STOCHASTIC

(Ill) have not, to the author's knowledge, been given before. These cases,
together with (Ii) subsume all the interesting recent results in the literature
[2, 10, 12, 13] in which convexity, pseudo-convexity or quasi-convexity
of certain specific functions were established. (The pseudo-convex case
is of particular interest since many of the theoretical and computational
results of nonlinear programming that were established for convex
functions also hold for pseudo-convex functions.)
Remark 2. Most of the computational algorithms of nonlinear
programming [7, 11, 6] require the convexity of the functions defining the
problem in order to guarantee convergence to a global minimum. Some
of these algorithms however will also converge to a global minimum if
we merely have pseudo-convex functions instead. For example the
Frank-Wolfe algorithm [7] which finds a global minimum of a continously
differentiable function 9 on a polyhedron will also converge to a
global minimum if we merely require 9 to be pseudo-convex on the
polyhedron rather than convex. (See the convergence proof of the
algorithm on page 90 of [5].) Result II of the theorem then extends the
Frank-Wolfe algorithm to functions 9 defined by 9{x) = cp(f(x), g(x))
where assumptions (i), (ii), (iii), or (iv) are satisfied and tp is pseudoconvex.
Some such functions are nonlinear fractional functions and
bi-nonlinear functions which are described in the next section.
Applications
We indicate now how our principal result includes the recent results
of [13,2,10,12] which establish the convexity, pseudo-convexity or
quasi-convexity of certain specific functions.
(A) Nonlinear fractional functions [13, 10, 12]. Let r be a convex set
in R", letp and0 concave > 0
(2) concave < 0 convex < 0
(3) convex < 0 convex > 0
(4) concave > 0 concave < 0
(5) linear linear ^ 0
(6) linear < 0 convex ^ 0
38 PART I. MATHEMATICAL TOOLS