Differential Equations Solutions 44

Differential Equations Solutions 44 - 54 Chapter 10....

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54 Chapter 10. Solutions: Numerical Methods for Constrained Optimization The solution is x = 0 , λ =[5 , 3] T . (d) Remembering to convert the frst constraint to Form, we get 10 01 2 x 1 2 x 2 T λ = · 2 x 1 x 2 +5 8 x 2 x 1 +3 ¸ , λ 0 , x 0 , 1 x 2 1 x 2 2 0 , λ 1 x 1 + λ 2 x 2 + λ 3 (1 x 2 1 x 2 2 )=0 . The solution is x = 0 , λ , 3 , 0] T . CHALLENGE 10.2. (Partial Solution) We need to veriFy that Z T xx L ( x, λ ) Z is positive semidefnite. (a) Z T xx L ( x, λ ) Z = H ( x )= · 2 1 18 ¸ . (b) xx L ( x, λ · 2 1 ¸ , and we can take Z T =[1 , 1]. (c) xx L ( x, λ · 2 1 ¸ . Both constraints are active at the optimal solution, so Z is the empty matrix and the optimality condition is satisfed trivially.
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