Optimization - theory and applications

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This work provides a comprehensive exploration of optimization problems across various mathematical domains. It begins with an introduction to optimization in elementary geometry, calculus of variations, approximation problems, linear programming, and optimal control, followed by a literature survey. The section on linear programming delves into the dual program's definition and interpretation, the FARKAS-Lemma, and CARATHEODORY's theorem, culminating in the strong duality theorem and its application to polyhedra. Next, it examines convexity in linear and normed linear spaces, detailing the separation of convex sets and the characteristics of convex functions. The exploration of convex optimization problems includes examples, the dual program's definition, the weak and strong duality theorems, and quadratic programming. The text further addresses necessary optimality conditions, covering GATEAUX and FRECHET differentials, LYUSTERNIK's theorem, LAGRANGE multipliers, and first and second-order optimality conditions in calculus of variations and optimal control theory. Finally, it discusses existence theorems for solutions to optimization problems, including functional analytic existence theorems and the existence of optimal controls. A symbol index concludes the work, providing clarity and reference for readers.