From constraints fusion to manifold optimization: A new directional transport manifold metaheuristic algorithm

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The ascent of geometry-based models and methodologies, exemplified by geometric deep learning and manifold numerical optimization algorithms, has inaugurated a novel domain across various applications that grapple with geometric data complexities, such as electroencephalogram signals represented by symmetric positive definite matrix manifold, hierarchical data represented by hyperbolic manifold. The imperative fuels this inevitable paradigm shift to encapsulate the intricacies and richness inherent in data, areas where traditional methods prove inadequate. While metaheuristic algorithms are renowned for their versatile adaptability across applications, offering practical solutions within reasonable timeframes. However, the conventional metaheuristic algorithms fail on manifold applications with meaningless solutions. From an extrinsic optimization perspective, we treat manifold optimization problems as general optimization problems with multiple fused constraints that limit the optimization path to the manifold. This study pioneered the proposal and implementation of a metaheuristic manifold optimization, introducing a novel directional transport operator to rectify previously identified issues. Through experimentation across five sets of 25 problems, comparing against five algorithms, including both gradient-free and gradient-dominant counterparts, our proposed algorithm emerges as the optimal performer within the gradient-free category, demonstrating competitiveness even against gradient-dominant algorithms. Furthermore, we applied the proposed algorithm to the robot dynamic manipulation problem, achieving a close-optimal solution that eludes gradient-dominant approaches. This paper delves into the inherent capabilities and establishes the generalization of a metaheuristic algorithm within non-Euclidean functional landscapes. The source code will be available at https://github.com/lingping-fuzzy/metaheuristic-manifold-optimization.