Preface -- Contents -- 1 Approximation in Orlicz Spaces -- 1.1 Introduction -- 1.2 A Brief Framework for Approximation in Orlicz Spaces -- 1.3 Approximation in C(T) -- 1.3.1 Chebyshev Approximations in C(T) -- 1.3.2 Approximation in Haar Subspaces -- 1.4 Discrete LΦ-approximations -- 1.4.1 Discrete Chebyshev Approximation -- 1.4.2 Linear LΦ-approximation for Finite Young Functions -- 1.5 Determination of the Linear LΦ-approximation -- 1.5.1 System of Equations for the Coefficients -- 1.5.2 The Method of Karlovitz -- 2 Polya Algorithms in Orlicz Spaces -- 2.1 The Classical Polya Algorithm -- 2.2 Generalized Polya Algorithm -- 2.3 Polya Algorithm for the Discrete Chebyshev Approximation -- 2.3.1 The Strict Approximation as the Limit of Polya Algorithms -- 2.3.2 About the Choice of the Young Functions -- 2.3.3 Numerical Execution of the Polya Algorithm -- 2.4 Stability of Polya Algorithms in Orlicz Spaces -- 2.5 Convergence Estimates and Robustness -- 2.5.1 Two-Stage Optimization -- 2.5.2 Convergence Estimates -- 2.6 A Polya-Remez Algorithm in C(T) -- 2.7 Semi-infinite Optimization Problems -- 2.7.1 Successive Approximation of the Restriction Set -- 3 Convex Sets and Convex Functions -- 3.1 Geometry of Convex Sets -- 3.2 Convex Functions -- 3.3 Difference Quotient and Directional Derivative -- 3.3.1 Geometry of the Right-sided Directional Derivative -- 3.4 Necessary and Sufficient Optimality Conditions -- 3.4.1 Necessary Optimality Conditions -- 3.4.2 Sufficient Condition: Characterization Theorem of Convex Optimization -- 3.5 Continuity of Convex Functions -- 3.6 Fréchet Differentiability -- 3.7 Convex Functions in Rn -- 3.8 Continuity of the Derivative -- 3.9 Separation Theorems -- 3.10 Subgradients -- 3.11 Conjugate Functions -- 3.12 Theorem of Fenchel.

3.13 Existence of Minimal Solutions for Convex Optimization -- 3.14 Lagrange Multipliers -- 4 Numerical Treatment of Non-linear Equations and Optimization Problems -- 4.1 Newton Method -- 4.2 Secant Methods -- 4.3 Global Convergence -- 4.3.1 Damped Newton Method -- 4.3.2 Globalization of Secant Methods for Equations -- 4.3.3 Secant Method for Minimization -- 4.4 A Matrix-free Newton Method -- 5 Stability and Two-stage Optimization Problems -- 5.1 Lower Semi-continuous Convergence and Stability -- 5.1.1 Lower Semi-equicontinuity and Lower Semi-continuous Convergence -- 5.1.2 Lower Semi-continuous Convergence and Convergence of Epigraphs -- 5.2 Stability for Monotone Convergence -- 5.3 Continuous Convergence and Stability for Convex Functions -- 5.3.1 Stability Theorems -- 5.4 Convex Operators -- 5.5 Quantitative Stability Considerations in Rn -- 5.6 Two-stage Optimization -- 5.6.1 Second Stages and Stability for Epsilon-solutions -- 5.7 Stability for Families of Non-linear Equations -- 5.7.1 Stability for Monotone Operators -- 5.7.2 Stability for Wider Classes of Operators -- 5.7.3 Two-stage Solutions -- 6 Orlicz Spaces -- 6.1 Young Functions -- 6.2 Modular and Luxemburg Norm -- 6.2.1 Examples of Orlicz Spaces -- 6.2.2 Structure of Orlicz Spaces -- 6.2.3 The Δ2-condition -- 6.3 Properties of the Modular -- 6.3.1 Convergence in Modular -- 6.3.2 Level Sets and Balls -- 6.3.3 Boundedness of the Modular -- 7 Orlicz Norm and Duality -- 7.1 The Orlicz Norm -- 7.2 Hölder's Inequality -- 7.3 Lower Semi-continuity and Duality of the Modular -- 7.4 Jensen's Integral Inequality and the Convergence in Measure -- 7.5 Equivalence of the Norms -- 7.6 Duality Theorems -- 7.7 Reflexivity -- 7.8 Separability and Bases of Orlicz Spaces -- 7.8.1 Separability -- 7.8.2 Bases -- 7.9 Amemiya formula and Orlicz Norm.

8 Differentiability and Convexity in Orlicz Spaces -- 8.1 Flat Convexity and Weak Differentiability -- 8.2 Flat Convexity and Gâteaux Differentiability of Orlicz Spaces -- 8.3 A-differentiability and B-convexity -- 8.4 Local Uniform Convexity, Strong Solvability and Fréchet Differentiability of the Conjugate -- 8.4.1 E-spaces -- 8.5 Fréchet differentiability and Local Uniform Convexity in Orlicz Spaces -- 8.5.1 Fréchet Differentiability of Modular and Luxemburg Norm -- 8.5.2 Fréchet Differentiability and Local Uniform Convexity -- 8.5.3 Fréchet Differentiability of the Orlicz Norm and Local Uniform Convexity of the Luxemburg Norm -- 8.5.4 Summary -- 8.6 Uniform Convexity and Uniform Differentiability -- 8.6.1 Uniform Convexity of the Orlicz Norm -- 8.6.2 Uniform Convexity of the Luxemburg Norm -- 8.7 Applications -- 8.7.1 Regularization of Tikhonov Type -- 8.7.2 Ritz's Method -- 8.7.3 A Greedy Algorithm in Orlicz Space -- 9 Variational Calculus -- 9.1 Introduction -- 9.1.1 Equivalent Variational Problems -- 9.1.2 Principle of Pointwise Minimization -- 9.1.3 Linear Supplement -- 9.2 Smoothness of Solutions -- 9.3 Weak Local Minima -- 9.3.1 Carathéodory Minimale -- 9.4 Strong Convexity and Strong Local Minima -- 9.4.1 Strong Local Minima -- 9.5 Necessary Conditions -- 9.5.1 The Jacobi Equation as a Necessary Condition -- 9.6 C1-variational Problems -- 9.7 Optimal Paths -- 9.8 Stability Considerations for Variational Problems -- 9.8.1 Parametric Treatment of the Dido problem -- 9.8.2 Dido problem -- 9.8.3 Global Optimal Paths -- 9.8.4 General Stability Theorems -- 9.8.5 Dido problem with Two-dimensional Quadratic Supplement -- 9.8.6 Stability in Orlicz-Sobolev Spaces -- 9.9 Parameter-free Approximation of Time Series Data by Monotone Functions.

9.9.1 Projection onto the Positive Cone in Sobolev Space -- 9.9.2 Regularization of Tikhonov-type -- 9.9.3 A Robust Variant -- 9.10 Optimal Control Problems -- 9.10.1 Minimal Time Problem as a Linear L1-approximation Problem -- Bibliography -- List of Symbols -- Index.