A THEORETICAL FRAMEWORK FOR AI-ASSISTED MATHEMATICAL OPTIMIZATION VIA ADAPTIVE OPERATORS

Abstract

Mathematical optimization plays a fundamental role in scientific computing, engineering design, and machine learning; however, classical optimization methods such as gradient descent often suffer from limitations including slow convergence, sensitivity to initial conditions, and dependence on fixed learning parameters. These challenges highlight the need for more adaptive and intelligent optimization frameworks. In this paper, we propose a novel theoretical approach to optimization by introducing an adaptive learning operator that models the role of artificial intelligence within the optimization process. The proposed operator dynamically adjusts the update step based on the current state of the system, replacing conventional fixed-step mechanisms. A modified iterative scheme is developed, and its convergence properties are rigorously analyzed under standard convexity assumptions. We establish a convergence theorem demonstrating that the proposed AI-assisted framework ensures monotonic decrease of the objective function and convergence to an optimal solution. The theoretical formulation provides a generalized foundation for integrating artificial intelligence into optimization algorithms, with potential applications in machine learning, control systems, data science, and complex engineering optimization problems.