UNIVERSITY OF MELBOURNE SCHOOL OF MATHEMATICS AND STATISTICS MAST30013 Techniques in Operations Research Semester 1, 2020 Assignment 3 Due: 4 pm, Thursday, 28 May – Solution must be typeset in LaTex. – Please submit your solution online by the due date. – Show all necessary working. 1. Consider the nonlinear program min x 1 2 xTQx+ cTx s.t. Ax ≤ b, where x, c ∈ Rn, b ∈ Rm ≥ 0 A ∈ Rm×n. (a) Write down the KKT conditions and find all points satisfying the conditions. (b) Check that one of the constraint qualifications holds. (c) Show that the KKT conditions are necessary and sufficient for a global minimiser if Q = QT > 0. 2. Consider the nonlinear program min x x32 − x1 − 2×2 s.t. x1 + x2 ≤ 1 x1, x2 ≥ 0. (a) Write down the log barrier penalty function Pk(x) with penalty parameter k. (b) Write down ∇Pk(x), and solve ∇Pk(x) = 0 to find any stationary points xk = (xk1, xk2) for Pk(x). (c) Find the limit x∗ = lim k→∞ xk. (d) Write down an estimate λk of the optimal Lagrange multiplier, and find the limit λ∗ = lim k→∞ λk. 3. Consider the nonlinear program min x x31 + 2x 2 1 − 10×1 + x22 − 8×2 s.t. x1 + x2 ≤ 2 x1, x2 ≥ 0. 1 of 2 (a) Verify that (x∗, λ∗) = ((−1 +√3, 3−√3) , (2 + 2√3, 0, 0)) is a KKT point. Does the KKT point correspond to a local minimum? (b) Show that the objective function is convex for all x in the constraint set. (c) Show that the Lagrangian Saddle Point inequalities hold for all x in the constraint set and λ ≥ 0, that is L(x∗, λ) ≤ L(x∗, λ∗) ≤ L(x, λ∗). 2 of 2
