Converting inequalities into equalities by adding more variablesSo my question is, it is valid to do so? What I'm trying to do is to find the range of x where the following is true: A x ≤ 0 , x ≥ 0where A is a specific m × n coefficient matrix and x is an n × 1 vector. But is this the same as solving the following: A ~ x ~ = 0 , x ~ ≥ 0where A ~ is an m × ( n + m ) matrix and x ~ is an ( n + m ) × 1 vector. By solution, I mean writing the elements of the original x in terms of the elements that are in x ~ but not x.

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Converting inequalities into equalities by adding more variablesSo my question is, it is valid to do so? What I&#039;m trying to do is to find the range of x where the following is true:  A  x  ≤  0  ,  x  ≥  0where   A is a specific   m  ×  n coefficient matrix and   x is an   n  ×  1 vector. But is this the same as solving the following:        A    ~        x    ~    =  0  ,      x    ~    ≥  0where         A    ~   is an   m  ×  (  n  +  m  ) matrix and       x    ~   is an   (  n  +  m  )  ×  1 vector. By solution, I mean writing the elements of the original   x in terms of the elements that are in       x    ~   but not   x.

Angelo Murray · Accepted Answer

We will show that the two problems are equivalent in the sense they define the same range of solutions.Define         A    ~    =      [                                    A                                  I                      ]   and         x    ~    =      [                                    x                                              t                      ]  Then, if   x is a solution of the inequality constrained problem :  A  x  ≤  0it suffices to take   t as   −  A  x, which is a positive vector by the inequality above. So we have :         A    ~          x    ~    =  A  x  +  t  =  A  x  −  A  x  =  0Conversely, if        A    ~          x    ~    =  0then we have :   A  x  +  t  =        A    ~          x    ~    =  0, or equivalently   A  x  =  −  t. Since   t is a positive vector, we can conclude that   A  x  ≤  0.

Converting inequalities into equalities by adding more variables So my question is, it is valid to

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