PFA

PFA


Optimization with restrictions 4. (IS) Consider the linear function of N variables ?: ℝ? → ℝ given by ?(?) = ∑ ?? ? ?=1 . Given the Euclidean norm ‖⋅‖2 in ℝ ?, define the set ? = {??ℝ?: ‖?‖2 = 1}. Answer: a) Using Weierstrass's theorem for the existence of a solution to optimization problems, prove that both problems ???????(?) and ???????(?) have a solution. b) Check the regularity conditions to apply the method of Lagrange multipliers to identify candidates for the solution of each of the problems listed in the previous letter is satisfied at any point in the set X. (Hint: to facilitate any calculation, you can write, de equivalent form, X = {x?ℝ?: 〈x, x〉 = 1}). c) Solve for Lagrange the problems of maximizing and minimizing f(x), each subject to ???. 5. (IS) Let ?: ℝ? → ℝ be a function given by f(x) = 〈x, x〉, and ?: ℝ? → ℝ another function expressed by ?(?) = ∑ ?? − 1 ? ?=1 . Using the Lagrange method, find the solution to the problem of minimizing ?(?) subject to ?(?) = 0.