Question 3: (7 points) You are to optimize a wind farm made of 5 wind turbines whose positions are described by their latitude and longitude: x1 and x2 are the latitude 1 and longitude of the first turbine, x3 and x4 of the second turbine, . . . , x9 and x10 of the fifth turbine. All positions are summed up in the vector x = (x1, . . . , x10) >. The positions are restricted to a rectangular domain bounded by x min and x max, which are known, x min i ≤ xi ≤ x max i , i = 1, . . . , 10. The optimization problem has two goals: 1) to maximize the average power produced by the farm, P(x); 2) to minimize the cost of the farm (that of connecting the turbines to the power network), C(x). Question 3.1 (1 point) Formulate the optimization problem as a single objective minimization problem. Many answers are possible to aggregate P(x) and C(x). Answer: Let P¯ and C¯ be typical order of magnitudes of the production and the cost, and be different from 0. The optimization problem is formulated as min xmin≤x≤xmax f(x) where f(x) = C(x) C¯ − P(x) P¯ Question 3.2 (6 points) Each evaluation of P(x) takes 30 minutes because it relies on a computationnally expensive fluid simulation, and each evaluation of C(x) takes of the order of 10−2 seconds. Propose an algorithm to approximate the solution to the optimization of the wind farm, as formulated in Question 3.1. Because of the time it takes to evaluate P(x), only about 100 wind farms can be simulated during the optimization. Describe the algorithm with some details in about 1 handwritten page. In particular, explain how the budget of 100 wind farm simulations will be respected.