SESG6018 (Design Search and Optimisation) Exam 2010/11
Duration 120 mins
Answer all four short questions in Part A and one of the three
long essay questions in Part B (only the first long essay question on
your script will be marked).
A total of 80 marks are available for this paper.
Part A – Short Questions – answer all four questions.
S1) The function f ( x1 , x2 )  2x1 2  x2 2 is to be minimized subject to the
constraints x1 x2  1 and x1  x2  3 for positive values of x1 and x2 . Use
the methods of classical calculus to decide which, if either, of the two
constraints is active and hence locate the minimum of the problem.
S2) The function f ( x1, x2 )  x1  x2  2x12  2x1 x2  x2 2 is to be minimized
using the method of conjugate gradients, starting from the point (0,0).
The first iteration of the scheme results in a step length of 1 in the
direction (-1, 1). Carry out one further iteration of the scheme, clearly
showing how you have decided the correct step length and search
direction and thus how you have derived the next iterate.
(10 marks)
S3) A multi-objective optimization problem is defined by two goal
functions of a single variable x . Both functions must be minimized and
are given by f 1 ( x)  x 3  3x  6 and f ( x)  3x 2  3x  5 . Find the two end
points of the Pareto front for this problem in terms of the design
variable x and equivalent function values f1 ( x) and f 2 ( x) . Also find the
point on the Pareto front where the goal functions are given equal
weight.
2
S4) A least squares parabolic curve fit to four data points may be
obtained using SVD. If the data points are given at x  1,0,1,2 write
down the matrix equation relating the function values of y  1,0,1,2 at
these x values and the coefficients of the fitting parabola a, b and c.
What then are the values of a, b and c for the parabolic curve fit given
these data? Derive an estimate of the function value at x  0.5 .
You may assume that if the SVD of
1

0

1

4
1 1
0 1
1
1







isU .w.V ' then it may
2 1
 0.25
 0.25  0.25 0.25 




0
.
55
0
.
15
0
.
35
0
.
05

be shown that V .diag(1/ w j ).U '  
.
0.55
0.45  0.15
 0.15




Part B – Long questions – answer only one of the three
questions, only the first answer on your script will be marked.
L1) Describe how design requirements may be codified as
optimization problems and the role parameterization plays in design
optimization & search. Pay particular attention to the differences
between goals and constraints, between bounds and limits and
deterministic versus probabilistic approaches. Address also issues of
flexibility, robustness, ease of use and development cost
L2) Describe the role of curve fitting in optimization methods, paying
particular attention to the differences between implicit and explicit
curve fits, local versus global approaches, interpolation versus
regression and the roles of experiment and surrogate design,
validation and updating. Give examples of different search methods to
illustrate your discussion.
L3) Describe the various types of optimizers available to tackle nonlinear search problems and the range of typical problem types
encountered in design. Pay particular attention to speed accuracy,
robustness and usability.