The Mathematics of Financial Modelingand Investment Management

7-Optimization Page 206 Wednesday, February 4, 2004 12:50 PM

206 The Mathematics of Financial Modeling and Investment Management

the above equations together may be written as

∇F = 0

or

∂F ∂F ∂F ∂F

--------- = ... = ---------= --------- = ... = ---------- =^0

∂x 1 ∂xn ∂λ 1 ∂λm

In other words, the method of Lagrange multipliers transforms a con-

strained optimization problem into an unconstrained optimization

problem. The method consists in replacing the original objective func-

tion f to be optimized subject to the constraints g with another objective

function

m

F = f – ∑λjgj

j = 1

to be optimized without constraints in the variables (x 1 ,...,xn,λ 1 ,...,λm).

The Lagrange multipliers are not only a mathematical device. In many

applications they have a useful physical or economic interpretation.

NUMERICAL ALGORITHMS

The method of Lagrange multiplers works with equality constraints,

that is, when the solution is constrained to stay on the surface defined

by the constraints. Optimization problems become more difficult if ine-

quality constraints are allowed. This means that the admissible solu-

tions must stay within the boundary defined by the constraints. In this

case, approximate numerical methods are often needed. Numerical

algorithms or “solvers” to many standard optimization problems are

available in many computer packages.

Linear Programming

The general form for a linear programming (LP) problem is as follows.

Minimize a linear objective function

fx( 1 , ..., xn ) = c 1 x 1 + ... + c xn n

or, in vector notation,