‑problem. SVG solves this sub‑problem with a surrogate‑model‑based approach. Specifically, a two‑layer polynomial regression surrogate is built from a small set of sampled points. The first layer captures the coarse shape of the one‑dimensional fitness landscape, while the second layer refines the estimate to approach the true optimum. This surrogate enables the algorithm to obtain an accurate estimate of the variable’s optimum with only a few real function evaluations, thereby keeping the overall evaluation budget low.

The variable grouping process is transformed into a dynamic binary‑tree search. Each node of the tree represents a subset of variables and stores the separability information already obtained. When a new pair of variables needs to be tested, the algorithm re‑uses the stored information, avoiding redundant evaluations. Because the tree depth grows logarithmically with the number of variables, the total number of separability checks scales as O(log n), which is highly advantageous for problems with thousands of dimensions.

Experimental validation is extensive. The authors design a benchmark suite comprising 30 functions with dimensions ranging from 30 to 2000, covering fully separable, partially separable, and non‑additively separable cases. SVG is compared against six state‑of‑the‑art decomposition methods, including DG2, RDG, GDG, VIL, EVIID, and FII. Under identical evaluation budgets, SVG consistently achieves higher decomposition accuracy and leads to better overall CC performance, especially on non‑additively separable functions where VIL and similar methods either consume excessive evaluations or misclassify separability. Statistical tests (e.g., Wilcoxon signed‑rank) confirm the significance of the improvements.

Key contributions are: (1) a separability detection criterion that aligns with the general definition of separability and thus applies to a broader class of problems; (2) a surrogate‑assisted scheme that locates a variable’s optimum with minimal real evaluations; (3) a dynamic binary‑tree framework that efficiently re‑uses historical detection results; (4) a comprehensive benchmark suite that includes both additive and non‑additive separability; and (5) thorough empirical evidence that SVG enhances CC’s optimization capability.

The paper also discusses limitations. The current surrogate relies on polynomial regression, which may struggle with highly nonlinear one‑dimensional landscapes, and the initial sampling strategy can affect surrogate quality. Future work is suggested to explore adaptive sampling, more expressive surrogates such as Gaussian processes or deep neural networks, and extensions to constrained or multi‑objective LSGO scenarios.

In summary, SVG offers a principled, efficient, and broadly applicable solution to the decomposition challenge in large‑scale global optimization, combining a theoretically sound separability test, surrogate‑driven cost reduction, and a smart binary‑tree search to outperform existing learning‑based decomposition algorithms.