<p>Global optimization problems are often addressed using the practical and efficient approach of evolutionary sophistication, which refers to advanced processes inspired by various systems, particularly those rooted in biological systems. However, these problems, like the original evolutionary systems that inspired them, become increasingly complex, particularly in terms of efficiency, scalability, and achieving a balance between exploitation and exploration. To address these challenges, this research introduces the Lagrange elementary optimization (LEO) algorithm, an evolutionary method that is self-adaptive and inspired by the exceptional accuracy of vaccinations, modeled using the albumin quotient of human blood. They develop intelligent agents using their fitness function value after gene crossing. These genes direct the search agents during both exploration and exploitation. The main objective of the LEO algorithm is presented in this paper along with the inspiration and motivation for the concept. To demonstrate its precision, the proposed algorithm is validated against a variety of test functions, including 19 traditional benchmark functions and the CEC-C06 2019 test functions. The results of LEO for 19 classic benchmark test functions are evaluated against DA, PSO, and GA separately, and then two other recent algorithms such as FDO and LPB are also included in the evaluation. In addition, the LEO is tested by ten functions on CEC-C06 2019 with DA, WOA, SSA, FDO, LPB, and FOX algorithms distinctly. The cumulative outcomes demonstrate LEO’s capacity to increase the starting population and move toward the global optimum. Different standard measurements are used to verify and prove the stability of LEO in both the exploration and exploitation phases. Moreover, Statistical analysis supports&#xa0;the findings results of the proposed research. Finally, novel applications in the real world are introduced to demonstrate the practicality of LEO.</p>