7-Optimization Page 201 Wednesday, February 4, 2004 12:50 PM

CHAPTER

7

Optimization

T

he concept of optimization is intrinsic to finance theory. The seminal

work of Harry Markowitz demonstrated that financial decision-mak-

ing is essentially a question of an optimal trade-off between risk and

returns. While Markowitz was developing his theory of investment in

the 1950s, as we will see in Chapter 16, Georg Dantzig, the father of

linear programming, was laying down the foundations of the modern

computerized approach to optimization.^1

Purely mathematical solutions to optimization problems were proposed

early in the history of calculus. In the eighteenth century, the French mathe-

matician Lagrange introduced a general methodology for finding the

maxima or minima of a multivariate function subject to constraints; the

Swiss-born mathematician Euler^2 introduced the mathematics of the calculus

of variations.^3 Nevertheless, no matter how important from the concep-

tual point of view, optimization had limited practical applications in

engineering, business, and financial planning until the recent develop-

ment of high-performance computing.

In modern terminology, an optimization problem is called a mathe-

matical programming problem. From an analytical perspective, a static

mathematical program attempts to identify the maxima or minima of a

function f(x 1 ,...,xn) of n real-valued variables, called the objective func-

tion, in a domain identified by a set of constraints. The latter might take

the general form of inequalities gi(x 1 ,...,xn) ≥ bi. Linear programming is

the specialization of mathematical programming to instances where

(^1) Dantzig and Markowitz worked together at the Rand Corporation in the 1950s.
(^2) Euler was born in Basel, Switzerland, but spent a large part of his long career in
Russia.
(^3) The calculus of variations played a fundamental role in the development of modern
science.
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