Variational Methods in Optimization

by Smith, Donald R.

• ISBN13: 9780486404554
• ISBN10: 0486404552
• Format: Paperback
• Copyright: 1998-06-19
• Publisher: Dover Publications

Summary

Highly readable text elucidates applications of the chain rule of differentiation, integration by parts, parametric curves, line integrals, double integrals, and elementary differential equations. Clear, well-illustrated treatment addresses optimization problems in a diverse array of fields. Only basic knowledge of calculus required. 1974 edition.

Table of Contents

Functionals

Introduction; Examples of Optimizational Problems

Vector Spaces

Functionals

Normed Vector Spaces

Continuous Functionals

Linear Functionals

A Fundamental Necessary Condition for an Extremum

Introduction

A Fundamental Necessary Condition for an Extremum

Some Remarks on the Gateaux Variation

Examples on the Calculation of Gateaux Variations

An Optimization Problem in Production Planning

Some Remarks on the Frechet Differential

The Euler-Lagrange Necessary Condition for an Extremum with Constraints

Extremum Problems with a Single Constraint

Weak Continuity of Variations

Statement of the Euler-Lagrange Multiplier Theorem for a Single Constraint

Three Examples, and Some Remarks on the Geometrical Significance of the Multiplier Theorem

Proof of the Euler-Lagrange Multiplier Theorem

The Euler-Lagrange Multiplier Theorem for Many Constraints

An Optimum Consumption Policy with Terminal Savings Constraint During a Period of Inflation

The Meaning of the Euler-Lagrange Multipliers

Chaplygin's Problem, or a Modern Version of Queen Dido's Problem

The John Multiplier Theorem

Applications of the Euler-Lagrange Multiplier Theorem in the Calculus of Variations

Problems with Fixed End Points

John Bernoulli's Brachistochrone Problem, and Brachistochrones Through the Earth

Geodesic Curves

Problems with Variable End Points

How to Design a Thrilling Chute-the-Chute

Functionals Involving Several Unknown Functions

Fermat's Principle in Geometrical Optics

Hamilton's Principle of Stationary Action; an Example on Small Vibrations

The McShane-Blankinship Curtain Rod Problem

Functionals Involving Higher-Order Derivatives

Functionals Involving Several Independent Variables; the Minimal Surface Problem

The Vibrating String

Applications of the Euler-Lagrange Multiplier Theorem to Problems with Global Pointwise Inequality Constraints

Slack Functions and Composite Curves

An Optimum Consumption Policy with Terminal Savings Constraint Without Extreme Hardship

A Problem in Production Planning with Inequality Constraints

Applications of the Euler-Lagrange Multiplier Theorem in Elementary Control Theory

Introduction

A Rocket Control Problem: Minimum Time

A Rocket Control Problem: Minimum Fuel

A More General Control Problem

A Simple Bang-Bang Problem

Some Remarks on the Maximum Principle and Dynamic Programming

The Variational Description of Sturm-Liouville Eigenvalues

Introduction to Sturm-Liouville Problems

The Relation Between the Lowest Eigenvalue and the Rayleigh Quotient

The Rayleigh-Ritz Method for the Lowest Eigenvalue

Higher Eigenvalues and the Rayleigh Quotient

The Courant Minimax Principle

Some Implications of the Courant Minimax Principle

Further Extensions of the Theory

Some General Remarks on the Ritz Method of Approximate Minimization

Some Remarks on the Use of the Second Variation in Extremum Problems

Higher-Order Variations

A Necessary Condition Involving the Second Variation at an Extremum

Sufficient Conditions for a Local Extremum

The Cauchy and Schwarz Inequalities

An Example on Normed Vector Spaces

An Integral Inequality

A Fundamental Lemma of the Calculus of Variations

Du Bois-Reymond's Derivation of the Euler-Lagrange Equation

A Useful Result from Calculus

The Construction of a Certain Function

The Fundamental Lemma for the Case of Several Independent Variables A

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