Optimization

A Theory of Necessary Conditions

Lucien W. Neustadt

• Publisher: Princeton University Press
• Year: 2015
• Print ISBN:
  9780691616834
  9780691644042
• eBook ISBN:
  9781400870530
• Topic: Technology & Engineering
• Subtopic: Mathematics General
• Index: Technology & Engineering

This book presents a comprehensive treatment of necessary conditions for general optimization problems. The presentation is carried out in the context of a general theory for extremal problems in a topological vector space setting.

Following a brief summary of the required background, generalized Lagrange multiplier rules are derived for optimization problems with equality and generalized "inequality" constraints. The treatment stresses the importance of the choice of the underlying set over which the optimization is to be performed, the delicate balance between differentiability-continuity requirements on the constraint functionals, and the manner in which the underlying set is approximated by a convex set. The generalized multiplier rules are used to derive abstract maximum principles for classes of optimization problems defined in terms of operator equations in a Banach space. It is shown that special cases include the usual maximum principles for general optimal control problems described in terms of diverse systems such as ordinary differential equations, functional differential equations, Volterra integral equations, and difference equations. Careful distinction is made throughout the analysis between "local" and "global" maximum principles.

Originally published in 1977.

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Table of contents

• Cover
• Contents
• Preface
• Summary Of Notation
• I: Mathematical Preliminaries
• II: A Basic Optimization Problem in Simplified Form
• III: A General Multiplier Rule
• IV: Optimization with Operator Equation Restrictions
• V: Optimal Control Problems with Ordinary Differential Equation Constraints
• VI: Optimal Control Problems with Parameters and Related Problems
• VII: Miscellaneous Optimal Control Problems
• Appendix: Volterra-Type Operators
• Notes and Historical Comments
• References
• Subject Index