Jim Lambers MAT 419/519 Summer Session 2011-12 Practice Final ...

Jim LambersMAT 419/519Summer Session 2011-12Practice Final ExamEach problem is worth 20 points.1. Compute the first two iterates x 1 , x 2 in the Steepest Descent sequence {x k } forf(x, y) = x 2 + 3y 2 − 2xystarting with the initial point x 0 = (1, 1).1

2. Compute the first two terms x 1 and x 2 of the BFGS sequence for minimizing the functionf(x, y) = x 2 − 2xy + 5y 2starting with the initial point x 0 = (2, 2) and D 0 = I. For each case, choose t k > 0 to be theexact minimizer of f(x) in the search direction from x k .2

3. Find the vector v ∈ R 3 of the formthat is closest to (1, 2, 3).v = α(3, −1, 2) + β(2, −4, 0)3

4. Find the point on the line prescribed by the two equationsx + 2y + 3z = 6x − y − z = 1that is closest to the origin.4

5. Apply the KKT Theorem to locate all solutions of the convex program⎧Minimize f(x, y) = e⎪⎨−x−ysubject to(P )e ⎪⎩x + e y ≤ 20x ≥ 05