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# L22_10 - Nonlinear Programming(contd Difficulties of NLP...

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Nonlinear Programming (cont’d)

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Difficulties of NLP Models Nonlinear Programs: Linear Program:
Graphical Analysis of Non-linear programs in two dimensions: An example 22 14 15 () xy −+ Minimize subject to (x - 8) 2 + (y - 9) 2 49 x 2 x 13 x + y 24

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Where is the optimal solution? 0 2 4 6 8 10 12 14 16 18 0 2 4 6 8 10 12 14 16 18 y x Note: the optimal solution is not at a corner point. It is where the isocontour first hits the feasible region.
Another example: 0 2 4 6 8 10 12 14 16 18 0 2 4 6 8 10 12 14 16 18 y x Minimize (x-8) 2 + (y-8) 2 Then the global unconstrained minimum is also feasible. The optimal solution is not on the boundary of the feasible region.

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Local vs. Global Optima There may be several locally optimal solutions. x z 1 0 z = f(x) max f(x) s.t. 0 x 1 A B C Def’n : Let x be a feasible solution, then –x is a global max if f(x) f(y) for every feasible y . is a local max if f(x) f(y) for every feasible y sufficiently close to x (i.e., x j - ε≤ y j x j + ε for all j and some small ε ).
When is a locally optimal solution also globally optimal? • For minimization problems – The objective function is convex.

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L22_10 - Nonlinear Programming(contd Difficulties of NLP...

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