ISE 530 Optimization Methods for Analytics Homework 2
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
ISE 530 Optimization Methods for Analytics
Homework 2
1. Assume that we haven customers in a plane, each with known coordinates (xi, yi), i = 1,...,n. We want to find coordinates (¯(x), ¯(y)) to open a store, so that the sum of the distances from (¯(x), ¯(y)) to each of the points (xi, yi) is minimized. Formulate this problem as a nonlinear optimization problem.
2. A company wants to maximize its revenue by adjusting the prices of its three prod-
ucts: Cola, Orange and Lemon. The demand for each product i ∈ {Cola, Orange, Lemon} is a function of its price pi and is given by Di = ai−bipi for some known parameters
ai, bi ∈ R. In addition, there are side constraints that need to be satisfied: a) Cola, as its premium product, needs to have a highest price.
b) The average price of a bottle needs to be at least 10 dollars.
c) The price of two bottles of Cola cannot exceed the prices of three bottles of any other product.
Formulate this problem as a quadratic optimization problem.
3. Consider the system of linear inequalities
x1+ 2x2+ 3x3+ 4x4 ≤ 5
x1 − x2 ≥ 0
x2 ≥ x3 + x4
x1 , x2 , x3 , x4 ≥ 0,
and consider the point ¯(x)⊤ = (−1, 0, 1, 2). Formulate a problem that finds the
closest point to ¯(x) that satisfies all constraints. Use any solver to find this point.
4. Consider the “small data” dataset posted with quiz 4. Formulate a problem of selecting three points (among the 15) to serve as centroids of clusters, such that sum of the distance between each point and its assigned centroid is minimized, with the additional constraint that each cluster has the same number of points.
5. Find an optimal solution of the optimization problem
minf(x) = (x1 + x2 )2 + (3 − 2x1 + x2 )2 + (1 + 2x2 )2
x
by setting the gradient to 0.
6. Compute the square root of 1013 (find a root of g(x) = x2 − 1013) using:
❼ Binary search (compute to 1 decimal digit of precision). Use a lower bound of 10 and an upper bound 50 in the initial iteration.
❼ Newton method.
Show the detail of all your iterations in all cases.
7. Find an optimal solution of the one-dimensional optimization problem
x(mi)nf(x) = x2 + 
,
i.e., find a point where f′ (x) = 0, using either binary search or Newton method. Detail each iteration of the algorithm.
8. We are given 7 points with the following values:
|
|
a1 |
a2 |
a3 |
a4 |
a5 |
a6 |
a7 |
|
Value |
1 |
2 |
4 |
8 |
9 |
10 |
11 |
Solve the problem
R(n) 
(ai− x)4
using
a) Binary search. Compute a solution within 0.1 of the optimal. You may assume that an optimal solution satisfies 1 ≤ x∗ ≤ 11.
b) Newton method (until convergence), starting from the point x0 = 1.
2023-12-16