Math 164: Homework #3, due on Wednesday, April 22
[1] Prove that a function fis concave if and only if −fis convex.
[2] Let fbe a convex function on the convex set Sof Rn. Let kbe a nonzero
scalar, and define g(x) = kf (x). Prove that if k > 0 then gis a convex
function on S, and if k < 0 then gis a concave function of S.
[3] Consider a feasible region Sdefined by a set of linear constraints
S={x:Ax ≤b},
where Ais an m×nmatrix and bis a column vector.
(a) Prove that Sis convex.
(b) Derive the conditions that must be satisfied by a feasible direction p
at a point x∈S.
[4] Consider the problem
minimize f(x)
subject to x1+ 2x2+ 3x3= 6, x1≥0, x2≥0, x3≥0.
(a) Find the sets of all feasible directions at points xa= (0,0,2)T,xb=
(3,0,1)T, and xc= (1,1,1)T.
(b) Using (a), verify that p= (3,0,−1)Tis a feasible direction for xc=
(1,1,1)T; then find an upper bound on the step length αso that xc+αp is
a feasible point, with p= (3,0,−1)T.
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