54Chapter 10.Solutions: Numerical Methods for Constrained OptimizationThe solution isx=0,λ=[5,3]T.(d) Remembering to convert the frst constraint to≥Form, we get⎡⎣1001−2x1−2x2⎤⎦Tλ=·2x1−x2+58x2−x1+3¸,λ≥0,x≥0,1−x21−x22≥0,λ1x1+λ2x2+λ3(1−x21−x22)=0.The solution isx=0,λ,3,0]T.CHALLENGE 10.2.(Partial Solution) We need to veriFy thatZT∇xxL(x, λ)Zis positive semidefnite.(a)ZT∇xxL(x, λ)Z=H(x)=·2−1−18¸.(b)∇xxL(x, λ·2−1−¸,and we can takeZT=[1,−1].(c)∇xxL(x, λ·2−1−¸.Both constraints are active at the optimal solution, soZis the empty matrix andthe optimality condition is satisfed trivially.
This is the end of the preview. Sign up
access the rest of the document.