[SOLVED] MATH2003J OPTIMIZATION IN ECONOMICS BDIC 2023/2024 SPRING Problem Sheet 10 Python

MATH2003J, OPTIMIZATION IN ECONOMICS,

BDIC 2023/2024, SPRING

Problem Sheet 10

Question 1:

Let f ∶ R2 → R be defined by f(x, y) = −20x − 10y2
and the constraints

x2 + y2
≤ 100 and x ≥ 0.

(I) Sketch the feasible set and show that it is closed and bounded. Conclude that f achieves both its maxima and minima under the above constraints.

(II) Using the Kuhn-Tucker method, find the extrema of f subject to the above constraints.

Question 2:

Let f ∶ R2 → R be defined by f(x, y) = 2x + y2 − 3 and consider the constraints

x2 + y2
≤ 10 and x ≥ 0.

(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.

Question 3:

Let f ∶ R2 → R be defined by f(x, y) = 2 + x + y
2
, and the constraints

x2 − 2x + y
2
≥ 0, x2 − 4x + y
2
≤ 0, x + y ≤ 3

(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.