The Mathematics of Financial Modelingand Investment Management

7-Optimization Page 212 Wednesday, February 4, 2004 12:50 PM

212 The Mathematics of Financial Modeling and Investment Management

■ If the matrix D is such that the problem is bilinear, that is, the variables

x can be split into two subvectors such that the problem is linear when

one of the two subvectors is fixed, then the QP problem is bilinear.

There are efficient algorithms for solving this problem.

■ If the matrix D is indefinite, that is, it has both positive and negative

eigenvalues, then the QP problem is very difficult to solve. Depending

on the matrix D, the complexity of the problem might grow exponen-

tially with the number of variables.

Many modern software optimization packages have solvers for several

of these problems.

CALCULUS OF VARIATIONS AND OPTIMAL CONTROL THEORY

We have thus far discussed the problem of finding the maxima or min-

ima of a function of n real variables. The solution to these problems is

typically one point in a domain. This formulation is sufficient for prob-

lems such as finding the optimal composition of a portfolio for a single

period of a finite horizon: An investment is made at the initial time and

a payoff is received at the end of the period. However, many other

important optimization problems in finance require finding an optimal

function or path throughout time and over multiple periods. The mathe-

matical foundation for problems whose solution requires finding an

optimal function or path of this kind is the calculus of variations. The

basic setting of the calculus of variations is the following. An infinite set

of admissible functions y = f(x), x 0 ≤x ≤x 1 is given. The end points

might vary from curve to curve. Let’s assume all curves are differentia-

ble in the given interval [x 0 ,x 1 ]. A function of three variables F(x,y,z) is

given such that the integral

x 1

Jy = ∫F xyy( ′ ,, ) xd

x 0

is well defined where y′= dy/dx. The value of J depends on the curve y. The

basic problem of the calculus of variations is to find the curve y = f(x) that

minimizes J. This problem could be easily reformulated in many variables.

One strategy for solving this problem is the following. Any solution

y = f(x) has the property that, if we slightly displace the curve y, the

integral assumes higher values. Therefore if we parameterize parallel

displacements with a variable ε(denoting by {yε} the collection of all