Find local minimizers for x12 subject to x12 2x1x2 x22 1

Find local minimizers for x_1^2 + subject to x_1^2 + 2x_1x2 + x_2^2 = 1, x_2^2 - x_2^2 lessthanorequalto 0.

Solution

x_1^2 + x_2^2 has a local minima at (x_1,x_2) = (0,0) in R^2

there doesn\'t exists any other local minima for this function

since it is a radial function and strictly increasing .

So subject to the constraints

x_1^2 + 2x_1x_2 +x_^2 = 1

(0,0) is not in the constraint region

So there doesn\'t exist any local minimizers