Dire Dawa University
School of Electrical & Computer Engineering
Mathematical Foundations in Engineering
Assignment on Optimization
Submission date: Feb 24, 2022
1. Consider the following LP problem
max f = 3x1 + 2x2
2x1 + x2 ≤ 4
s.t.
2x1 + x2 ≤ 2
x1 − x2 ≤ 1
x1 , x2 ≥ 0.
(a) Solve the problem graphically
(b) Verify the solution using Excel’s solver package.
2. A farmer has 100 acres of land on which she plans to grow wheat and corn. Each acre
of wheat requires 4 hours of labor and $20 of capital, and each acre of corn requires
16 hours of labor and $40 of capital. The farmer has at most 800 hours of labor and
$2400 of capital available. If the profit from an acre of wheat is $80 and from an acre
of corn is $100, how many acres of each crop should she plant to maximize her profit?
Use graphical method to solve the problem.
3. Consider the function of three variables given by
f (x1 , x2 , x3 ) = x21 − x1 − x1 x2 + x22 − x2 + x43 − 4x3
(a) Compute the gradient ∇f (x1 , x2 , x3 )
(b) Compute the Hessian matrix H(x1 , x2 , x3 )
(c) Use the gradient to find a local extremum of f
(d) Compute the three principal minors of the Hessian matrix and use them to identify the extremum as a local minimum or a local maximum.
4. For each of the following, determine whether the function is convex, concave, or neither
(a) f = x1 x2 − x21 − x22
(b) f = 10x1 + 20x2
(c) f = x41 + x1 x2
(d) f = −x21 − x1 x2 − 2x22 .
5. Let the following function be defined for all points (x, y) in the plane
f (x, y) = 2xy − x4 − x2 − y 2
(a) Write the gradient of the function f
(b) Write the Hessian matrix of f
(c) Is the function f convex, concave or neither?
(d) Use the gradient to find a local extremum of f
(e) Identify this extremum as a minimum, a maximum or neither.
6. Solve the problem
max f = 2x + y
s.t.
4x2 + y 2 = 8
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7. Solve the following constrained optimization problem using the method of Lagrange
multipliers
max f = ln x + 2 ln y + 3 ln z
s.t.
x + y + z = 60
8. Find the two points on the ellipse given by x21 + 4x22 = 4 that are at minimum distance
from the point (1, 0). Formulate the problem as a minimization problem and solve it
by solving the Lagrangian equations. [Hint: To minimize the distance d between two
points, one can also minimize d2 . The formula for the distance between points (x1 , x2 )
and (y1 , y2 ) is d2 = (x1 − y1 )2 + (x2 − y2 )2 .]
9. Solve the problem
min f = (x − 2)2 + 2(y − 1)2
s.t.
x + 4y ≤ 3
x≥y
10. Record’m Records needs to produce 100 gold records at one or more of its three studios.
The cost of producing x records at studio 1 is 10x; the cost of producing y records at
studio 2 is 2y 2 ; the cost of producing z records at studio 3 is z 2 + 8z.
(a) Formulate and solve the nonlinear program of producing the 100 records at minimum cost using the Lagrangian method.
(b) Verify your answer using Excel’s Solver package.
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