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1. Consider the following optimization problem (P): (P) minimize = i=1 xi 2 0, i=1, = , n Xiwhere ai ER,i=1, , n are fixed constants. (a) (20pts) Write down the KKT conditions for problem (P). (b) (20pts) Determine if the KKT point in Part (a) is an optimal solution. Justify your answer. (c) (20pts) Formulate the direction finding problem (DFP) for problem (P): n (P) minimize x2 subject to xi=1 = i=1 Xi 20, i = 1, n Xin where ai € R, i = 1, , n are fixed constants. (d) (20pts) Write down the KKT conditions for problem (DFP) in Part (c). (e) (20pts) Formulate the dual problem (D) of problem (P): (P) minimize [xt subject to xi=1 i=1 i=1,..., xiwhere ai € R, i = 1,...,n are fixed constants.

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