Test functions The following objective functions represent different types of optimization test functions: quadratic, large-dimensional, etc. 1. f(x) = -(i-x;), where n = 50. The optimal solution is x* = 0. 2. f(x) = 212 [100(i+1 2?)2 + (1 I;)?], where n = 10. The optimal solution is x* = (1, ..., 1]" 3. f(x) = (1.5 11 + 2112)2 + (2.25 Il + 1123)2 + (2.625 11 + 1103)2. The optimal solution x* = [3,0.5] METHODS Your task is to implement the following optimization algorithms with backtracking line search: Steepest descent method Newtons method Quasi-Newton method (DFP or BFGS) Conjugate gradient method (Polak-Ribiere or Fletcher-Reeves ) QUESTIONS, For all three test functions and all methods, create a table reporting the number of function calls and iterations required for convergence. You may include any other metric that can help analyse the methods. It may be helpful to test the impact of the initial condition by trying more than one starting point. Sometimes just using one point can be misleading. Discuss your results. [10 marks] Provide the convergence plot for all three test functions. For each test function, plot the results of all methods in the same set of axis. A convergence plot should show iterations on the x-axis and a convergence metric with a log scale on the y-axis. You may use the convergence metric f(x) f(x*)], or any other metric such as || Vf(x) || (since f(x*) is usually not available.) [10 marks]