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. A
particular emphasis is placed on the geometrical aspects of strong
solvability of a convex optimization problem: it turns out that
this property is equivalent to local uniform convexity of the
corresponding convex function. This treatise also provides 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. From the contents: Approximation and Polya Algorithms in
Orlicz Spaces Convex Sets and Convex Functions Numerical Treatment
of Non-linear Equations and Optimization Problems Stability and
Two-stage Optimization Problems Orlicz Spaces, Orlicz Norm and
Duality Differentiability and Convexity in Orlicz Spaces
Variational Calculus
General
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