Pray Algorithm Optimization (PAO): An Astronomically Inspired Metaheuristic Based on Solar Altitude and Islamic Prayer Times for Global Optimization and Engineering Design

Most population-based metaheuristics govern the exploration–exploitation balance with a monotonically decreasing parameter, locking the search into fine-tuning early with no principled way to recover diversity. This paper introduces Pray Algorithm Optimization (PAO), whose schedule is derived from an external physical model rather than a decay rule chosen by hand: the run is mapped to a solar day, and the instantaneous solar altitude places five operators at the canonical Islamic prayer times, producing a non-monotone profile that restores diversity at fixed, planned points. PAO is validated against 42 competing algorithms over 75 benchmark functions — the classical 23-function set and the CEC-2017, CEC-2019 and CEC-2022 suites — with 30 runs of 500 iterations, and on 22 constrained engineering design problems. It ranks first of 43 algorithms on CEC-2017 (mean rank 2.17), CEC-2019 (4.55), CEC-2022 (4.38) and the engineering suite (2.96), and second on the classical set, where widespread ties at the global optimum compress the ranking. Friedman tests reject rank equality on every collection (p < 10⁻²⁶), and Wilcoxon win rates reach 86–94 % on the modern suites. On the engineering problems, PAO recovers or improves the reference optimum on 18 of 22 designs, attains the best solution found by any algorithm on 20, and is statistically net-positive against all 42 competitors. Diversity analysis shows an adaptive balance, from 4 % exploration on unimodal landscapes to 52 % on deceptive ones, while an ablation indicates the schedule acts chiefly as a variance-reduction mechanism. PAO ranks 36–41 of 43 in speed, though its complexity stays in the same class as standard swarm methods. Source code is available from the corresponding author.