[SOLVED] Optimization and Algorithms 2022 Exam Python

Optimization and Algorithms

February 22, 2022

Exam

1. Simple convex function. (3 points) One of the following six functions R → R is convex:

(A) (1 − (x − 1)+)+

(B) |(x − 1)+ − 1|

(C) -(1 − (x − 1)+)+

(D) ((x − 1)+ − 1)+

(E) -((x − 1)+ − 1)+

(F) -|(x − 1)+ − 1|

Which one?

Write your answer (A, B, C, D, E, or F) in the box at the top of page 1

2. Least-squares. (2 points) Consider the following six optimization problems:

In each of the six problems above, the variable to optimize is x ∈ Rn . The matrix A and the vector b are given. The scalar ρ > 0 is also given.

One of the optimization problems above is a least-squares problem.

Which one?

Write your answer (A, B, C, D, E, or F) in the box at the top of page 1

3. Convex function. (3 points) Let f : Rn → R be a convex function. One of the following functions is guaranteed to be convex:

(A) |f(x)|

(B) f(x) + (f(x))2

(C) (f(x))2

(D) f(x)(f(x))2

(E) |f(x)| + (f(x))2

(F) f(x) + |f(x)|

Which one?

Write your answer (A, B, C, D, E, or F) in the box at the top of page 1

4. Robust portfolio selection. (4 points) A problem that often occurs in finance has the following form.

where the variable to optimize is x ∈ Rn .

The matrices V1 ∈ Rp×n , V2 ∈ Rp×n , and D ∈ Rp×p are given, the matrix D being diagonal with positive entries in the diagonal:

with di > 0 for i = 1, . . . , p.

The vectors µ1 ∈ Rn , µ2 ∈ Rn and the scalar α ∈ R are given. Finally, recall that the symbol 1 stands for the vector of dimension n with all components equal to one:

Show that the optimization problem (1) is convex.

5. Mahalanobis projection. (4 points) Consider the optimization problem

where the variable to optimize is x ∈ Rn . The vector µ ∈ Rn and the matrix Σ ∈ Rn×n are given, with Σ being symmetric and positive definite.

Show that the optimal value of problem (2) is

6. Strictly convex functions. (4 points) Suppose that the functions f1 : Rn → R and f2 : Rn → R are both convex, and let f : Rn → R be defined as f(x) = max{f1(x), f2(x)} for each x ∈ Rn . Is the function f strictly convex? If you think the answer is ‘yes’, then prove it; if you think the answer is ‘no’, then give a counter-example.