Variational Methods In Optimization

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. Gives an elementary exposition of an extension of the standard differentiation method which leads to an increase in both the number and type of problems that can be solved. Covers problems with fixed endpoints, variable endpoints, isoperimetric constraints, global inequality constraints, the inverse function theorem, and the chain rule of differentiation. Intended for use as a stand-alone text or as a supplement to courses in applied economics, the calculus of variations, optimal control theory, or terminal calculus. An unabridged republication of the work originally published in 1974 by Prentice-Hall, Inc. Annotation c. by Book News, Inc., Portland, Or. 1. Functionals ? 1.1 Introduction; Examples of Optimizational Problems ? 1.2 Vector Spaces ? 1.3 Functionals ? 1.4 Normed Vector Spaces ? 1.5 Continuous Functionals ? 1.6 Linear Functionals 2. A Fundamental Necessary Condition for an Extremum ? 2.1 Introduction ? 2.2 A Fundamental Necessary Condition for an Extremum ? 2.3 Some Remarks on the G?teaux Variation ? 2.4 Examples on the Calculation of G?teaux Variations ? 2.5 An Optimization Problem in Production Planning ? 2.6 Some Remarks on the Fr?chet Differential 3. The Euler-Lagrange Necessary Condition for an Extremum with Constraints ? 3.1 Extremum Problems with a Single Constraint ? 3.2 Weak Continuity of Variations ? 3.3 Statement of the Euler-Lagrange Multiplier Theorem for a Single Constraint ? 3.4 Three Examples, and Some Remarks on the Geometrical Significance of the Multiplier Theorem ? 3.5 Proof of the Euler-Lagrange Multiplier Theorem ? 3.6 The Euler-Lagrange Multiplier Theorem for Many