Given a system of inequalities A x &#x2264;<!-- ≤ --> b , how can I derive the upper and

Nicholas Cruz

Nicholas Cruz

Answered question

2022-05-28

Given a system of inequalities A x b , how can I derive the upper and lower bound on x?
I have the following system of inequalities A x b, where x 1 and x 2 are unknown, and the a's and и's are constants.
[ a 11 a 12 a 21 a 22 ] [ x 1 x 2 ] [ b 1 b 2 ]

Answer & Explanation

floygdarvn

floygdarvn

Beginner2022-05-29Added 12 answers

Given
a 11 x 1 + a 12 x 2 b 1 a 21 x 1 + a 22 x 2 b 2
Suppose a 11 and a 21 > 0 ,, then
x 1 b 1 a 12 x 2 a 11 x 1 b 2 a 22 x 2 a 21
Suppose a 11 and a 21 < 0 ,, then
x 1 a 12 x 2 b 1 a 11 x 1 a 22 x 2 b 2 a 21
The upper bound of x 1 is m i n ( b 1 a 12 x 2 a 11 , b 2 a 22 x 2 a 21 ) , and the lower bound of x 1 is m a x ( a 12 x 2 b 1 a 11 , a 22 x 2 b 2 a 21 )
reryfaikear

reryfaikear

Beginner2022-05-30Added 3 answers

You are looking for Fourier–Motzkin elimination, of which you are performing just the first step. The next step is concluding that if the lower bound on x 1 is not greater than the upper bound on x 1 , you know that a feasible x 1 exists, and you no longer need to specify its value, leaving you with inequalities for just x 2 .

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