MATH-2640 MATH-264001
This question paper consists of 3 printed pages, each of which is identiﬁed by the reference MATH-2640 Only approved basic scientiﬁc calculators may be used.

c UNIVERSITY OF LEEDS Examination for the Module MATH-2640 (January 2003)

Introduction to Optimisation
Time allowed: 2 hours Attempt four questions. All questions carry equal marks. In all questions, you may assume that all functions f (x1 , . . . , xn ) under consideration are sufﬁciently ∂2f ∂2f continuous to satisfy Young’s theorem: fxi xj = fxj xi or ∂xi ∂xj = ∂xj ∂xi . The following abbreviations, consistent with those used in the course, are used throughout for commonly occurring optimisation terminology: LPM – leading principal minor; PM – (non-leading) principal minor; CQ – constraint qualiﬁcation; FOC – ﬁrst-order conditions; NDCQ – non-degenerate constraint qualiﬁcation; CSC – complementary slackness condition; NNC – non-negativity constraint.

Q1 (a) You are given that the formula for the total differential at the point x0 of a function f of n variables x1 , . . . , xn is 1 δf (x0 ) = δx· f (x0 ) + 2 (δx)T H(x0 )(δx) + O |δx|3 ,

where x = (x1 , . . . , xn )T , the Hessian of f at x0 .

∂ ∂ ≡ ( ∂x1 , . . . , ∂xn )T is the n-dimensional gradient operator and H(x0 ) is

(i) Deﬁne: the total differential in terms of f , x0 and δx; the Hessian matrix H in terms of f and x0 ; the kth LPM of the Hessian H. (ii) What is meant by saying that x∗ is a stationary point of f ? What then is the formula for the total differential δf (x∗ )? (iii) State the rules governing the LPMs of the Hessian H(x∗ ) by which we can classify the deﬁniteness of H and therefore whether a stationary point is a local maximum, local minimum or saddle point. (iv) If x0 is the only point for which say about x0 ? (b) Locate and, using the Hessian, classify all stationary points of the function f (x1 , x2 ) = x3 + x3 − 3x1 x2 . 1 2 1 Continued ... f (x0 ) = 0, and if the elements of H are constant,...

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...CANDIDATES 1. 2. 3. This question paper consists of seven (7) questions. Answer five (5) questions only in the Answer Booklet. Start each answer on a new page. Do not bring any material into the examination room unless permission is given by the invigilator. Please check to make sure that this examination pack consists of: i) ii) the Question Paper an Answer Booklet - provided by the Faculty
DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO
This examination...

...2 HOURS
INSTRUCTIONS TO CANDIDATES This question paper consists of four (4) questions. Answer ALL questions. Start each answer on a new page. Do not bring any material into the examination room unless permission is given by the invigilator. Please check to make sure that this examination pack consists of: i) ii) the Question Paper an Answer Booklet - provided by the Faculty
DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO
This examination paper...

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