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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.