Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: THEend8_
MATH3171/5171 Linear and Discrete Optimization Modelling
Quiz This sheet must be filled in and returned in your answer booklet Student’s
family name Given name or initials Student number • UNSW approved Calculators are permitted
• Time Allowed: 15 minutes 1. [3 marks] Determine whether the following matrix is
positive semi-definite, positive definite, negative semi-definite, negative definite or indefinite.
Provide reasons for your answer. 10 2 12 9 −4 1 −4 11 . 2. [7 marks] Consider f(x) = xTAx
where A is an (n × n) symmetric matrix and x = (x1, . . . , xn) ∈ Rn. Suppose that A is positive definite. i)
Is f a coercive function on Rn? Justify your answer ii) Hence or otherwise,
show that a global minimizer exists for the following opti- mization problem (P )
Minimizex∈Rn xTAx subject to ‖x‖22 ≥ 1. State clearly any facts you have used. 3. [5 marks]
Let C be a given (p × n) matrix. Consider the function f : Rp×n → R defined by f(X) = trace(CTX)
for all X ∈ Rp×n, where trace(M) denotes the trace of a matrix M . Show that f is convex function on Rp×n.