The output variable y (a real number) of a system depends on the input variable x (a real number). We have collected observations y~_i , i = 1,.

The output variable y (a real number) of a system depends on the input variable x (a real number). We have collected observations y~_i , i = 1,...,N , for N different settings x_i of the input variable, with x_1 < x_2 < ··· < x_N.We know that the dependence of y on x should be convex and nondecreasing. Formulate the problem of finding new values y_i such that this condition is satisfied and the sum of the squares of adjustments is minimized. Solve the problem for N = 4, input values x_i equal to 1, 2, 4, and 10, and the corresponding observations y~_i equal to 2, 1, 5, and 9.