A Particle Dynamical System Algorithm to Find the Sparse Linear Complementary Solutions

The Linear Complementarity Problem (LCP) offers a comprehensive modeling framework for addressing a wide range of optimization problems. In many real-world applications, finding an LCP solution with a sparse structure is often necessary. To address this problem, we introduce an innovative global optimization framework named the Particle Dynamical System Algorithm (PDSA), which consists of two components. The first component is a dynamical system (DS) inspired by the Absolute Value Equation (AVE), proven to have equilibria corresponding to LCP solutions, with additional relaxing regulators that enhance coverage rate and stability. The second component is an Adaptive Oscillated Particle Swarm Optimization (AOPSO) designed to globally enhance sparsity in LCP solutions, addressing the complexities posed by non-convex and non-smooth regulation models. Within this framework, the DS achieves optimality, while the AOPSO promotes solution sparsity. We compared our proposed DS with relaxing regulators to two classic efficient DSs, fully validating the effectiveness of our approach and underscoring the significant role of the introduced relaxing regulators in improving the convergence rate. Our newly developed variant of PSO, AOPSO, was compared with three classic and state-of-the-art variants on fourteen benchmark functions, demonstrating its competitive performance. Finally, we performed experiments on seven test examples and an application in portfolio selection, showing that the proposed PDSA algorithm surpasses other competitors in finding sparse LCP solutions.