STOCHASTIC

the property that linear interpolations never underestimate the functions.
Alternatively, linear supports never overestimate the functions. These and
related characterizations are illustrated in Exercise ME-8. There are two
important properties possessed by problems of minimizing convex functions
subject to constraints having the form that a set of convex functions not
exceed zero. First, the set of feasible points determined by the constraints
is a convex set. Second, local minima are always global minima. For applications,
certain generalizations of the convexity concept are needed.
Mangasarian's first article introduces pseudo-convex functions. These
functions are differentiable and possess both of the properties mentioned
above, yet are not necessarily convex. They are essentially defined by the
property that if a directional derivative is positive (points up), then the function
continues to increase in the given direction. An even wider class of functions,
called quasi-convex, is defined by retaining only the first property. There are
several alternative ways to define quasi-convex functions and they are explored
in Exercise ME-11. These generalized functions along with other related
functions are discussed in Exercises CR-15 and ME-9. In Exercises CR-8
and 9 the reader is asked to investigate the convexity and generalized convexity
properties of simple functions in one and several dimensions, respectively.
Exercise ME-12 shows how one can verify whether or not a given function
is a convex or generalized convex function by examining a Hessian matrix
of second partial derivatives or a Hessian matrix bordered by first
partials.
The most useful results concerning minimization problems are surely the
Kuhn-Tucker necessary and sufficient conditions. In Mangasarian's article
it is shown that the Kuhn-Tucker conditions are sufficient if the objective is
pseudo-convex and the constraints are quasi-concave. The conditions are
necessary if a mild constraint qualification is met. This is discussed in Exercise
ME-13, where a proof of their necessity that utilizes Farkas' lemma is outlined.
Exercises CR-10 and 13 illustrate how the Kuhn-Tucker conditions may be
used- to solve simple mean-variance tradeoff models and other optimization
problems, respectively. The determination of the correct sign of the multipliers
is considered in Exercise CR-16. The Kuhn-Tucker conditions are intimately
related to the Lagrange function. Exercise ME-10 illustrates the relationship
between saddle points of the Lagrangian and solutions of the minimization
problem. The primal problem may be considered as a minimax of the
Lagrangian where the min is with respect to original (primal) variables, and
the max is with respect to Lagrange (dual) variables. A dual problem results
when one maximins. When the functions are convex and differentiable one
obtains the Wolfe dual. Mangasarian presents some results related to this
dual problem in his first paper; other results are developed in Exercise ME-14.
The reader is asked to determine the actual dual problems in some special
INTRODUCTION 5