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dikcijom2k 2022-07-12 Answered
Consider this system:
e i = b i j = 1 n a i , j x j ( i = 1 , 2 , . . . m )
where a i , j ( ( 1 i m ,   1   j n )) and ( 1 i m ) are given. The problem is to find an assignment of values to the variables x 1 , . . . , x n that minimizes max | e j |. Express this problem as a linear program in the standard form.
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Answers (1)

Tristin Case
Answered 2022-07-13 Author has 15 answers
Let e = [ e 1 e m ] , e = [ e 1 e m ] , A = [ a 11 a 1 n a m 1 a m n ] , x = [ x 1 x n ]
Then you can equivalently right your system as:
e = b A x
Let, scalar t be the value of max j | e j |, then you will have:
e j t     j = 1 , , m e j t     j = 1 , , m
Now let 1 = [ 1 1 ] be a vector of m stackes 1's, then these constraints can be written as:
e t 1 e t 1
where is the element-wise less than/or equal. Equivalently you have:
b A x t 1 A x b t 1
the linear program that you are looking for will be:
max x , t     t subject to b A x t 1 A x b t 1
max x , t     t subject to A x t 1 b A x t 1 b
If you want to standardize it further, do the following: let
z = [ x T     t ] T , c = [ 0 0 n  times     1 ] T , d = [ b T     b T ] T , H = [ A 1 A 1 ] , now it can be written as:
max z     c T z subject to H z d

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