Constrained Hybrid Metaheuristic: A Universal Framework for Continuous Optimisation

Constrained Optimization Problems are crucial in fields such as engineering, economics, and robotics, where high-dimensional search spaces and complex objectives and constraints are common. Numerical optimization methods, including Feasible Direction, Interior Point, and Sequential Quadratic Programming, have shown strong performance in finding feasible local optima, but require accurate analytical gradients and effective initialization, which can be challenging in real-world settings. Evolutionary Algorithms, on the other hand, offer gradient-free search and robustness to noisy landscapes, managing to detect global optima more often than numerical methods, but they often suffer from high computational costs and slow convergence. In this work, we propose a Hybrid Augmented Lagrangian (HyAL) method that integrates the AL framework's constraint-handling strengths with the exploratory power of population-based search. Our approach employs evolutionary techniques to solve subproblems within the AL iterations, promoting exploration and aiding in the escape from local optima. We conduct extensive experiments on benchmark optimization problems, comparing our method against state-of-the-art optimizers, including IPOPT and CMA-ES, and a standalone evolutionary optimization baseline (with constraint enforcement via penalties). In addition, we evaluate four population-based methods integrated within the AL framework to study the effect of different evolutionary solvers. Our results show that HyAL consistently produces high-quality solutions across the benchmark suite. It outperforms purely evolutionary approaches and scales more effectively to high-dimensional constrained problems, where evolutionary-only methods often struggle. HyAL also surpasses state-of-the-art numerical optimization algorithms on complex landscapes containing numerous local minima and saddle points.

When gradient information is unavailable, black-box optimization (BBO) methods provide a practical alternative. While Evolution Strategies (ES), Consensus-Based Optimization (CBO), Optimization via Integration (OVI), and related methods have each been studied independently, their connections remain underexplored. We unify these approaches within a common theoretical framework, revealing that they differ primarily in two design choices: fitness aggregation (controlling sharpness preference) and consensus scope (controlling modality). Leveraging these insights, we introduce hybrid optimizers that interpolate between existing methods. Our ES-OVI hybrid allows explicit control over the preference for flat minima, enabling a trade-off between performance and robustness in continuous control tasks. Our CBO-OVI hybrids combine the higher-dimensional efficiency of parametric methods with the multimodal capabilities of particle-based approaches, achieving competitive results on language model merging under limited evaluation budgets. We validate our methods on standard BBO benchmarks and higher-dimensional locomotion tasks, demonstrating that the hybrid methods can outperform their constituent algorithms.

The present paper introduces a new representation-driven approach to per-instance algorithm selection, applied to black-box optimization, for automatically choosing the most promising solver from a fixed portfolio. Prior work in continuous optimization largely relies on numerical descriptors, including Exploratory Landscape Analysis features and learned embeddings such as Deep-ELA. This work studies a complementary representation: contour-map visualizations of probed landscapes. A CNN regressor takes multiple instance-specific contour views (stacked or encoded per view and aggregated) and predicts per-solver performance, enabling selection by the predicted best value. On the standard BBOB 2009 single-objective protocol, the resulting selectors significantly outperform the single best solver (SBS) and are competitive with feature-based baselines. A subsequent bi-objective evaluation under the DeepELA setting further indicates that the same image-based principle can be competitive when using windowed contour views. Overall, the results suggest that simple vision models can exploit spatial structure in probed landscapes for algorithm selection without handcrafted ELA features.