Distributed Optimization and Machine Learning - Week 0 Quiz Solutions

Question 1

Let A = \begin{pmatrix} 2 & 1 & 0 \\ 1 & k & 1 \\ 0 & 1 & 2 \end{pmatrix}. Which of the following are true?

A.The determinant of A is 4k-4

B.A is invertible if and only if k ≠ 1

C.If k=1 , the nullity of A is 1

D.The trace of A is k+4

Correct Answers: The determinant of A is 4k-4, A is invertible if and only if k ≠ 1, If k=1 , the nullity of A is 1, The trace of A is k+4

Question 2

Let B = \begin{pmatrix} 3 & 1 & 0 \\ 1 & \alpha & 1 \\ 0 & 1 & 3 \end{pmatrix}. Which of the following statements are true? Hint: A matrix B is symmetric if B = B^T. For positive definiteness, check the signs of the leading principal minors. The sum of eigenvalues equals the trace.

A.B is always a symmetric matrix

B.The determinant of B is 9α - 6

C.B is positive definite if α > 2/3

D.The sum of the eigenvalues of B is 6+α

Correct Answers: B is always a symmetric matrix, The determinant of B is 9α - 6, B is positive definite if α > 2/3, The sum of the eigenvalues of B is 6+α

Question 3

Consider the differential equation \dot{x}(t) = -3x(t) + 6 with x(0) = x_0. Which of the following are true?

A.The solution x(t) approaches 2 as t → ∞

B.If x₀ = 2 , then x(t) = 2 for all t ≥ 0

C.The particular solution is x(t) = (x₀ - 2)e-3t + 2

D.If the constant term were 0 , the solution would be x(t) = x₀ e3t

Correct Answers: The solution x(t) approaches 2 as t → ∞, If x₀ = 2 , then x(t) = 2 for all t ≥ 0, The particular solution is x(t) = (x₀ - 2)e-3t + 2

Question 4

Which of the following functions are convex on their specified domains? Hint: For a twice-differentiable f(x) on \mathbb{R}, f is convex if f''(x) \geq 0. For multivariable functions the Hessian must be positive semi-definite. The pointwise maximum of convex functions is convex.

A.f(x) = e2x for x ∈ \mathbb{R}

B.f(x) = x³ for x ∈ \mathbb{R}

C.f(x,y) = x² + 4y² for (x,y) ∈ \mathbb{R}²

D.f(x) = \max\{x, 2x-1\} for x ∈ \mathbb{R}

Correct Answers: f(x) = e2x for x ∈ \mathbb{R}, f(x,y) = x² + 4y² for (x,y) ∈ \mathbb{R}², f(x) = \max\{x, 2x-1\} for x ∈ \mathbb{R}

Question 5

Let f(x,y) = x^2 + y^3 - 4x - 3y + 5. Which of the following are true?

A.∇ f(x,y) = [2x-4, 3y²-3]T

B.The point (2,1) is a stationary point

C.The Hessian matrix ∇² f(x,y) is \begin{pmatrix} 2 & 0 \\ 0 & 6y \end{pmatrix}

D.The point (2,1) is a local minimizer

Correct Answers: ∇ f(x,y) = [2x-4, 3y²-3]T, The point (2,1) is a stationary point, The Hessian matrix ∇² f(x,y) is \begin{pmatrix} 2 & 0 \\ 0 & 6y \end{pmatrix}, The point (2,1) is a local minimizer

Question 6

If an interior point x^* is a local extremum (minimum or maximum) of a differentiable function f(x), then:

A.∇ f(x^*) = 0

B.∇ f(x^*) ≠ 0

C.For f: \mathbb{R} → \mathbb{R} , f'(x^*) = 0

D.f(x^*) must be the global extremum value of f(x)

Correct Answers: ∇ f(x^*) = 0, For f: \mathbb{R} → \mathbb{R} , f'(x^*) = 0

Question 7

Which of the following is true about a convex function f: \mathbb{R} \to \mathbb{R}?

A.Any local minimum is also a global minimum

B.The set of all global minimizers is a convex set

C.It is always strictly increasing

D.It must be continuous on the interior of its domain

Correct Answers: Any local minimum is also a global minimum, The set of all global minimizers is a convex set, It must be continuous on the interior of its domain

Question 8

For a twice-differentiable function f: \mathbb{R} \to \mathbb{R}, if f'(a) = 0 and f''(a) > 0 at a point x=a, then x=a is a:

A.Local minimizer

B.Local maximizer

C.Inflection point where concavity changes

D.Point where the test is inconclusive

Correct Answer: Local minimizer

Practice: Week 0: Assignment 0