STOCHASTIC

CHOOSING INVESTMENT PORTFOLIOS
Lemma 3 Suppose/: A -»R, where A is a convex subset of R", is differentiable
at x, and there is a direction d such that Vf(x)'d > 0. Then a 0
exists such that for all z,a ^ T > 0,/(x + •«/) >/(X).
PTOO/ See Zangwill ([32, p. 24].
Theorem 5 Suppose 0(x): A->P, where A is a convex subset of iJ", is
strictly pseudo-concave. Then the maximum of 9, if it exists, is attained at
most at one point x e A.
Proof Case 1: 3xe A such that V0(x) = 0. Then by the strict pseudoconcavity
of 0, i.e., Vd(x)'(y-x) 5£ 0 => 0(y) < 9(x) for all ye A, x is the
unique maximizer.
Case 2: ^xeA such that V0(x) = 0. Suppose x maximizes 9 over A.
Then \/y # x, V0(x)'(.V—x) ^0 => 9(y)