MAT591Optimization代写 MAT591Homework代写

MAT591: Optimization (Spring 2020) Homework 3

MAT591Optimization代写 Problem 1 (20 points)：Consider a random variable x ∈ Rᵏ with values in a fifinite {a1, a2, . . . , an}, and with distribution

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Consider a random variable x ∈ Rᵏ with values in a fifinite {a1, a2, . . . , an}, and with distribution

pᵢ = prob(x = aᵢ), i = 1, 2, . . . , n.

Show that a lower bound on the covariance of X,

S ≤ E(X − EX)(X − EX)ᵀ

is a convex constraint in p.

(a) Show that if P is not positive semi-defifinite, i.e. the objective function f is not convex then the problem is unbounded below

(b) Suppose that P ≥ 0 (so f is convex) but the optimality condition P x∗ = −q does not have a solution. Show that the problem is unbounded below.

Solution: [Type your solution here.]

Assume that rank(A) = n and b /∈ R(A)

(a) Show that f is closed

(b) Show that the minimizer x∗ of f is given by

x∗ = x₁ + tx₂

where x₁ = (Aᵀ A)﹣¹Aᵀ b, x₂ = (Aᵀ A)﹣¹c and t ∈ R can be calculated by solving a quadratic equation.

Solution: [Type your solution here.]

Solution: [Type your solution here.]

Complete the following questions.

(a) Explain how to fifind a steepest descent direction in the ℓ∞-norm, and give a simple interpreta-tion.

(b) Write down the defifinition of the Newton decrement λ(x) and work out its explicit form.

Solution: [Type your solution here.]