Problem 1 (40 points) For the following NLPs determine (without proof, sketch of feasible set is...

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Problem 1 (40 points) For the following NLPs determine (without proof, sketch of feasible set is sufficient) the optimal solution x, determine the tangent cone T(F,x) and the linearized cone L(F,), and determine whether T(F,x) = L(Fx). 1. 2. minimize -x1, subject to x+x2 0, -x2Y, x1 x2 2 minimize x1, subject to -(x1 1)+x2 0, Y-x10, -x20 For both problems consider the cases y = 0 and y=1. Problem 1 (40 points) For the following NLPs determine (without proof, sketch of feasible set is sufficient) the optimal solution x, determine the tangent cone T(F,x) and the linearized cone L(F,), and determine whether T(F,x) = L(Fx). 1. 2. minimize -x1, subject to x+x2 0, -x2Y, x1 x2 2 minimize x1, subject to -(x1 1)+x2 0, Y-x10, -x20 For both problems consider the cases y = 0 and y=1.