Nov 30 fix up

# Nov 30 fix up - guarantee that j x is positive for at least...

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Fix Up Question: The data in the optimization problem that appears below are positive numbers 1 a through m a and positive numbers 1 b through m b . What is its optimal solution? Why? Minimize = m 1 j 2 j j ) x ( a , subject to λ : 100 x b m 1 j j j = = . Answer: The objective is convex and differentiable, and the constraints are linear, which assures us that Hypothesis #1 is satisfied. Thus, the KKT conditions are satisfied at x if and only if x is an optimal solution. Taking partial derivatives and using the ross-over table shows us that the KKT conditions are: j x : j j j x a 2 b = λ for j = 1, 2, …, m . The constraint of the NLP and the fact that 1 b through m b are positive

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Unformatted text preview: guarantee that j x is positive for at least one j, say, for j = 2. The equation 2 2 2 x a 2 b = Î» demonstrates that Î» is positive and that 2 2 2 a 2 b x Î» = . And the equation that is complementary to j x guarantees (1) j j j a 2 b x Î» = for j = 1, 2, â€¦, m . Multiply equation (1) by j b , sum over j, and then substitute into the constraint of the nonlinear program to see that ] a / ) b [( 200 k m 1 k 2 k âˆ‘ = Î» = , and then use (1) to see that the optimal solution to this nonlinear program is ] a / ) b [( a / b 100 x k m 1 k 2 k j j j âˆ‘ = = for j = 1, 2, â€¦, m ....
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Nov 30 fix up - guarantee that j x is positive for at least...

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