Problem 2 Let D be a given n x n diagonal matrix with strictly positive diagonal elements di > 0, i = 1, 2, ..., n and l

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Problem 2 Let D Be A Given N X N Diagonal Matrix With Strictly Positive Diagonal Elements Di 0 I 1 2 N And L 1 (149.74 KiB) Viewed 20 times

Problem 2 Let D be a given n x n diagonal matrix with strictly positive diagonal elements di > 0, i = 1, 2, ..., n and let the set F be defined by F = {x + R" |x2x < 1} = Given a point xo that does not belong to F consider the following quadratic minimization problem with the quadratic constraint (P) min 3(x– xo)"(x– xo) XO subject to x E F Tasks: 1. (5 points) Derive the dual function L*(a) for the problem (P) and formulate the dual problem (D). 2. (4 points) Derive an explicit expression for the derivative L4 (9) of the dual objective function, valid for all 10. 3. (3 points) Then show that L4(0) > 0 and that L'4() is strictly decreasing for all 1 > 0 and that there is a number 1 > 0 such that L4 (9) <0. Here we denote the derivative d L!.(A) = ÓL.(1) di 6+ ( 4. (3 points) From the previous point deduce that there is a unique solution )* to the dual problem. Show that the point x* that minimizes the Lagrangian L(x, \*) is the optimal solution of the primal problem (P). Hint: Use the KKT (optimality) conditions.