Two simple yet powerful optimization algorithms, named the Best-Mean-Random (BMR) and Best-Worst-Random (BWR) algorithms, are developed and presented in this paper to handle both constrained and unconstrained optimization problems. These algorithms are free of metaphors and algorithm-specific parameters. The BMR algorithm is based on the best, mean, and random solutions of the population generated for solving a given problem, and the BWR algorithm is based on the best, worst, and random solutions. The performances of the proposed two algorithms are investigated by implementing them on 26 real-life nonconvex constrained optimization problems given in the Congress on Evolutionary Computation (CEC) 2020 competition, and comparisons are made with those of the other prominent optimization algorithms. The performances on 12 constrained engineering problems are also investigated, and the results are compared with those of very recent algorithms (in some cases, compared with more than 30 algorithms). Furthermore, computational experiments are conducted on 30 unconstrained standard benchmark optimization problems, including 5 recently developed benchmark problems with distinct characteristics. The results demonstrated the superior competitiveness and superiority of the proposed simple algorithms. The optimization research community may gain an advantage by adapting these algorithms to solve various constrained and unconstrained real-life optimization problems across various scientific and engineering disciplines. The codes of the BMR and BWR algorithms are available at https://sites.google.com/view/bmr-bwr-optimization-algorithm/home?authuser=0.