<![CDATA[Lagrangian duality theory constitutes a fundamental framework in nonlinear optimization analysis, connecting primal problems with dual problems through the Lagrange function. This article presents a comprehensive narrative review of Lagrangian duality theory development from the perspective of modern geometric analysis. We explore the geometric structure of primal-dual spaces, Karush-Kuhn-Tucker (KKT) optimality conditions, and strong and weak duality theorems in the context of constrained optimization. The discussion encompasses geometric interpretations of saddle points, convexity in duality theory, and the duality gap as a solution quality measure. Furthermore, we analyze theoretical applications from functional analysis and topology perspectives, including the role of reflexive Banach spaces and Fréchet differentiability in characterizing optimal solutions. The findings demonstrate that geometric approaches provide profound insights into optimization problem structures and open new perspectives in numerical algorithm development. This review contributes to a more robust theoretical understanding of the mathematical foundations of nonlinear optimization and its relevance in contemporary applied mathematics.]]>
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