Exploring the Connections Between the Implicit Function Theorem and Optimization Methods

Abstract

Background and aims. The implicit function theorem is a powerful tool for solving non-linear optimization problems. It provides conditions under which first-order optimality conditions define an implicit function for each element of the optimal vector of the decision variables. In this article, we explore the connections between the implicit function theorem and optimization methods and compare them with other theories in the same field, such as non-smooth implicit differentiation, algebraic functions, and inverse functions. Methods. We present a comprehensive comparative analysis of the implicit function theorem and other mathematical theories related to optimization. We also provide examples and applications of the implicit function theorem to various optimization problems. Results. Our analysis shows that the implicit function theorem is a powerful and versatile tool for solving a wide range of optimization problems. Conclusion. We demonstrate the theorem's superiority over other theories in terms of accuracy, efficiency, and applicability.