数值优化代写 NUMERICAL OPTIMISATION ASSIGNMENT 8

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.

  1. (a)  State the KKT conditions for this problem.
  2. (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.