Threshold-Based Particle Comparison Criterion for Multi-Objective Constrained Optimization

Building upon the original text, we incorporate detailed explanations about multi-objective constrained optimization problems. When addressing multi-objective constrained optimization problems, we implement a threshold-based particle comparison criterion. This criterion algorithmically preserves infeasible solution particles that exhibit small ordinal values while maintaining constraint violations within specified tolerance thresholds. The implementation typically involves comparing particles' constraint violation magnitudes against a dynamically adjusted threshold value, allowing controlled migration from infeasible to feasible regions in the solution space. Furthermore, we introduce a novel crowding distance function that computationally assigns higher values to points located in sparsely populated regions and adjacent to Pareto front boundaries. This is achieved through density estimation techniques that calculate the relative distribution of solutions in objective space. Points with larger crowding distance values receive higher selection probabilities during the optimization process, promoting diversity preservation in the solution set. This crowding distance mechanism serves as a crucial component in our hybrid particle swarm optimization algorithm for solving multi-objective constrained optimization problems. Through these modifications and enhancements, we have expanded the original content to provide more comprehensive explanations of the methodologies employed in handling multi-objective constrained optimization problems, including specific algorithmic implementations and selection mechanisms.