Optimization and Algorithms
November 10, 2021
Quiz
1. A basic question. (3 points) Task 1 of the project is about solving the problem
What does Ex(t) represent?
(A) The velocity of the target at time t
(B) The position of our vehicle at time t
(C) The control signal at time t
(D) The velocity of our vehicle at time t
(E) The position of the target at time t
(F) The state of our vehicle at time t
Write your answer (A, B, C, D, E, or F) here: B
2. Levenberg-Marquardt. (4 points) Suppose we use the Levenberg-Marquardt (LM) method to address the optimization problem
where f : Rn → R is given by Assume that the function fp : Rn → R is differentiable for 1 ≤ p ≤ P.
Suppose the current iterate is xk. To get the next iterate, the LM method starts by solving one of the following problems:
Which problem?
Write your answer (A, B, C, D, E, or F) here: C
3. Gradient. (3 points) Consider the function f : R3 → R,
The gradient ∇f (4, 0, 3) is one of the following vectors:
Which one?
Write your answer (A, B, C, D, E, or F) here: F
4. Random target. (3 points) Task 5 of the project is about analyzing an optimization problem of the form.
Recall that A, B, and xinitial are given constants. Also, λ > 0, p1 > 0, p2 > 0, and p1 + p2 = 1.
One of the following statements about problem (3) is true:
(A) The function f is strongly convex
(B) The function f is a quadratic
(C) The solution of the optimization problem (3) when p1 = 0.7, p2 = 0.3, and λ = 1 is the same as the solution of the optimization problem (3) when p1 = 0.7, p2 = 0.3, and λ = 10
(D) The solution of the optimization problem (3) when p1 = 0.7, p2 = 0.3, and λ = 1 is the same as the solution of the optimization problem (3) when p1 = 0.2, p2 = 0.8, and λ = 1
(E) We have x*1
(t) = x*2
(t) and u*1
(t) = u*2
(t) for t = 1, 2, . . . , 24, where the symbol (x*1
, u*1
, x*2
, u*2
) denotes the solution of optimization problem (3)
(F) The function f is not convex
Which one?
Write your answer (A, B, C, D, E, or F) here: D
5. Convexity. (4 points) Consider the optimization problem
where A ∈ Rm×n and b ∈ Rn
. Consider the following choices for the form. of function f:
(A) f(x) = ||x − c||
(B) f(x) = (cTx)2
(C) f(x) = ecTx + ||c||2
(D) f(x) = cTx
(E) f(x) = (x − c)
Tx
(F) f(x) = ||x|| + cTx.
For one of the six forms of f above, the optimization problem (4) is guaranteed to have a unique global minimizer, regardless of how the constants A, b, and c are chosen.
For which form?
Write your answer (A, B, C, D, E, or F) here: E
6. Trade-off. (3 points) Consider the optimization problem
where f and g are nonnegative functions Rn → R (that is, f(x) ≥ 0 and g(x) ≥ 0 for all x ∈ Rn
), and ρ is a positive number.
Suppose that x1 is a global minimizer for (5) when ρ = ρ1, and suppose that x2 is a global minimizer for (5) when ρ = ρ2. Consider that ρ2 > ρ1 > 0.
One of the following inequalities is guaranteed to be true:
(A) f(x1) + ρ1g(x1) > f(x2) + ρ2g(x2)
(B) f(x2) > f(x1)
(C) g(x2) ≥ g(x1)
(D) g(x2) > g(x1)
(E) f(x1) + ρ1g(x1) < f(x2) + ρ2g(x2)
(F) f(x2) ≥ f(x1)
Which inequality?
Write your answer (A, B, C, D, E, or F) here: F