Convexity Properties of Some Implicit Functions

Abstract

We consider implicit functions y = y(x) defined by a system of equations

G_i(x,y) = 0

, i=1,...,m. In the case of convex differentiable functions

G_i

we establish some sufficient conditions under which the component function

y_k(x)

is convex or concave. Examples show that without these assumptions

y_k(x)

can be nonconvex and nonconcave. For the special case with additive separated convex functions

G_i(x,y) = g_i(x) + h_i(y)

additional results concerning the gradient vectors of

g_i

and

h_i

are obtained which can be applied to the differentiable continuation of convex marginal functions in parametric optimization.

Contact details are reproduced from the original publication and may be historical.

Fakultät für Mathematik, Technische Universität, 09107 Chemnitz, Germany

26B10; 26B25, 52A20, 90C25

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Published by Heldermann Verlag, 2001. Rights now held by Banach Press LLC.