ISE 426: Homework 1
Due Date: Wednesday, September 20th, by 11:59pm on Coursesite
Total 100 points
1. (6 points) Function Identification: For each of the plots below, state whether f (x) is a function or
not. Briefly justify your answer.
2. (16 points) Convex Functions: Recall the following formal definition of a convex function: A function
f (x) is convex, if for every x0 , x00 ∈ X (domain of the function), and α ∈ [0, 1], f (αx0 + (1 − α)x00 ) ≤
αf (x0 ) + (1 − α)f (x00 ). Use this definition to prove whether the following functions are convex or not
Hint: To prove that a function is not convex find explicit x0 , x00 , α that would not satisfy the definition.
(a) (2 points) For any x ∈ IR consider the unidimensional function f (x) = ax + b, for any constants
a, b ∈ IR.
(b) (4 points) For any x ∈ IRn consider the multidimensional function f (x) = a| x + b, where x =
(x1 , x2 , . . . , xn ), and a ∈ IRn is constant n-dimensional vector and b ∈ IR is a constant.
(c) (4 points) For any x ∈ IR consider the unidimensional function f (x) = |x|. Hint: Use the triangle
inequality; namely, |a + b| ≤ |a| + |b|.
(d) (2 points) For any x ∈ IR consider the unidimensional function f (x) = −x2 .
(e) (4 points) For any x ∈ IRn let f (x) be a convex function. Now, for any for any x ∈ IRn let
g(x) = af (x) where is a non-negative constant (a ≥ 0). Show whether g(x) is convex or not.
3. (16 points) Convex Sets: Recall the following formal definition of a convex set: A set S is convex, if
for every x0 ∈ S, x00 ∈ S, and α ∈ [0, 1], αx0 + (1 − α)x00 ∈ S. Use this definition to prove whether the
following sets are convex or not Hint: To prove that a set is not convex find explicit x0 , x00 , α that would
not satisfy the definition.
(a) (2 points) S = {(x, y) ∈ IR2 : x2 + y 2 = 1}
(b) (4 points) S = {(x, y) ∈ IR2 : x ≤ 1, y ≤ 1, x ≥ 0, y ≥ 0}
(c) (4 points) S = {(x1 , x2 ) : 2x1 − 3x2 = 1}
(d) (4 points) S = {(x1 , x2 ) : x1 x2 ≤ 1, x1 ≥ 0, x2 ≥ 0}
(e) (2 points) S is the star set depicted in the graph below:
4. (16 points) Gradients and Hessians: For each of the functions f (x) below, compute the gradient
∇f (x) and the Hessian ∇2 f (x).
(a) (3 points) f (x1 , x2 ) = ex1 sin(x2 )
(b) (3 points) f (x1 , x2 , x3 ) = x21 + 5x22 + 4x1 x2 − 2x2 x3
(c) (3 points) f (x1 , x2 ) = 5x21 + 3x42
√
(d) (3 points) f (x1 , x2 ) = ln( x1 x2 ), x1 > 0, x2 > 0
p
(e) (4 points) f (x1 , x2 ) = tan(x1 ) + x2
5. (16 points) Convex functions and PSD Hessians: Recall that a function f (x) defined in a domain
X is convex on that domain if the function’s Hessian ∇2 f (x) is positive semidefinite (PSD) for all x ∈ X .
Use this fact, and the characterizations of PSD matrices discussed in class (avoid using the computation
of eigenvalues) to find out if the following functions (the ones for which you computed the Hessian in the
previous question) are convex.
(a) (3 points) f (x1 , x2 ) = ex1 sin(x2 )
(b) (3 points) f (x1 , x2 , x3 ) = x21 + 5x22 + 4x1 x2 − 2x2 x3
(c) (3 points) f (x1 , x2 ) = 5x21 + 3x42
√
(d) (3 points) f (x1 , x2 ) = ln( x1 x2 ), x1 > 0, x2 > 0
p
(e) (4 points) f (x1 , x2 ) = tan(x1 ) + x2
6. (10 points) Simple Optimization Problem: Pintuco produces both Interior and Exterior paint from
two raw materials called M1 and M2 (guess they want to keep them secret!). The following table provides
the units of the raw materials that are needed to make the interior and exterior paint, as well as the
maximum daily availability of these raw materials.
Tons of raw material per ton of
Exterior Paint Interior Paint
Raw Material M1
Raw Material M2
6
1
Maximum daily
availability (Tons)
4
2
24
6
A market survey indicates that the daily demand for interior paint cannot exceed that of exterior paint
by more than one (1) Ton. Also, the daily maximum demand of interior paint is two (2) tons. Finally,
Pintuco also know that Interior paint gives them a profit of $5000 per Ton, and Exterior paint gives them
a profit of $4000 per Ton.
Pintuco wants to determine the optimum (best) production quantities of Interior and Exterior paint
that maximizes the total daily profit. For that purpose, formulate a Linear Mathematical Programming
formulation for this decision-making problem:
(a) Decision variables.
(b) Objective.
(c) Constraints.
7. (10 points) Simple Optimization Problem: Box Allocation Problem: In class, we discussed the
optimization problem of locating boxes of given height into two bins such that the bin with the maximum
height has the minimum possible height, and discussed the solution of this problem in different ways.
Write a Mathematical Programming formulation for this problem:
(a) Decision variables. Hint: there should be one decision variable for each box
(b) Objective.
(c) Constraints.
not to be marked Can you write the mathematical formulation for he more general case of m bins and
n boxes with heights hi , i = 1, . . . , n?
8. (10 points) Back to Pintuco Problem: Determine the best feasible solution among the following
(possibly infeasible) solutions of the Pintuco Problem you formulated above:
Solution
a
b
c
d
e
Interior Paint (Tons)
Exterior Paint (Tons)
1
2
3
2
2
4
2
1.5
1
-1