APM462: Homework 1
Due at the beginning of class1
Mon Jan 22, 2018
Recall that a function of the form. f(x) = 12x Qx b x, where Q is positive de nite n n
symmetric matrix and x2Rn, can be written in the form. f(x) = 12(x x ) Q(x x ) 12x Qx ,
where x = Q 1b. This is basically what is called "completing the square". The rst 3 questions
explore di erent functions which essentially have this form.
(1) Let 2R and let f : R2 !R be given as f (x;y) = 2x2 + 2y2 +
xy y.
(a) For each nd the point satisfying the rst order condition for
a local minimum of f .
(b) For which do the points you found in (a) satisfy the second
order condition for a local minimum of f ?
(c) Prove that the candidate you found in (b) for alocal minimun
of f is in fact a global minimum. (Prove this in two di erent
ways: by completing the square and by showing that f is a
convex function.)
(2) Find the local minimum point(s) for this function by nding the
point(s) which satisfy the rst order condition for a local minimum.
f(x;y;z) = (x y2)2 + 34(y 2)2 +z2 3
Prove that your solution is actually a global minimum for f : R3 !
R. (Prove this in two di erent ways: by completing the square and
by showing that f is a convex function.)
(3) Show that any n n matrix of the form. xxT, where xT = (x1;:::;xn)
is a row vector, is positive semide nite.
(4) (a) Let f(x) = b Ax, where A is an n m matrix, x2Rm, and
b2Rn. Show that rf(x) = ATb, where AT is the transpose of
A.
(b) Let f(x) = x Ax, where A is an n n matrix and x 2 Rn.
Show that rf(x) = (A+AT)x.
(5) Let f : R2n !R be de ned as f(x;y) = 12jAx Byj2, where A;B
are m n matrices, x;y2Rn.
(a) Find rf(x;y) and r2f(x;y).
1Copyright c 2017 J. Korman. Sharing this material publically is not allowed without
permission of author.
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(b) Let (x0;y0) be such that Ax0 = By0. Show that (x0;y0) satis es
the necessary rst and second order conditions for a minimizer
of f(x) on = Rn.
(6) Assume that g is a convex function on Rn, that f is a linear function
of a single variable, and in addition thatf is a nondecreasing function
(which means that f(r) f(s) whenever r s).
(a) Show thatF := f g is convex by directly verifying the convexity
inequality
F( x+ (1 )y) F(x) + (1 )F(y):
Explain where each hypothesis (convexity of g, convexity of f,
and the fact that f is nondecreasing) is used in your reasoning.
(The notation F = f g means that F(x) = f(g(x)).)
(b) Now assume that f and g are both C2. Express the matrix of
second derivatives r2F(x) in terms of f and g. Prove directly
(without using part (a)) thatr2F(x) is positive semide nite at
every x. Hint: see problem (3) above.
Discussion: Expressingr2F in terms of f and g is basically an exercise in using
the chain rule for functions of several variables. If you nd it at all di cult, then
review the chain rule until you have completely mastered it!
When showing thatr2F is positive semide nite, please explain again, as you did
in part (a), where each hypothesis is used in your reasoning.
(7) Show that for any unit vector u, uTr2f(x)u, is equal to the second
directional derivative of f at x in the direction u: d2ds2 s=0f(x+su).
(8) Suppose that f : (a;b) !R is C1 on the open interval (a;b) R
and that f0(x) > 0 for all x2(a;b) except for nitely many points
at which f0(x) = 0. Show that f is strictly increasing on (a;b).