How to find the minimum difference between two linear functions?

fjn6869930

Z = aX-bY (a, b are constants, X, Y are integers and a certain range exists) Find the minimum value of Z?

lhempire

From the equation point of view, it is a plane in a three-dimensional space. After limiting the range of x and y, a parallelogram of a space is obtained.
z is the smallest, that is, the lowest point of the space bounding box of the parallelogram
Since it is a linear relationship, the extreme values ​​all occur on the boundary. Calculate z using four boundary points in the x and y range.

zhangjk

This is the simplest linear programming (LP) problem and can be solved in many ways, such as the "graphing method".

fjn6869930

Still do not understand

muscle1

Just judge the sign of a and b, if a is less than or equal to 0, take the largest plant of X, otherwise X takes the minimum value; if b is less than or equal to 0, take the smallest plant of Y, otherwise Y takes the maximum value. The Z value obtained in this way must be the smallest.

muscle1

That is to say, Z is composed of two parts, aX and -bY. If both parts get the minimum value, the value of Z must be the smallest, and the minimum value of the two parts within a certain range is very easy to find, which is the above Too.

suny00059

Code running on Linux:

#include <iostream>

using namespace std;

int main ()
{
int a, b;
int X_min, X_max;
int Y_min, Y_max;
int Z;
int X, Y;
int doonce = 0;

cout << "Input a, b" << endl;
cin >> a >> b;
cout << "Input X_min, X_max" << endl;
cin >> X_min >> X_max;
cout << "Input Y_min, Y_max" << endl;
cin >> Y_min >> Y_max;
The
for (X = X_min; X <= X_max; X ++)
{
for (Y = Y_min; Y <= Y_max; Y ++)
{
if (doonce == 0)
{
Z = a * X-b * Y;
doonce = 1;
}
if (Z <(a * X-b * Y))
{
Z = a * X-b * Y;
}
}
}
cout << "The min of a * X-b * Y =" << Z << endl;
return 0;
}

suny00059

I forgot to add the absolute value

suny00059

By the way, the less-than sign is changed to a greater-than sign, otherwise it is the maximum value

fjn6869930

Z = aX-bY (a, b are constants, real numbers, X, Y are integers and there is a certain range) What is the minimum value of | Z |?
I forgot to add the absolute value in the front. Is this difficult to find the minimum value of | Z |?

If only the minimum value of Z is sought, as in the answer oflhempire, that is, the minimum value of ax minus the maximum value of by.

Thanks to thesuny00059program, in fact, in my case, X and Y may be a larger range, or may need to select two or more numbers in a set of real numbers as a and b to calculate So using the exhaustive method may result in orders of magnitude of calculation time. Now I want to find a way to find the minimum or close to the minimum | Z |.