THE UNIVERSITY OF NEW SOUTH WALES SCHOOL OF MATHEMATICS AND STATISTICS MATH3161/MATH5165–OPTIMIZATION SAMPLE CLASS TEST 2 Version 2 (1) TIME ALLOWED – 50 Minutes (2) TOTAL NUMBER OF QUESTIONS – 3 (3) ANSWER ALL QUESTIONS (4) THE QUESTIONS ARE NOT OF EQUAL VALUE (5) ALL STUDENTS MAY ATTEMPT ALL QUESTIONS. MARKS GAINED ON ANY QUESTION WILL BE COUNTED. (6) THIS PAPER MAY BE RETAINED BY THE CANDIDATE All answers must be written in ink. Except where they are expressly required pencils may only be used for drawing, sketching or graphical work. MATH3161/MATH5165–OPTIMIZATION SAMPLE CLASS TEST 2 Page 2 1. [12 marks] Consider the equality constrained optimization problem (EP) minimize 2 x∈R x21 + (x2 − 1)2 − βx21 + x2 = 0, 0 ∗ where β ∈ R is a parameter. Let x = . 0 subject to i) Show that x∗ is a regular feasible point for (EP). ii) Verify that for all β the point x∗ is a constrained stationary point for (EP). iii) Using the second-order sufficient optimality conditions find the set of values of β for which x∗ is a strict local minimizer for (EP). iv)* Is x∗ a strict local minimizer for (EP) for the value β = 21 ? Give reasons for your answer. 2. [13 marks] Consider the following optimization problem 42 −5x21 + 6x1 x2 + 2x22 − x2 + 1 (P) minimize 2 x∈R 5 subject to 0 ≤ x1 ≤ 1 0 ≤ x2 ≤ x21 1 ∗ and let x = 3 . 5 i) Sketch the feasible region Ω of (P). ii) Is the problem (P) a convex programming problem? Give reasons for your answer. iii) Write the problem (P) in standard form. iv) Verify that x∗ is a feasible point for (P) and then find the active constraints at x∗ . v) Show that the point x∗ is a regular point for (P). vi) Verify that the point x∗ is a constrained stationary point for (P). vii) Using second-order sufficient conditions, determine whether or not x∗ is a strict local minimizer for (P). 3. [5 marks] Consider applying the method of steepest descent with exact line searches to q(x) = 21 xT Gx + dT x + c, where d is a constant n × 1 vector, G is a positive definite constant n × n symmetric matrix and c is a scalar. Let x∗ be the minimizer of q(x). Let x(1) be the initial point and x(1) 6= x∗ . i) Show that the initial search direction is s(1) = G(x∗ − x(1) ). ii) If the next iterate x(2) = x∗ then show that the step length α(1) is an eigenvalue of G−1 .