STOCHASTIC

Reprinted from Cahiers du Centre d'Etudes de Recherche Operationelle 12,114-122 (1970).
ABSTRACT
CONVEXITY, PSEUDO-CONVEXITY
AND QUASI-CONVEXITY
OF COMPOSITE FUNCTIONS
O.L. MANGASARIAN
Mathematics Research Center
University of Wisconsin, Madison I U.S.A.)
A number of recent results which establish the convexity, pseudo-convexity
or quasi-convexity of ceitain functions are shown to be special cases of the
fact that under suitable conditions a composite function is convex, pseudoconvex
or quasi-convex.
Many of the theoretical results and computational algorithms of
nonlinear programming require that the functions entering the problem
be convex, pseudo-convex or quasi-convex [8, 1, 14, 9, 7, 11, 6]. Accordingly,
a number of papers have been recently published which establish
the convexity, pseudo-convexity or quasi-convexity of certain specific
functions [13, 2, 10,12] and the convexity of a general composite function
[3]. All these results are shown to be simple consequences of the
fact that under suitable conditions a general composite function is
convex, pseudo-convex or quasi-convex.
Preliminaries
R" denotes the n-dimensional Euclidean space, small Latin letters
denote vectors or vector functions, and small Greek letters denote real
numbers or numerical functions. Subscripts denote components of
vectors, superscripts denote specific vectors. If x and y are in R" then xy
denotes their scalar product xly1 + ...+x„y„. If 6 is a differentiable
numerical function on R" that is 9:R"-+R, then W0(x) denotes the
M-dimensional vector of the first partial derivatives of 6, d0(jc)/dx,-,
i = 1, ..., n. If/is a differentiable m-dimensional vector function on R",
that is f:R" -»BT, then Vf(x) denotes the m x n Jacobian matrix of the
first partial derivatives df,(x)/dxj, i= 1, ..., m, j= 1, ..., n. If