Optimization in Function Spaces: With Stability Considerations in Orlicz Spaces
This is an essentially self-contained book on the theory of convex functions and convex optimization in Banach spaces, with a special interest in Orlicz spaces. Approximate algorithms based on the stability principles and the solution of the corresponding nonlinear equations are developed in this text. A synopsis of the geometry of Banach spaces, aspects of stability and the duality of different levels of differentiability and convexity is developed. And it is provided a novel approach to the fundamental theorems of Variational Calculus based on the principle of pointwise minimization of the Lagrangian on the one hand and convexification by quadratic supplements using the classical Legendre-Ricatti equation on the other. The reader should be familiar with the concepts of mathematical analysis and linear algebra. Some awareness of the principles of measure theory will turn out to be helpful. The book is suitable for students of the second half of undergraduate studies, and it provides a rich set of material for a master course on linear and nonlinear functional analysis. Additionally it offers novel aspects at the advanced level.
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1 Approximation in Orlicz Spaces
2 Polya Algorithms in Orlicz Spaces
3 Convex Sets and Convex Functions
4 Numerical Treatment of Nonlinear Equations and Optimization Problems
5 Stability and Twostage Optimization Problems
6 Orlicz Spaces
7 Orlicz Norm and Duality
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according to Theorem Apparently arbitrary Banach space best approximation best Chebyshev approximation bounded boundedness Chebyshev approximation condition conjugate continuous convergence contradiction converges pointwise convex functions convex set convex subset Corollary corresponding defined derivative E-space equation extremaloid f(xk f(xn finite measure finite Young function flat convex following theorem Fréchet differentiable function f Gateaux derivative hence holds hyperplane implies Lemma Let f Let further level sets linear locally uniformly convex lower semi-continuous Luxemburg norm matrix measure space metric space minimal solution Minkowski functional modular monotonically increasing non-empty normed space Orlicz norm Orlicz spaces point of accumulation pointwise convergence Polya algorithm positive numbers Proof purely atomic measure reflexive Remark resp restriction set satisfies sequence xn Stability Theorem strictly convex strong minimum subgradient inequality subspace tending to zero variational problem vector space