Universität Konstanz Sommersemester 2015 Fachbereich Mathematik und Statistik

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Universität Konstanz Sommersemester 2015 Fachbereich Mathematik und Statistik

Prof. Dr. Stefan Volkwein Sabrina Rogg

Optimierung

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

Sheet 3

Tutorial: 26th May Exercise 8

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|| 2 if and only if the Gaussian normal equation A ^> Ax ₀ = A ^> b holds.

Exercise 9

Use the characterization given in Exercise 8 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

²

5 X

i=1

(γ _i − γ _x (t _i )) ² .

Exercise 10

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

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

Determine the global minimal point of the distance function

(s, t) 7→ ||x(s) − y(t)|| ₂ .

Universität Konstanz Sommersemester 2015 Fachbereich Mathematik und Statistik

Universität Konstanz Sommersemester 2015 Fachbereich Mathematik und Statistik

Prof. Dr. Stefan Volkwein Sabrina Rogg

Optimierung

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

Sheet 3

Tutorial: 26th May Exercise 8

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|| 2 if and only if the Gaussian normal equation A > Ax 0 = A > b holds.

Exercise 9

Use the characterization given in Exercise 8 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 10

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

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

Determine the global minimal point of the distance function

(s, t) 7→ ||x(s) − y(t)|| 2 .

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|| 2 if and only if the Gaussian normal equation A ^> Ax ₀ = A ^> b holds.

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

γ _i 30 35 38 42 44

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

x ^∗ = argmin

(γ _i − γ _x (t _i )) ² .

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

(s, t) 7→ ||x(s) − y(t)|| ₂ .