Q1. Find the range of values of the parameter a for which the function

f(x₁, x₂, x₃) = 2x₁x₃ - x₁² - x₂² - 5x₃² - 2ax₁x₂ - 4x₂x₃ is concave.

Q2. Consider the function

f(x) = ½x^TQx - x^Tb,

where Q = Q^T > 0 and x, b ∈ Rⁿ. Define the function Φ: R → R by Φ(α) = f(x + αd), where x, d ∈ Rⁿ are fixed vectors and d ≠ 0. Show that Φ(α) is a strictly convex quadratic function of α.

Q3. Show that f(x) = x₁x₂ is a convex function on Ω = {[a, ma]T: a ∈ R}, where m is any given nonnegative constant.

Q4. Suppose that the set Ω = {x : h(x) = c} is convex, where h : Rⁿ → R and c ∈ R. Show that h is convex and concave over Ω.

Q5. Find all sub-gradients of

f(x) = |x|, x ∈ R,

at x = 0 and at x = 1.

Q6. Let Ω ⊂ Rⁿ be a convex set, and f_i: Ω → R, i = 1, . . . , l be convex functions. Show that max {f₁, . . . , f_l} is a convex function.

Note: The notation max{f₁, . . . , f_l} denotes a function from Ω to R such that for each x ∈ Ω, its value is the largest among the numbers f_i(x), i = 1, . . . ,l.

Q7. Let Ω ⊂ Rⁿ be an open convex set. Show that a symmetric matrix Q ∈ Rⁿ is positive semi definite if and only if for each x, y ∈ Ω, (x-y)^TQ(x - y) ≥ 0. Show that a similar result for positive definiteness holds if we replace the "≥" by ">" in the inequality above.

Q8. Consider the problem

minimize ½||Ax-b||²

subject to x₁ + · · · + x_n = 1

               x1, . . . ,x_n ≥ 0

Is the problem a convex optimization problem? If yes, give a complete proof. If no, explain why not, giving complete explanations.

Q9. Consider the optimization problem

minimize f(x)

subject to x ∈ Ω,

where f(x) = x₁x²₂, where x = [x₁, x₂]^T, and Ω = {x ∈ R²: x₁ = x₂, x₁ ≥ 0}. Show that the problem is a convex optimization problem.

Q10. Consider the convex optimization problem

minimize f (x)

subject to x ∈ Ω.