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For which triples (a, b, c) of real numbers is the function f (x) = ax2

INSTRUCTIONS TO CANDIDATES
ANSWER ALL QUESTIONS

Problems

Question 1.

(a) Prove  that,  if  f, g  :  Rn  ! R are  convex  functions  and  t  ≥  0,  then  the  function  h(x)  =

f (x)+ tg(x) is convex.

(b) For which triples (a, b, c) of real numbers is the function f (x) = ax2 + bx + c convex? Prove your answer without using any calculus.

(c) For which integers n ≥ 1 is the function f (x) = xn convex? You may use the  fact  that  a function f  : R ! R such that f 00(x) ≥ 0 for all x is convex, but not the converse of this fact.

(d) Is it true that the product of two convex functions f, g : R ! R is always convex? Prove or give a counterexample.

(e) Bonus: Give an example of a function f : R ! R such that f ( 1 x1 + 1 x2) 1 f (x1)+ 1 f (x2)

for all x1, x2 2 R, but f is not convex. (Hint : this is a very very weird function. The fact that R is a vector space over Q is relevant to its construction.).

 

Question 2.

 

Let f : Rn ! R be a convex function. In this question, we want to

 

prove that if M 2 R and f (x) M for all x, then f is a constant function.

(a) Prove that f (x¯ + t(y¯ — x¯)) ≥ f (x¯)+ t(f (y¯) — f (x¯)) for all x¯, y¯ 2 Rn  and t ≥ 1.

(b) Suppose that f (x)      M  for all x 2 Rn, but f  is not constant.  Then there exist a, b 2 Rn  with f (a) < f(b). Use this fact, together with part (a), to find some c with f (c) > M , giving a contradiction.

Question 3.

(a) Let f, g1,. .., gm be convex functions from Rn ! R.  Prove that the set of optimal solutions to the NLP {min f (x) : g1(x) 0, g2(x) 0,. .., gm(x) 0} is a convex set.

(b) Let f  : Rn  ! R be a convex function.  Prove that, if λ1, λ2,. .., λk satisfy Pk λi = 1 and

 

λi ≥ 0 for all i, then

 

 

k

f λixi  

i=1

 

 

k

λif (xi).

i=1

 

(c) Let  f, g  :  Rn  ! R be  convex  functions  with  epigraphs  epi(f )  and  epi(g).   Find  a  function

h : Rn ! R for which epi(h) = epi(f ) \ epi(g), and prove your answer.

 

 

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