Optimization in function spaces
With Stability Considerations in Orlicz Spaces
This self-contained book focuses on the theory of convex functions and optimization in Banach spaces, particularly Orlicz spaces. It develops approximate algorithms rooted in stability principles and the resolution of related nonlinear equations. The text presents a synopsis of Banach space geometry, stability aspects, and the duality of differentiability and convexity. A key emphasis is on the geometrical aspects of strong solvability in convex optimization, revealing its equivalence to local uniform convexity of the corresponding convex function. The book introduces a novel perspective on fundamental theorems of Variational Calculus, highlighting pointwise minimization of the Lagrangian and convexification through quadratic supplements using the classical Legendre-Ricatti equation. Readers should possess knowledge of mathematical analysis and linear algebra, with familiarity in measure theory being beneficial. This work is suitable for advanced undergraduate students and serves as rich material for a master’s course in linear and nonlinear functional analysis, while also offering advanced insights. Key topics include approximation and Polya algorithms in Orlicz spaces, convex sets and functions, numerical methods for nonlinear equations and optimization, stability in two-stage optimization, and differentiability and convexity in Orlicz spaces.
