4. a) Given the quadratic function: f(x) = {x} + 2x + 3x3 + x1 x3 + 2x2x2 3x1 x3 2 i) Express f(x) in the following form, f(x) = C + BX + {x"Q X ii) Minimise f(x) using the conjugate gradient method with an initial starting point, X = [0,0]T iii) Find the minimum of f(x) analytically. 8 b) A Liverpool based radio channel provides services in most Northern cities in the UK. The channel is planning to expand its service to four cities in the Midlands by designing a new broadcasting network. To provide a good quality service, the channel needs a new transmission tower, which will transmit radio frequency to the pre-existing smaller towers in those cities. The new tower to be constructed should cover areas within a radius of K km. Thus, the new tower must be located within K km of each of those existing towers. The problem is to determine the tower location that minimises the total distance from the new tower to each of the existing towers. The location of each city can be calculated using two dimensional coordinates (x, y) from a given reference point as follows: Table Q4 City y 10 15 2 3 4 20 45 25 10 20 55 6 Formulate the problem stated above as a formal optimisation problem by identifying the variables, the objective function and the constraints. 4. a) Given the quadratic function: f(x) = {x} + 2x + 3x3 + x1 x3 + 2x2x2 3x1 x3 2 i) Express f(x) in the following form, f(x) = C + BX + {x"Q X ii) Minimise f(x) using the conjugate gradient method with an initial starting point, X = [0,0]T iii) Find the minimum of f(x) analytically. 8 b) A Liverpool based radio channel provides services in most Northern cities in the UK. The channel is planning to expand its service to four cities in the Midlands by designing a new broadcasting network. To provide a good quality service, the channel needs a new transmission tower, which will transmit radio frequency to the pre-existing smaller towers in those cities. The new tower to be constructed should cover areas within a radius of K km. Thus, the new tower must be located within K km of each of those existing towers. The problem is to determine the tower location that minimises the total distance from the new tower to each of the existing towers. The location of each city can be calculated using two dimensional coordinates (x, y) from a given reference point as follows: Table Q4 City y 10 15 2 3 4 20 45 25 10 20 55 6 Formulate the problem stated above as a formal optimisation problem by identifying the variables, the objective function and the constraints