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NUMERICAL OPTIMISATION TUTORIAL 2 MARTA BETCKE KIKO RUL·LAN (a)  Code backtracking line search, steepest descent and Newton’s algorithms. See Cody Courseworks for more guidance. Submit your implementation via Cody Coursework. [30pt] (b)  Apply steepest descent and Newton’s algorithms (with backtracking line search) to minimise the Rosenbrock function f(x)=100(y−x2)2 +(1−x)2. Set the initial point x0 =

NUMERICAL OPTIMISATION ASSIGNMENT 1 MARTA BETCKE KIKO RUL·LAN EXERCISE 1. Given the following function f(x,y)=2x+4y+x2 −2y2 (a)  Visualise the function and its contours. Submit your solutions via Turnitin. (b)  Calculate the contours analytically. Submit your solutions via Turnitin. (c)  Calculate the gradient analytically. Find the stationary points and classify them i.e. are them minima, maxima

EXERCISE 1. NUMERICAL OPTIMISATION ASSIGNMENT 0: EXAMPLE MARTA BETCKE KIKO RUL·LAN (a)  Write a Matlab function that implements the Rosenbrock function f(x,y)=100(y−x2)2 +(1−x)2. Be careful to implement a function that can be evaluated at many points simulta- neously. Submit your implementation via Cody Coursework. (b)  Create a two dimensional grid using Matlab’s command meshgrid. Plot

EXERCISE 1 subject to the constraint x2 Ax ≤ b, NUMERICAL OPTIMISATION ASSIGNMENT 8 MARTA BETCKE KIKO RUL·LAN Consider a problem to minimise the function minf(x)= 1xTGx+cTx where G ∈ Rn×n symmetric positive semidefinite, A ∈ Rm×n, c ∈ Rn, b ∈ Rm. (a)  State the KKT conditions for this problem. (b)  Rewrite the constraint

EXERCISE 1 NUMERICAL OPTIMISATION ASSIGNMENT 5 MARTA BETCKE KIKO RUL·LAN (a)  Implement the BFGS method by modifying the descentLineSearch function. More help is pro- vided inside Cody Coursework. Submit your solution via Cody Coursework. [20pt] (b)  Make your implementation efficient as explained in the lecture i.e. avoid explicitly forming the inverse Hessian matrix Hk. Copy

EXERCISE 1 EXERCISE 3 Consider a model NUMERICAL OPTIMISATION ASSIGNMENT 7 MARTA BETCKE KIKO RUL·LAN Implement the Gauss-Newton method for solution of nonlinear least square problems. As Gauss-Newton is a line search method, it can be easiest implemented inside the function descentLineSearch.m. More help is provided in Cody Coursework. Submit your implementation via Cody Coursework.