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.
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