- GRADIENTS OF CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS 433
for all VE TpM. It follows that
min{(Dvf) (p): VE s;-^1 }::; 0.
3. Generalized gradients of convex functions on Riemannian
manifolds
If a function f is differentiable at a point p E M, then we may define itsgradient '! f (p), which is a tangent vector at p. In this section we discuss the
notion and properties of generalized gradients for locally Lipschitz convex
functions on Riemannian manifolds.
In this section (Mn,g) shall denote a connected complete Riemannian
manifold.3.1. Definition of generalized gradient.
Let C C M be a connected locally convex set and let f : C -+ IR be
a convex function. Given p E C, we assume that f is Lipschitz in some
neighborhood of p. By Lemma H.27(i), Dvf (p) exists and is a Lipschitz
function of V on TpC and we can extend· D f (p) to a Lipschitz function on
its closure TpC.
We make the following definition of a generalized gradient of f at p.
DEFINITION H.29 (Generalized gradient). If p E C and if a convex func-
tion f : C -+ Ris Lipschitz in some neighborhood of p, .then there exists a
unit vector W E T pC n s;-^1 such that
(H.16) (Dw f) (p) =min { (Dv f) (p) : v E TpC n s;-:-^1 } ~ (Dminf) (p).
For any such W, the vector
(H.17) 'VJ (p) = (Dw f) (p)\ WE TpC
is called a generalized gradient of f at p.
Note that Wis a direction of steepest descent and
(H.18) \'VJ (p)\ = \(Dminf) (p)\.Hence a generalized gradient is a direction of steepest descent times the
absolute value of the rate of steepest descent.
The generalized gradient has the following, elementary properties.
LEMMA H.30. Suppose f : C -+ IR is a convex function which is Lipschitz
in a neighborhood of a point p E C.(i) If f is differentiable at p, then '! f (p) is equal to the negative of
the (standard) gradient of f.
(ii) Since W in (H.16) is not necessarily unique, neither is 'V f (p).