1. Having been set different priorities by different stakeholders, a manager realises they need to maximise functions f and g where f(x,y)=9-1, g(x, y) = 9-(1-1) - y. The manager believes that one of the stakeholders will be satisfied provided that f(x,y) 5. (i) Explain why we can model this program by min: F(x, y) = x + y-22-8-r ln(-x+4), for small r. [10 marks] (ii) Give one advantage and one disadvantage of the Barzilai-Borwein step size compared to Cauchy's step size. [8 marks] (iii). Set r= 0.1, and apply one iteration of the Gradient Descent Method, with Barzilai-Borwein step size to approximate the minimum value of F. Use the initial values Xo= To) - (1) - () - (0). = = Yo You may use the following formulae: Axk-X-Xk 1 AF(x) = VF(x) - VF(x* 1), Yk (AF(x), Ax) AF(x)- [16 marks (iv) Find the exact values of r and y that maximise g subject to the constraint f> 5, and compare with your approximate solution. 10 marks 1. Having been set different priorities by different stakeholders, a manager realises they need to maximise functions f and g where f(x,y)=9-1, g(x, y) = 9-(1-1) - y. The manager believes that one of the stakeholders will be satisfied provided that f(x,y) 5. (i) Explain why we can model this program by min: F(x, y) = x + y-22-8-r ln(-x+4), for small r. [10 marks] (ii) Give one advantage and one disadvantage of the Barzilai-Borwein step size compared to Cauchy's step size. [8 marks] (iii). Set r= 0.1, and apply one iteration of the Gradient Descent Method, with Barzilai-Borwein step size to approximate the minimum value of F. Use the initial values Xo= To) - (1) - () - (0). = = Yo You may use the following formulae: Axk-X-Xk 1 AF(x) = VF(x) - VF(x* 1), Yk (AF(x), Ax) AF(x)- [16 marks (iv) Find the exact values of r and y that maximise g subject to the constraint f> 5, and compare with your approximate solution. 10 marks