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 using a vector of slack variables y ∈ Rm, y ≥ 0 and give the corresponding
KKT conditions.
(c) Formulate the dual to the problem in (b) and discuss its properties.
EXERCISE 2
Solve the following constraint minimisation problem: minf(x,y)=(x−2y)2+(x−2)2, x−y=4.
(x,y) (a) Formulate the KKT system.
(b) Solve the KKT system with a method of your choice. Explain briefly your approach.
[20pt]
[20pt] [20pt]
[20pt] [20pt]
Remark. Submit your solutions via Turnitin. This submission should not be longer than 4 pages.