matlab数值优化代写: NUMERICAL OPTIMISATION ASSIGNMENT 1

NUMERICAL OPTIMISATION ASSIGNMENT 1

MARTA BETCKE KIKO RUL·LAN

EXERCISE 1. Given the following function f(x,y)=2x+4y+x2 −2y2

1. (a)  Visualise the function and its contours. Submit your solutions via Turnitin.
2. (b)  Calculate the contours analytically. Submit your solutions via Turnitin.
3. (c)  Calculate the gradient analytically. Find the stationary points and classify them i.e. are them minima, maxima or something else?
   Submit your solutions via Turnitin.

EXERCISE 2.

1. (a)  Show that A = BTB is symmetric positive semidefinite for all B ∈ Rn×n. Hint: use the Rayleigh quotient representation of the eigenvalue Ax = λx.
   Submit your solutions via Turnitin.
2. (b)  Let f(x) = xTAx with A symmetric positive semidefinite matrix A ∈ Rn×n. Show that f(x) is convex on the domain Rn. Hint: you may want show the equivalent inequality instead

   f(y + α(x − y)) − αf(x) − (1 − α)f(y) ≤ 0. Submit your solutions via Turnitin.

Remark. The submission to Turnitin should not be longer than 5 pages. Avoid submitting more code than needed (if any) and focus on explaining your results.

1