7-Optimization Page 216 Wednesday, February 4, 2004 12:50 PM
216 The Mathematics of Financial Modeling and Investment ManagementSUMMARY
■ Optimizing means finding the maxima or minima of a function or of a
functional.
■ Optimization is a fundamental principle of financial decision-making
insofar as financial decisions are an optimal trade-off between risk and
return.
■ The partial derivatives of an unconstrained function vanish at maxima
and minima.
■ The maxima and minima of a function subject to equality constraints
can be found equating to zero the derivatives of the corresponding
Lagrangian function, which is the sum of the original function and of a
linear combination of the constraints.
■ If constraints are linear inequalities, the problem can be solved numeri-
cally with the techniques of linear programming, quadratic program-
ming, or nonlinear mathematical programming.
■ There are two major solution strategies for a linear programming prob-
lem: the simplex method and the interior points method.
■ The simplex method searches for a solution by moving on the vertices
of the simplex, that is, the area identified by the constraint equations.
■ The interior points method allows movement in the interior points of
the area identified by the constraint equations.
■ Quadratic and, more in general, nonlinear optimization problems are
more difficult to solve and more computationally intensive.
■ Functionals are functions defined on other functions.
■ Calculus of variations deals with the problem of finding those func-
tions that optimize a functional.
■ Control theory deals with the problem of optimizing a functional by
controlling some of the variables while other variables are subject to
exogenous dynamics.
■ Bellmann’s Dynamic Programming and Pontryagin’s Maximum Princi-
ple are the key mathematical tools of control theory.
■ Multistage stochastic programming is a set of numerical techniques for
finding the maxima and minima of a functional defined on a stochastic
process.
■ Multistage stochastic optimization is based on formalizing the rules for
recourse, that is, how decisions are made at each stage and on describ-
ing possible scenarios.