程序代写代做代考 NUMERICAL OPTIMISATION

NUMERICAL OPTIMISATION

ASSIGNMENT 1

MARTA BETCKE
KIKO RUL·LAN

EXERCISE 1. Given the following function

f(x, y) = 2x+ 4y + x2 − 2y2

(a) Visualise the function and its contours.
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(b) Calculate the contours analytically.
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(c) Calculate the gradient analytically. Find the stationary points and classify them
i.e. are them minima, maxima or something else?
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EXERCISE 2.

(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.
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(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.
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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.

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