Suppose we want to find a point on the unit circle centered at the origin to...

 

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Suppose we want to find a point on the unit circle centered at the origin to minimize -x? -2x+y. That is, we wish to solve the following equality constrained problem: -x - 2x + y x? + y? = 1 P min s.t Is the problem convex? Show that x)= (1,0)" is a KKT point of P. Then use the second-order conditions to show that x is indeed a strict local minimizer. Is it a global minimizer? Why or why not? (Note that it can also be shown, although you don't have to do it, that there are exactly 4 stationary (KKT) points, and the other three are x2)= (-1,0)", x6)= (-1/2, V3/2)", x(4) = (-1/2, -V3/2)". You may use this knowledge to answer the last part of this question.) Suppose we want to find a point on the unit circle centered at the origin to minimize -x? -2x+y. That is, we wish to solve the following equality constrained problem: -x - 2x + y x? + y? = 1 P min s.t Is the problem convex? Show that x)= (1,0)" is a KKT point of P. Then use the second-order conditions to show that x is indeed a strict local minimizer. Is it a global minimizer? Why or why not? (Note that it can also be shown, although you don't have to do it, that there are exactly 4 stationary (KKT) points, and the other three are x2)= (-1,0)", x6)= (-1/2, V3/2)", x(4) = (-1/2, -V3/2)". You may use this knowledge to answer the last part of this question.)