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Final Exam / Alternative solution to the optimization in Problem 2 on the practice final
« on: April 23, 2022, 02:19:38 PM »
Could I have solved the maximization/minimization using Lagrange multipliers? In particular, define $g_1(x,y) = y-x$, $g_2(x,y) = y+x$, and $g_3(x,y) = -(x^2+y^2)+1$. Then, a solution $(x^*,y^*)$ necessarily satisfies $$\nabla{u} + \lambda_1\nabla{g_1} + \lambda_2\nabla{g_2} + \lambda_3\nabla{g_3} = 0$$ and $$\lambda_1{g_1} = 0, \lambda_2{g_2}=0, \lambda_3{g_3}=0$$
for some $\lambda_{i} \geq 0$.
Then, after finding the points $(x^*, y^*)$, I need to verify that $$\nabla^2{u} + \lambda_1\nabla^2{g_1} + \lambda_2\nabla^2{g_2} + \lambda_3\nabla^2{g_3} $$ is positive definite on the tangent space $T_{x^*,y^*}D$.
Would this approach also work?
for some $\lambda_{i} \geq 0$.
Then, after finding the points $(x^*, y^*)$, I need to verify that $$\nabla^2{u} + \lambda_1\nabla^2{g_1} + \lambda_2\nabla^2{g_2} + \lambda_3\nabla^2{g_3} $$ is positive definite on the tangent space $T_{x^*,y^*}D$.
Would this approach also work?