Homework3_2011 - (b) Let f ( x ) = 1 2 x T A T Ax b T Ax ....

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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. (b) Find all local mimina of f ( x, y ) = 1 2 x 2 + x cos y . (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 ) | 0 < x < 2 π, 0 < y < 2 π } . 2. Use optimality conditions to show that for all x > 0 we have 1 x + x 2 . 3. Find all the descent directions of the following function at the point (0 , 0) T , f ( x 1 , x 2 ) = 100( x 2 x 2 1 ) 2 + (1 x 1 ) 2 . 4. Suppose that x k and x k +1 are two consecutive points generated by the steepest descent algorithm with an exact line search, e.g., minimization rule. Show that f ( x k ) T f ( x k +1 ) = 0 . 5. Consider the problem of ±nding a solution to the system of linear equations: ( OP ) : Find x that solves Ax = b, where A is a nonsingular square m × m matrix and b R m . (a) Show that A T A is nonsingular and positive de±nite.
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Unformatted text preview: (b) Let f ( x ) = 1 2 x T A T Ax b T Ax . Show that solving ( OP ) is equivalent to minimizing f ( x ). What is f ( x )? (c) Given x and d , nd explicitly that solves: min f ( x + d ) . (d) Formulate an algorithm that solves (OP) using steepest descent. What is the step size at each iteration? 1 6. (Steepest Descent) Implement the steepest descent algorithm to minimize (a) f ( x ) = 5 x 2 1 + x 2 2 + 4 x 1 x 2 14 x 1 6 x 2 + 20 as presented in class. Check your results with the handout. (b) f ( x ) = 1 2 x T Qx c T x + 10 for (i) x = b 40 100 B , Q = b 20 5 5 2 B , c = b 14 6 B . (ii) x = b 40 100 B , Q = b 20 5 5 16 B , c = b 14 6 B . Set the stop criterion = 10-6 . Make the tables and Fgures as shown in the handout. Attach your Matlab code. 2...
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Homework3_2011 - (b) Let f ( x ) = 1 2 x T A T Ax b T Ax ....

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