Approximation and Optimization of Discrete and Differential Inclusions

Dedication

Preface

Acknowledgments

About the Author

1. Convex Sets and Functions

1.1 Introduction

1.2 Some Basic Properties of Convex Sets

1.3 Convex Cones and Dual Cones

1.4 The Main Properties of Convex Functions

1.5 Conjugate of Convex Function

1.6 Directional Derivatives and Subdifferentials

2. Multivalued Locally Adjoint Mappings

2.1 Introduction

2.2 Locally Adjoint Mappings to Convex Multivalued Mappings

2.3 The Calculus of Locally Adjoint Mappings

2.4 Locally Adjoint Mappings in Concrete Cases

2.5 Duality Theorems for Convex Multivalued Mappings

3. Mathematical Programming and Multivalued Mappings

3.1 Introduction

3.2 Necessary Conditions for an Extremum in Convex Programming Problems

3.3 Lagrangian and Duality in Convex Programming Problems

3.4 Cone of Tangent Directions and Locally Tents

3.5 CUA of Functions

3.6 LAM in the Nonconvex Case

3.7 Necessary Conditions for an Extremum in Nonconvex Problems

4. Optimization of Ordinary Discrete and Differential Inclusions and -Transversality Conditions

4.1 Introduction

4.2 Optimization of Ordinary Discrete Inclusions

4.3 Polyhedral Optimization of Discrete and Differential Inclusions

4.4 Polyhedral Adjoint Differential Inclusions and the Finiteness of Switching Numbers

4.5 Bolza Problems for Differential Inclusions with State Constraints

4.6 Optimal Control of Hereditary Functional-Differential Inclusions with Varying Time Interval and State Constraints

4.7 Optimal Control of HODI of Bolza Type with Varying Time Interval

5. On Duality of Ordinary Discrete and Differential Inclusions with Convex Structures

5.1 Introduction

5.2 Duality in Mathematical Programs with Equilibrium Constraints

5.3 Duality in Problems Governed by Polyhedral Maps

5.4 Duality in Problems Described by Convex Discrete Inclusions

5.5 The Main Duality Results in Problems with Convex Differential Inclusions

6. Optimization of Discrete and Differential Inclusions with Distributed Parameters via Approximation

6.1 Introduction

6.2 The Optimality Principle of Boundary-Value Problems for Discrete-Approximation and First-Order Partial Differential Inclusions and Duality

6.3 Optimal Control of the Cauchy Problem for First-Order Discrete and Partial Differential Inclusions

6.4 Optimal Control of Darboux-Type Discrete-Approximation and Differential Inclusions with Set-Valued Boundary Conditions and Duality

6.5 Optimal Control of the Elliptic-Type Discrete and Differential Inclusions with Dirichlet and Neumann Boundary Conditions via Approximation

6.6 Optimization of Discrete-Approximation and Differential Inclusions of Parabolic Type and Duality

6.7 Optimization of the First Boundary Value Problem for Hyperbolic-Type Discrete-Approximation and Differential Inclusions

References

Glossary of Notations