Synopsis

Abstract

This chapter introduces a book that focuses on variational methods. Variational methods refer to the technique of optimization in which the object is to find the maximum or minimum of integral involving unknown functions. The technique is central to the study of functional analysis in the same way that the theory of maxima and minima are central to the study of calculus. Calculus of variations had its beginnings in the seventeenth century when Newton used it for choosing the shape of a ship's hull to assure minimum drag of water. Several great mathematicians, including Jean Bernoulli, Leibnitz, and Euler, contributed to its development. The concept of variation was introduced by Lagrange. The book introduces the basic ideas of the classical theory of calculus of variations and obtains the necessary conditions for an optimum. Such conditions are known as Euler or Euler-Lagrange equations. The book also describes variational techniques used in stochastic control problems, controlled Markov chains, and stopping problems, as well as the application of variational methods to many other statistical problems, such as in obtaining efficiencies of nonparametric tests. © 1976, Academic Press, Inc.