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, what does this
Q2 (a) Write down the Lagrangian, L(x, y, z; µ), used to determine solution candidates for ﬁnding a maximum of f (x, y, z) = x2 + y 2 + z 2 subject to the single equality constraint h(x, y, z) ≡ x + 2y − z = 1. What is the constraint qualiﬁcation in this case? Does it affect solution candidates (x∗ , y ∗ , z ∗ ) of the FOC? Solve the FOC to ﬁnd solution candidates for stationary points, which you should classify. (b) Write down the objective function f (x, y) and Lagrangian L(x, y; µ), needed to determine solution candidates for the nearest and furthest points, to the point (x, y) = (1, 2), which lie on the circle of radius 1 centred at (−1, 0). What is the constraint qualiﬁcation in this case? Does it affect potential solution candidates? √ Find all solution candidates from the FOC derived from L. In particular, deduce that µ = 1 ± 2 2. Sketch the circle and on it draw these solution candidates. Deduce that the nearest and furthest points √ √ on the circle are respectively at a distance 2 2 − 1 and 2 2 + 1 from the point (1, 2). 2 Q3 Via the following steps, you are to maximise the function f (x, y, z) = x2 yz 2 subject to the inequality constraint g(x, y, z) ≡ x + y + 3z ≤ 2, together with the NNC x, y, z ≥ 0. Write down the four constraints in a form suitable for maximisation. (i) What can you say about the signs of the Lagrange multipliers in this inequality-constraint problem? Derive three FOC from the Lagrangian, together with four CSC, and so show that the maximiser is 4 (x∗ ; λ∗ ) = ( 4 , 2 , 15 ; 3254 , 0, 0, 0), giving f (x∗ ) = 2 5 5 8
29 . 32 55
(ii) Which constraints are binding? Evaluate the Jacobian derivative, Dg b (x), for the system of binding constraints and determine its rank. What does this tell you about the NDCQ? (iii) What are n and k for this problem? State (without proof) the conditions on the LPMs of the bordered Hessian, H, which ensure that H is negative deﬁnite. You are given that, in this case,...