OPTIMISATION PROJECT PROF. KARIM DJOUANI AND PROF. YSKANDAR HAMAM Contents 1. Unconstrained Optimisation 2. Non-convex case 3. BFGS Method 1 2 3...

Given the unconstrained optimisation problem:minx f(x)with x 2 Rn and f : Rn ! Rm(1) Give the expression of the rst and second derivative of f(x)(2) Give the necessary and sucient condition for the minimum of the functionf(x)Let us consider the Rosenbrock banana functionf(x) = 100(x2 ô€€€ x21)2 + (1 ô€€€ x2)2(1) Compute the gradient and the Hessian of f(x)(2) Give the necessary conditions for a local minimum(3) is x = [1; 1]T a local minimum?(4) Detail the Newton method algorithm(5) Write two matlab scripts one for the function one for the Newton method(6) Using Matlab, give the 10 rst iterations of the Newton method startingfrom the initial point x = [ô€€€1;ô€€€2]T(7) Plot the function using ezcontour(8) Plot the results of the 10 iterations on the same graph(9) Compute the norm of the error (x ô€€€ x) at each iteration and determinethe rate of the convergence. Justify theoretically your answer.1