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MATH2003J Optimisation in Economics SPRING TRIMESTER EXAMINATION - 2020/2021
发布时间：2024-05-29
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SPRING TRIMESTER EXAMINATION - 2020/2021
MATH2003J Optimisation in Economics
1. (a) Determine whether each of the following statements is True or False. No explanation is needed when answering 1(a)(i) to 1(a)(v).
(i) The set S = [2, 5] ∩ Z is convex. [1]
(ii) The following is a pure integer linear programming problem:
Maximize z = 2x1 + 5x2
subject to x1 + 2x2 ≤ 8
3x1 + 8x2 ≤ 25
x1 , x2 ≥ 0, x1 ∈ Z [1]
(iii) If the feasible set of the LP-relaxation of an integer linear programming problem is empty, then the feasible set of this integer linear program- ming problem is also empty. [1]
(iv) For the integer linear programming problem:
Maximize z = 25x1 + 50x2
subject to 5x1 + 12x2 ≤ 80
30x1 + 18x2 ≤ 220
x1 , x2 ≥ 0, x1 , x2 ∈ Z,
its LP-relaxation is:
Maximize z = 25x1 + 50x2
subject to 5x1 + 12x2 ≤ 80
30x1 + 18x2 ≤ 220
(v) The set S = {(x, y) ∈ R 2 : 16x 2 + y 2 < 16} is closed. [1]
(b) Determine whether each of the following statements is True or False.
Briefly justify your answers to questions 1(b)(i) to 1(b)(v).
(i) The function f : R → R defined by f(x) = 3x4 − 16x3 + 24x2 has a local minimum at x = 0. [3]
(ii) The function g : R2 → R defined by g(x, y) = x3 + 27y3 − 3x2 − 36y + 8 has two critical points. [3]
(iii) This question refers to 1(b)(ii). The function g has a saddle point at (2, (iv) The set S = {(x, y, z) ∈ R3 : x2 + y2 + z2 < 16 and x > 0} is bounded. [3]
(v) The following is a linear programming problem in standard form for the simplex method:
Maximize z = x1+ 3x2
subject to 5x1 + 6x2 ≤ 20
2. Solve the following linear programming problem by the simplex method:
Minimize z = −8x1 − 5x2
subject to 8x1 + x2 ≥ 72
2x1 + x2 ≤ 21
x1, x2 ≥ 0. [20]
3. (a) Consider the following linear programming problem:
Maximize z = 5x1 + 6x2
subject to x1 + 3x2 ≤ 15
2x1 + x2 ≤ 15
x1, x2 ≥ 0.
(i) Solve the above problem with the simplex method. [8]
(ii) Formulate the dual problem. [5]
(iii) Determine the optimal solution to the dual problem from the optimal tableau of the original problem. [3]
(b) Consider the symmetric matrix
A = 3
Determine whether it is positive definite, negative definite or neither of them. [4]
4. Letf : R2 → R bedefined byf(x‘ i) = 12x2 −3i2 +12 and consider the constraints
x2 + i2 ≤ 9 and x ≤ i.
(a) Sketch the feasible set in the plane and explain why f attains extrema (maximum and minimum) subject to the above constraints.
(b) Use the Kuhn-Tucker method to find the maximum and the minimum of f subject to the above constraints.
5. (a) Consider the following integer linear programming problem:
Maximize Σ = 15x + 30i
subject to x + 3i ≤ 18
x + i ≤ 8
x‘ i ≥ 0‘ x‘ i ∈ Z.
(i) Formulate the LP-relaxation of the above integer linear programming problem. Solve this LP-relaxation by graphical method. [Q]
(ii) From the results in 5(a)(i), what can you conclude on the optimal value of the objective function of the above integer linear programming problem?
(b) Give an example of a convex set in R2 which is different from R2, the empty set and sets consisting of one single point. Justify your answer. [Q]
(c) Let f : R3 → R be defined by
f(x‘ i‘ Σ) = x2 + 3i2 + 3Σ4 − 6.
Determine whether f is convex. [Q]
