Nonlinear Geometry of Norm-Attaining Functionals: Variational Principles, Subdifferential Calculus, and Polynomial Optimization in Locally Convex Spaces

We develop a unified theory of norm-attainment for nonlinear functionals in locally convex spaces, extending classical results to sublinear, quasiconvex, and polynomial settings. Our main contributions include: (1) nonlinear Bishop-Phelps theorems establishing density of norm-attaining functionals, (2) a subdifferential characterization of attainment via interiority conditions, (3) a Krein-Milman principle for convex functionals on compact sets, and (4) a complete solution to the polynomial norm-attainment problem through tensor product geometry. The work combines innovative applications of Choquet theory, variational analysis, and complex-geometric methods to reveal new connections between functional analysis and optimization. Key applications address stochastic variational principles and reproducing kernel Hilbert space optimization, with tools applicable to PDE constraints and high-dimensional data science. These results collectively bridge fundamental gaps between linear and nonlinear functional analysis while providing fresh geometric insight into infinite-dimensional phenomena.