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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 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.

NUMERICAL OPTIMISATION ASSIGNMENT 6 MARTA BETCKE KIKO RUL·LAN Minimal Surface Cost Function [Adapted from Exercise 7.7 from Nocedal-Wright] The minimum surface problem is a classical application of the calculus of variations and can be found in many textbooks. We wish to find the surface of minimum area, defined on the unit square, that inter- polates

NUMERICAL OPTIMISATION – ASSIGNMENT 4 MARTA BETCKE KIKO RUL·LAN EXERCISE 1 Consider the linear system Ax = b with A ∈ Rn×n, x ∈ Rn, b ∈ Rn. (a) Implement the linear preconditioned Conjugate Gradient method. Submit your implementation via Cody Coursework. Consider the starting point x0 = (0, . . . , 0)T ,

EXERCISE 1 NUMERICAL OPTIMISATION ASSIGNMENT 3 MARTA BETCKE KIKO RUL·LAN Derive the 2D subspace trust region method for convex functions (with s.p.d. Hessian). Note that (i)  when p is constraint to a subspace V = span(g, B−1g), it can be expressed as a linear combination of basis vectors p = V a. You can use