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Consider f : R n → R and X ⊂ R n nonempty. Show that the set of solutions to max x∈X f (x)

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Scientific Computing 2

Summer term 2017 Prof. Dr. Ira Neitzel Christopher Kacwin

Sheet 0 Submission on -.

Exercise 1. (minimization/maximization problems)

Consider f : R n → R and X ⊂ R n nonempty. Show that the set of solutions to max x∈X f (x)

and

min x∈X (−f (x)) are identical.

(0 points) Exercise 2. (sublevel sets)

Consider the functions

f : R 2 → R , x

y

7→ x 2 − y 2 . and

g : R 2 → R , x

y

7→ x 2 + y 2 .

a) Draw the level sets of f and g. Do they attain their minima/maxima on D := {(x, y) ∈ R 2 | x 2 + y 2 ≤ 1}? Add them to your drawings.

b) Show that a continuous function f : R n → R with lim kxk→∞ f (x) = ∞ has compact sublevel sets N f (w) = {x ∈ R n |f (x) ≤ w}.

c) Show that convex functions f : R n → R have convex sublevel sets. Does the converse also hold true?

(0 points) Exercise 3. (linear regression)

a) Let A ∈ R m,n with m > n and rang A = n, b ∈ R m and consider f : R n 7→ R , f (x) = 1

2 kAx − bk 2 2 . (1)

Assume that the QR-decomposition A = QR is known, where Q ∈ R m,n , Q T Q = I n , and R ∈ R n,n is an upper triangular matrix.

Calculate an optimal solution to

min

x∈ R

2

f(x).

b) Calculate the gradient and Hessian of f . Show that the Hessian is positive semi- definite, and further positive definite iff A is injective.

1

(2)

c) Formulate the linear regression problem in the form (1). show that for m ∈ N the matrix

H = 2

m

X

i=1

ξ i 2

m

X

i=1

ξ i

m

X

i=1

ξ i m

is positive definite if at least two of the ξ i are different. Discuss the relation of A in (1) and H. Solve the linear regression model using the measurements:

ξ i −5 −1 0 1 5 η i 1 4 5 6 9

(0 points) Exercise 4. (a test case)

Consider the following problem:

Find a point x ∈ R 2 such that it minimizes the sum of distances to three given points x 1 , x 2 , x 3 ∈ R 2 .

a) Formulate this problem as an optimization problem and show that a solution x exists. Is it unique?

b) Let x 6= x i , i = 1 . . . 3. Characterize x with the first order necessary condition for optimality and geometrical considerations.

(0 points)

2

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