The University of Queensland
School of Information Technology & Electrical Engineering
Engg7302 Advanced Computational Techniques in Engineering Tutorial OPT1
These exercises relate to material in Lectures @ optimization introduction and Lagrange multipliers
Exercises:
OPT1.1
Revise important definitions, and practice analytical optimisation.
Heath, Chapter 6, Exercises 6.1, 6.2, 6.4, 6.5, 6.6
If you are having trouble with the calculus, you may want to do some revision questions from an engineering calculus text book (i.e: Kreizig, Stewart).
OPT1.2 (See also, Question 5, second Semester Examinations, November 2007) Consider the function f(x) = 5x” + x# + 3x x + 5.
!”!”
(a) Calculate the gradient of this function, ∇f(x).
(b) f(x) has two critical points at x! = [0 0] and x” = [− $ , #]. Calculate the %& %
Hessian matrix, H’ and evaluate it at each of the critical points.
(c) It can be shown that H’ (x”) is positive definite, while H’ (x!) is not positive
definite. What type of critical points at x! and x”?
OPT1.3 (See also, Question 5, second Semester Examinations, November 2007) Consider the constrained optimisation problem
Minimize
Subject to
f(x) = 3x” + x” !”
h!(x)=x” −x! +2=0
(a) Use the Lagrangian function to find the solution to this constrained optimisation problem.
Consider the constrained optimisation problem Minimize
f(x) = 3x” + x” !”
Subject to
h!(x)=x” −x! +2≤0
h”(x) = x” + x! + 2 ≤ 0
(b) Write down the Lagrangian function for this problem.
(c) Draw a diagram showing the constraints and contours of objective function in the
(x!, x”) solution space. Indicate by shading the feasible region of the space.