Prof. Friedrich Eisenbrand Question session: 27.10.10 Location: MA A3 31 Discussion: 03.11.10

Exercises Optimization Methods in Finance Fall 2010 Sheet 3

Exercises marked with (*) qualify for one bonus point, if correctly presented in the discussion session Exercise 3.1 (*) Consider the optimization problem

2

min x · 1

µ´ µ 

´x 2 x 4 0 Ê x ¾

i) Analysis of primal problem. Give the feasible set, the optimal value and the optimal solution.

2

ii) Lagrangian and dual function. Plot the function x · 1 versus x. One the same plot, show the

; λ µ

feasible set, optimal point and value, and plot the Lagrangian L´x versus x for a few positive

£

λ ´ ; λ µ λ  values of . Verify the lower bound property (p  infx L x for 0). Derive and sketch the Lagrange dual function g.

iii) Lagrange dual problem. State the dual problem, and verify that it is a concave maximization

problem. Find the dual optimal value and dual optimum solution λ £ . Does strong duality hold?

Exercise 3.2 (*) In this exercise, we want to show an example of a convex program, where strong duality fails. Consi- der the optimization problem

min e x

2 

x =y 0

; µ ¾ ´x y D

2

f´ ; µ ¾ Ê j > g with D := x y y 0 .

i) Verify that this is a convex optimization problem. Find the optimal value. £ ii) Give the Lagrange dual problem, and find the optimal solution λ £ and optimum value d of the dual program. What is the optimal duality gap?

iii) Does Slater’s condition hold for this problem? Exercise 3.3 (*) In this exercise, we want to argue, why the RWMA (which can minimize convex functions over

Σm m m f ;:::; g = fλ ¾ Ê j λ = ; λ  g the simplex := conv e e ∑ 1 0 ) can also be used to optimize 1 m i=1 i i over general polytopes. Here, we are motivated, since the minimum variance portfolio problem is a

N N N N

¾ Ê j = ;  ;  g

convex optimization problem over the domain fx ∑ x 1 ∑ r x r x 0 which is ∑ = i=1 i i 1 i i N intersected with the half-space ∑ r x  r. i=1 i i

n m m

¾ Ê = f ;:::; g = f λ j λ = ; λ ;:::; λ  g

Let v ;:::; v and let Q : conv v v : ∑ v ∑ 1 0 . Define = 1 m 1 m i=1 i i i 1 i 1 m

m m

´λ µ = λ ! Ê q : Σ ! Q with q ∑ x . Let f : Q be a convex function. Show that i=1 i i

m

Ê ´λ µ = ´ ´λ µµ i) The function g : Σ ! with g f q is convex.

ii) One has

λ µ = ´ µ min g´ min f x m Σ ¾ λ ¾ x Q

N N N 2

¾ Ê j  g · iii) Describe ∑ \fx ∑ r x r as the convex hull of at most N N points and con- i=1 i i

T

j ¾ clude that the RWMA can be used to solve the portfolio optimization problem minfx Qx x

N N N

¾ Ê j  g g ∑ \fx ∑ r x r . i=1 i i

Exercise 3.4 (*)

n

Ê ;:::; ! Ê Let D  be a convex set and f0 fm : D be convex functions. Show that the set

m

f´ ; µ ¾ Ê ¢ Ê j 9 ¾ ´ µ  ; ´ µ  g A = u t x D : fi x ui f0 x t is convex.

Exercise 3.5 (*)

n

! Ê  Ê Let f : D be a convex function for some convex domain D . Show that

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µ ´ µ  ¾ i) The function f ´x is convex, given that f x 0 for all x D.

m¢n m

· µ ¾ Ê ¾ Ê ii) f ´Ax b is convex for any A and b .

n T n¢n

! Ê ´ µ = ¡ ¡ ¾ Ê ;  Conclude that a function f : Ê with f x x Q x and Q Q 0 is convex.