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HW5 - ELE 704 Optimization HW5 Due 27 March 2007 1 Suppose...

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ELE 704 Optimization HW5 Due 27 March 2007 1. Suppose that x ( k ) and x ( k +1) are two consecutive points generated by the gradient descent algorithm with exact line search. Show that T f x ( k ) · f x ( k +1) · = 0 2. For the following quadratic problem f ( x ) = 1 2 x T Qx + b T x + c (a) Derive a closed form expression of the steepest descent direction (for Euclidean norm), i.e. write an expression which gives you the nega- tive gradient of f at point x ( k ) , when you enter the parameters Q , b and c, and the current position x ( k ) . (b) Write the corresponding MATLAB function for part ( a ) which ac- cepts the parameters Q , b , c ,and x ( k ) as inputs. (c) Derive a closed form expression of the exact line search algorithm, i.e. write an expression which gives you the step size α when you enter the parameters Q , b and c, the search direction d , and the current position x ( k ) . (d) Write the corresponding MATLAB function for part ( c ) which ac- cepts the parameters Q , b , c , d and x ( k ) as inputs.
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