BEACON-M: Continuous Many-Objective Benchmark Problems via Gaussian Process Random Fourier Features with Explicit, Adjustable Correlation

Correlations between objectives are natural in real-world multi-objective optimisation problems. They fundamentally shape the objective landscape and influence the search behaviour of a multi-objective optimiser. Existing continuous multi-objective benchmarks rarely provide the mechanisms to define them explicitly. We present BEACON-M, a continuous many-objective benchmark generator that constructs smooth objectives via Gaussian Process Random Fourier Features and offers explicit user-defined correlations. These correlations can be embedded via two distinct structured correlation matrix formats: single-group and two-group structures. BEACON-M instances are reproducible by construction, and their correlation structures satisfy analytically derived validity constraints. Beyond the generator itself, we analyse and characterise the feasible correlation region and show how correlations reshape dominance structure and objective landscapes. We also provide concrete guidelines for practitioners and algorithm developers to use these problems. To demonstrate the benchmark's use, we run experiments with state-of-the-art multi-objective evolutionary algorithms (dominance, indicator and decomposition-based) with different numbers of decision variables and objectives. Performance depends on the interaction between problem dimensionality (both in the objective and decision spaces) and the correlation structure and strength; no single algorithm is best across all settings, and which method performs best changes systematically with these factors. These findings demonstrate that such problems can inform the development of new algorithms, both Bayesian and evolutionary, and the selection of existing ones when solving a real-world multi-objective optimisation problem.