Given n data points {(xi, yi) : i = , n}, linear regression fits the line y = ax + b to data points by minimizing the square of the distance between the data points and the line. that is, it solves: min a, b n sigma j = 1 (yj - ax3 - b)^2

here we consider an alternative regression model based on rectilinear distance along the y - axis and between two points e = (ex, ey) and f = (fx, fy) as |ey - fy| where ex, fx and ey, fy are the x and y coordinates of the points e and f respectively. note that this distance computes distances between two points only along the y axis (i. e. |ey - fy|) and disregards the distance along the x axis (i. e. |ex - fx|). for example, if e = (3,7) and f = (4,5), the rectilinear distance along y is |ey - fy| = |7 - 5| = 2.

1. formulate a non-linear program to find a line y = ax + b closest (in rectilinear distance along y) to a given set of data points {(xi, yi) : i = n}.
2. reformulate the non-linear program as a linear program to find a line y = ax + b closest (in rectilinear distance along y) to a given set of data points {(xi, yi) : i = n}.
3. what is the optimal solution in the case n = 2?