Find the range of values of the parameter a for which the


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) = ½xTQx - xTb,

where Q = QT > 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 fi: Ω → R, i = 1, . . . , l be convex functions. Show that max {f1, . . . , fl} is a convex function.

Note: The notation max{f1, . . . , fl} denotes a function from Ω to R such that for each x ∈ Ω, its value is the largest among the numbers fi(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 + · · · + xn = 1

               x1, . . . ,xn ≥ 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.

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Engineering Mathematics: Find the range of values of the parameter a for which the
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