Universität Konstanz Sommersemester 2013 Fachbereich Mathematik und Statistik

Universität Konstanz Sommersemester 2013 Fachbereich Mathematik und Statistik

Prof. Dr. Stefan Volkwein

Roberta Mancini, Stefan Trenz, Carmen Gräßle, Marco Menner, Kevin Sieg

Optimierung

http://www.math.uni-konstanz.de/numerik/personen/volkwein/teaching/

Sheet 3

Deadline for hand-in: 2013/05/27 at lecture Exercise 7

Let b ∈ R ⁿ and A ∈ R ^n×n .

Show that x ₀ ∈ R ⁿ is a minimal point of ϕ : R ⁿ → R , x 7→ ϕ(x) := ||Ax − b|| if and only if the Gaussian normal equation A ^∗ Ax ₀ = A ^∗ b holds.

Here, A ^∗ ∈ R ^n×n denotes the transposed matrix of A, i.e. ∀x, y ∈ R ⁿ : hAx, yi = hx, A ^∗ yi.

Exercise 8

Use the characterization given in Exercise 7 to solve the following linear regression prob- lem:

Find a vector of parameters x = (x 1 , x 2 ) ∈ R ² such that the corresponding regression line γ _x : R → R , defined by γ _x (t) := x ₁ + tx ₂ , approximates the following measuring points

t _i 1975 1980 1985 1990 1995

γ _i 30 35 38 42 44

optimally, i.e. such that x ^∗ = (x ^∗ ₁ , x ^∗ ₂ ) solves the optimization problem:

x ^∗ = argmin

x∈ R

5

X

i=1

(γ _i − γ _x (t _i )) ² .

Exercise 9 (4 Points)

Let a, b, c, d vectors in R ⁿ , with b, d linear independent. Consider the parametrization of two straight lines x(s) : R 7→ R ⁿ and y(t) : R 7→ R ⁿ with

x(s) := a + sb; y(t) := c + td (s, t ∈ R ).

Find the critical points of the distance function

(s, t) 7→ ||x(s) − y(t)||

and characterize them.