STOCHASTIC

. CONVEXITY AND THE KUHN-TUCKER
CONDITIONS
J.SIAM CONTROL
Ser. A, Vol. 3, No. 2
Printed in U.S.A., 1065
PSEUDO-CONVEX FUNCTIONS*
O. L. MANGASARIANf
Abstract. The purpose of this work is to introduce pseudo-convex functions and
to describe some of their properties and applications. The class of ail pseudo-convex
functions over a convex set, C includes the class of all differentiate convex functions
on C and is included in the class of all differentiable quasi-convex functions on C.
An interesting property of pseudo-convex functions is that a local condition, such as
the vanishing of the gradient, is a global optimality condition. One of the main
results of this work consists of showing that the Kuhn-Tucker differential conditions
are sufficient for optimality when the objective function is pseudo-convex and the
constraints are quasi-convex. Other results of this work are a strict converse duality
theorem for mathematical programming and a stability criterion for ordinary differential
equations.
1. Introduction. Throughout this work, we shall be concerned with the
real, scalar, single-valued, differentiable function 0(x) defined on the nonempty
open set D in the ni-dimensional Euclidean space E m . We let C be a
subset of D and let V^ denote the m X 1 partial differential operator
v. = l± ... A
\_dxr ' ' dx, •
where the prime denotes the transpose. We say that 8(x) is pseudo-convex
on C if for every x and x" in C,
(1.1) (x - x'YvJix 1 ) g 0 implies 0(x 2 ) g B{x).
We say that 6(x) is pseudo-concave on C if for every x and x in C,
(1.2) (x 2 - x)\j{x) § 0 implies 0(x 2 ) i 0(x').
Thus 0(x) is pseudo-concave if and only if — 6(x) is pseudo-convex. In
the subsequent paragraphs we shall confine our remarks to pseudo-convex
functions. Analogous results hold for pseudo-concave functions by the
appropriate multiplication by —1.
We shall relate the pseudo-convexity concept to the previously established
notions of convexity, quasi-convexity [1], [2] and strict quasi-convexity
[3], [5].
The function 0(x) is said to be convex on C, [2], if C is convex and if for
every x' and x 2 in C,
(1.3) 0(Xx' + (1 - X)x 2 ) g X0(x') + (1 - X)0(x 2 )
* Received by the editors March 4, 19G5.
f Shell Development Company, Emeryville, California.
CONVEXITY AND THE KUHN-TUCKER CONDITIONS