MATH2640 Introduction to Optimisation
(i) Feed and cows are the inputs that best explain milk production. The production Y of milk
(in litres per day) is modelled by the Cobb-Douglas function as Y (X1, X2) = pXa
X1 the feed given per day and X2 the number of cows involved. Using linear regression (least
squares) these coecients have been determined to be p = 445.69(litres/day), a = 0.346 and
b = 0.542. The average price for a litre of milk is c3 = £5.38 and the cost of a kg of feed
c1 = £4.00, while the price of one cow per day is c2 = £1.36/d, yielding the constraint
c1X1 + c2X2 = c3. Calculate and find the optimium output of litres of milk per day using
constrained optimisation. Hence, find expressions for the (maximum) values Y ⇤, X⇤
1 , X⇤
the corresponding value ⇤ of the Lagrange multiplier. Check also that the NDCQ is satisfied.
Provide both algebraic expressions and numerical values of your outcomes.
(ii) Minimise f(x, y, z) = 1
2×2 + y2 + ( z
2 )2 subject to the constraints given by the intersection of
the two planes x + y z = 1 and x y z = 2. Check that the NDCQ is satisfied at the
stationary point. What is the “distance” from the origin to that point under the alternative
distance norm implied by the function f(x, y, z)? Make a sketch to illustrate the geometry of
the situation (optional).
A2. Use Bordered Hessians to determine the sign properties (definiteness) of the following constrained quadratic form:
Q(x1, x2, x3)=2×2
2 + x2
3 + kx1x3 kx1x2
subject to the constraints x1 + 2×2 + x3 = 0 and x1 + x2 x3 = 0, and with parameter k. Verify
the result by eliminating two of the variables using the constraints and determine the sign property
of the reduced quadratic form as function of k.
i) Write down the Lagrangian, and hence find the two stationary points of the problem
f(x, y, z) = 4
3×3 y2 + 2z2 + 2xz, subject to g(x, y, z) = x + y + z 1=0.
ii) Find the Bordered Hessian for this problem, and evaluate the required leading principal
minors for the (two) solutions.
a) Use the Lagrangian approach to find the maximum values of f(x, y) = x + y2 subject to the
constraint h(x, y) = x2 + y2 = 1.
b) Find the points on the ellipse in the xy-plane, given by x2 + xy + y2 = 9, closest to and
farthest away from the origin, and hence find the maximum and minimum distances of the
ellipse to the origin. Make a sketch of the situation in the xy-plane. [Hint: use x2 + y2 as the
c) Use the Lagrangian approach to find the distance of the plane 4x + 2y + z = 5 to the origin,
and compare it to the result known from geometric considerations. [Hint: use x2 + y2 + z2 as
Use the method of Lagrange multipliers to solve the following constrained optimisation problems. In each case, check that the non-degenerate constraint qualification (NDCQ) is satisfied at
the stationary points.
a) Find the maximum and minimum values of f(x, y, z) = x2 + y + z subject to the constraints
x2 + y2 + z2 = 50 and y + z = 6.
b) Maximise f(x, y, z) = xz + yz subject to y2 + z2 = 1 and xz = 3.
B3. Use Bordered Hessians to determine the sign properties (definiteness) of the following constrained quadratic forms:
a) Q(x, y) = x2 + 2xy y2 subject to x y = 0,
b) Q(x, y, z) = 6y2 + 3z2 + 8xy + 2yz 2xz subject to x + y z = 0,
c) Q(x1, x2, x3) = x2
3 + 4x1x2 + 2x1x3 subject to x1+x2+x3 = 0 and x1x2x3 = 0.
In the first two cases verify, by elimination of one of the variables, that the thus obtained reduced
quadratic form has the sign behaviour as predicted by the bordered Hessian approach.
a) Write down the Lagrangian, and hence find all the stationary points of the problem of maximising the function
f(x, y, z)=2x + 4y + 3z2 , subject to h(x, y, z) = x2 + y2 + z2 = 1 .
b) Find the Bordered Hessian for this problem, and evaluate the relevant leading principal minors
for both solutions. Hence classify the stationary points found in part (a).
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