UNSW School of Mathematics and Statistics MATH3171/5171 Linear and Discrete Optimization Modelling Term 3, 2021 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.