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1991 | Book

No Major Constraints Read chapter Read first chapter

Extrema of Smooth Functions

With Examples from Economic Theory

Author: Professor Dr. Mohamed Ali El-Hodiri

Publisher: Springer Berlin Heidelberg

Included in: Professional Book Archive

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About this book

It is not an exaggeration to state that most problems dealt with in economic theory can be formulated as problems in optimization theory. This holds true for the paradigm of "behavioral" optimization in the pursuit of individual self interests and societally efficient resource allocation, as well as for equilibrium paradigms where existence and stability problems in dynamics can often be stated as "potential" problems in optimization. For this reason, books in mathematical economics and in mathematics for economists devote considerable attention to optimization theory. However, with very few exceptions, the reader who is interested in further study is left with the impression that there is no further place to go to and that what is in these second hand sources is all these is available as far as the subject of optimization theory is concerned. On the other hand the main results from mathematics are often carelessly stated or, more often than not, they do not get to be formally stated at all. Furthermore, it should be well understood that economic theory in general and, mathematical economics in particular, must be classified as special types of applied mathematics or, more precisely, of motivated mathematics since tools of mathematical analysis are used to prove theorems in an economics context in the manner in which probability theory may be classified. Hence, rigor and correct scholarship are of utmost importance and can not be subject to compromise.

Table of Contents

Frontmatter

Finite Dimensional Problems

Frontmatter
Chapter 1. No Major Constraints
Abstract
The mathematical content of this chapter is well known, and can be found in any reasonable book on multivariable calculus, e.g. Fleming (1977). Nevertheless, the chapter serves two purposes. First, it presents a wish list, an ideal for which we address the smooth optimization problem comprehensively. We pose certain questions more easily asked now than later. Second, the chapter samples some of the applications of smooth optimization theory to economic analysis.
Mohamed Ali El-Hodiri
Chapter 2. Equality Constraints
Abstract
By way of introducing the problem we deal with in this chapter, consider the problem of maximizing a smooth function f(x1, x2) subject to g(x1, x2) = 0, where g is also smooth and where f and g are real valued. Suppose x̂ provides a local solution to this problem. If (ĝx1,ĝx2) ≠ 0 then we can apply the implicit function theorem to solve for, say, x2 uniquely in terms of x1. Thus we have g(x1,ξ(x1)) ≡ 0 in a neighborhood of x̂1. So the constraint is always satisfied in that neighborhood. Our problem now is to maximize φ(x1)= f(xl, ξ(x1)) locally and with no constraints in a sense to be made precise presently. By the 1st order necessary condition of Chapter 1 we have: f̂1 + f̂2ξ̂′ = 0. But g(x1, ξ(x1)) is a constant function around x̂1. Thus ĝ1 + ĝ2ξ̂′ = 0. Solving for ξ̂′ we get: ξ̂′ = -ĝ1/ĝ2.
Mohamed Ali El-Hodiri
Chapter 3. Inequalities as Added Constraints
Abstract
We discuss here the problem of maximizing a real valued function f: Rn → R subject to equality constraints g(x) = 0, where g: Rn → Rm and subject to inequality constraints h(x) ≥ 0, where h: Rn → R1. We refer to the problem as the Equality-Inequality Constrained Maximization Problem (EICP). Denoting by C1 the set {x ∈ Rn|g(x) = 0} and by C2 the set {x ∈ Rn|h(x) ≥ 0), we define a local solution of EICP as a point x̂ ∈ C1 ∩ C2 such that f(x̂) ≥ f(x) for x ∈ N(x̂) ∩ C1 ∩ C2 where N(x̂) = Rn then x is a global solution of EICP. In either case we write x̂ = argu max(f) on N ∩ C1 ∩ C2.
Mohamed Ali El-Hodiri
Chapter 4. Extensions and Applications
Abstract
In this chapter we extend the results of Chapter 3 to deal with the case of vector maxima. But first we relate the results of Chapter 3 to saddle value problems. As we have pointed out, no new results are presented. But we have a unified and, hopefully, more direct treatment of the problems.
Mohamed Ali El-Hodiri

Variational Problems

Frontmatter
Chapter 5. The Problem of Bolza with Equality Constraints
Abstract
We shall state, here, some theorems that characterize solutions to the problem of Bolza in the calculus of variations. The proofs of some of these theorems will only be briefly outlined. The reader may refer to Bliss (1930), (1938), and (1946) and to Pars (1962) for a more detailed presentation. By way of introduction we discuss an unconstrained problem in the calculus variation, in section 1. In section 2, we state the problem of Bolza. In section 3 we discuss first order necessary conditions and in section 4 we state the necessary conditions of Weierstrass, Clebsch, and Mayer. In section 5, we state second order sufficient conditions. In section 6, we characterize solutions to problem A’.
Mohamed Ali El-Hodiri
Chapter 6. The Problem of Bolza with Equality-Inequality Constraints
Abstract
In this chapter, we study the problem of Bolza with added inequality constraints. We shall use the theorems of chapter 5 to prove our characterization of the present problem: Problem A″.
Mohamed Ali El-Hodiri
Chapter 7. Extensions and Applications
Abstract
In this chapter we note some extensions of the theorems of chapter 6 and some applications. We first state a problem in optimal control and characterize its solutions as applications of chapter 6. The optimal control problem with scalar criterion is presented in section 1. In section 2 we present extensions of the control problem to: a) problems with time lags, b) problems with bounded state variables and, c) problems with finite vector criteria. For these problems we discuss only the first order necessary conditions and the Weierstrass conditions. We then present some economic applications.
Mohamed Ali El-Hodiri
Backmatter
Metadata
Title
Extrema of Smooth Functions
Author
Professor Dr. Mohamed Ali El-Hodiri
Copyright Year
1991
Publisher
Springer Berlin Heidelberg
Electronic ISBN
978-3-642-76793-7
Print ISBN
978-3-642-76795-1
DOI
https://doi.org/10.1007/978-3-642-76793-7