Nonsmooth Optimization

The following sections are included:

• Preface

• List of Symbols

• Contents

The following sections are included:

• Notations

• Basic Results

The following sections are included:

• Functions, Derivatives and Subgradients

• Sets, Tangents and Normals

• Links between Geometry and Differentiation

The following sections are included:

• Generalization of Derivative

• Sub differential Calculus

• ε-sub differentials
  • Convex Case
  • Nonconvex Case

• Generalized Jacobians

The following sections are included:

• Generalization of Tangents and Normals

• Epigraphs and Level Sets of Nonsmooth Functions

The following sections are included:

• Optimality Conditions

• Linearization of Unconstrained Problem

• Linearization of Constrained Problem

Consider the following nonlinear constrained optimization problem

The following sections are included:

• Preliminaries
  • Smooth Minimization Methods
  • Subgradient Methods

• Lemaréchal's ɛ-steepest Descent Method
  • Theoretical Foundation
  • Implementation

• Kiwiel's Generalized Cutting Plane Method
  • The Cutting Plane Method
  • A Generalization of the Cutting Plane Method

• Proximal Bundle and Bundle Trust Region Methods

• The Tilted Proximal Bundle Method

The following sections are included:

• Direction Finding
  • Derivation of the Direction Finding Problem
  • Subgradient Aggregation
  • Nonconvexity

• Line Search

• Stopping Criterion

• Weight Updating

• Algorithm

The following sections are included:

• Test Examples

• Numerical Results

The improvment of industrial processes as well as a new generation of CAD systems (such as computer aided structural design) lead us to use methods of optimal control. In optimal control problems the objective is to optimize (minimize or maximize) certain criteria involving the solution of the state system (partial differential equation for example) with respect to coefficient or the right hand side of the state system (distributed control problems), boundary values (boundary control problems) or with respect to the domain of definition of the state system (optimal shape design). When the discretization of the state system has been done (typically by finite element method in space and finite difference method in time) we are in a position to apply methods of optimization. In practice we often meet optimal control problems which lead to nonsmooth optimization…

The following sections are included:

• Basics for Sensitivity Analysis

• Preliminaries from Functional Analysis

The following sections are included:

• Introduction

• Setting of the Problem. Existence of the Solution

• Practical Model Problems
  • Axially Loaded Rod with Stress Constraints
  • Clamped Beam with Displacement Constraints
  • Clamped Beam with Obstacle

The following sections are included:

• Introduction

• Setting of the Abstract Problem

• Optimal Shape Design with Unilateral Boundary Value Problem
  • Setting of the Problem
  • Discretization of the Problem
  • Design Sensitivity Analysis
  • Numerical Results

• Design of Optimal Covering
  • Setting of the Problem
  • Discretization of the Problem
  • Design Sensitivity Analysis
  • Numerical Results

The following sections are included:

• Introduction

• Setting of the Abstract Problem

• Simulation of the Continuous Casting Process
  • Setting of the Problem
  • Penalization of the State Constraints
  • Discretization of the Problem
  • Sensitivity Analysis
  • Numerical Results

The following sections are included:

• References

• Index