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,...

(1)