Download SESG6018_2007

SESG6018 (Design Search and Optimisation) Exam 2007/08
Answer all of the following four short questions (total time
allowed 1 hour) and then one of the long essay questions (total
time allowed 1 hour).
Short Questions – answer all four questions.
S1) The function f ( x1, x2 )  x1  x2  2x12  2x1 x2  x23 is to be
minimized using the method of steepest descent, starting from
the point (0,1). Carry out one iteration of the scheme, clearly
showing how you have decided the correct step length and
search direction and thus how you have derived the new iterate.
S2) 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 f1 ( x)  2 x 4  8x  5 and
f ( x)  3x 2  8x  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
S3) The function f ( x1, x2 )  1 x12  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.
S4) Produce two new members of a population from two parents
using single point cross-over and one bit random mutation of
both children, for a binary encoded Genetic Algorithm with 6 bits.
The two parents are 27 and 42 and the upper and lower bounds
on the variables are 0 and 63. The next three random numbers
available from your random number generator, which generates
numbers in the interval 0-1, are assumed to be 0.3772, 0.1397
and 0.7129 .
Long questions – answer only one of the three questions, only
the first answer on your script will be marked.
L1) Describe how the requirement for robust designs may be
codified as design search and optimization problems. Pay
attention to the differences between a robust final design and the
robustness of the process used to achieve that design.
L2) Describe the way that design requirements may be codified
as optimization problems. Pay particular attention to the
differences between goals and constraints, between bounds and
limits and deterministic versus probabilistic approaches.
L3) Describe the various types of optimizers available to tackle
non-linear search problems and the range of typical problem
types encountered in design. Pay particular attention to speed
accuracy, robustness and usability.