Problem 1 Optimization over the simplex (14 points) In this exercise you will prove two optimization results over the simplex that we used multiple times in the lectures. These results will also help you solve the other problems in this homework. The K - 1 dimensional simplex is simply the set of all distributions over K elements, denoted by A = [qER | 4 2 0, Vk and [ 19% = 1}. 1.1 Let a1,..., ag be K positive numbers. Prove that the solution of the following optimization problem arg max _ ax In 9% qEA is q' such that of = , (that is, of