Homework3_2011_sol - ESE 415 Optimization Assignment 3 Due:...

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Unformatted text preview: ESE 415 Optimization Assignment 3 Due: March 2, 2011 1. In each of the following problems fully justify your answer using optimality conditions. (a) Show that f ( x,y ) = ( x 2 4) 2 + y 2 has two global minima and one stationary point, which is neither a local maximum nor a local minimum. Solution: At critical points, f ( x,y ) = [ 4 x ( x 2 4) 2 y ] = [ ] (0 , 0) (2 , 0) ( 2 , 0) H ( x,y ) = [ 12 x 2 16 0 2 ] H (0 , 0) = [ 16 0 2 ] H (2 , 0) = [ 32 0 2 ] H ( 2 , 0) = [ 32 0 2 ] H (0 , 0) indefinite neither max or min H (2 , 0) > minimum H ( 2 , 0) > minimum f (2 , 0) = f ( 2 , 0) both (2,0) and (-2,0) are global minimums (b) Find all local mimina of f ( x,y ) = 1 2 x 2 + x cos y . Solution: At critical points f ( x,y ) = [ x + cos ( y ) xsin ( y ) ] = [ ] (0 , 2 + n )where n Z , ( 1 ,n )where n even, (1 ,n ) where n odd H ( x,y ) = [ 1 sin ( y ) sin ( y ) xcos ( y ) ] H (0 , 2 + n ) = [ 1 1 1 ] ,n Z H ( 1 ,n ) = [ 1 0 0 1 ] ,n even H (1 ,n ) = [ 1 0 0 1 ] ,n odd H (0 , 2 + n ) indefinite neither max or min H ( 1 ,n ) > local minimum H (1 ,n ) > local minimum (c) Find all local minima and all local maxima of f ( x,y ) = sin x + sin y + sin( x + y ) within the set { ( x,y ) | < x < 2 , < y < 2 } . 1 Solution: At critical points f ( x,y ) = [ cos ( x ) + cos ( x + y ) cos ( y ) + cos ( x + y ) ] = [ ] ( , ) , ( 3 , 3 ) , ( 5 3 , 5 3 ) H ( x,y ) = [ sin ( x ) sin ( x + y ) sin ( x + y ) sin ( x + y ) sin ( y ) sin ( x + y ) ] H ( , ) = [ 0 0 0 0 ] H ( 3 , 3 ) = [ 3 3 2 3 2 3 ] H ( 5 3 , 5 3 ) = [ 3 3 2 3 2 3 ] H ( , ) neither max or min H ( 3 , 3 ) < local maximum H ( 5 3 , 5 3 ) > local minimum 2. Use optimality conditions to show that for all x > 0 we have 1 x + x 2 . Solution: Let f ( x ) = 1 x + x . Then f ( x ) = 1 x 2 + 1 = 0, hence x 2 = 1 so that x = 1, because x > 0. Now H ( x ) = 2 x 3 so H (1) = 2 > 0, and hence the global minimum occurs at x = 1. Because f (1) = 2, then x > ,f ( x ) = 1 x + x 2....
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Homework3_2011_sol - ESE 415 Optimization Assignment 3 Due:...

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