Indeed many authors have various versions of standard form LP problems, but I am wondering if the following is an essential trait:
In an LP problem we have min(f(x)) with constraint,say, $Ax\geq b$. My question is whether the RHS has to be positive when changing to standard form eg. $3x-2y\leq -5\Rightarrow -3x+2y-s= 5$?
I don't think so because the simplex method relies on comparing the pivots i.e. ratios of coeff/RHS, and so any sign difference cancels out.
The simplex method starts with a basic feasible solution. The RHS can be negative in the standard form, but when we want to find a basic feasible solution it is much easier to have non-negative RHSs. Here are some references:
Linear and Nonlinear Programming (David G. Luenberger, Yinyu Ye):
Linear Programming and Network Flows (Hanif D. Sherali, John J. Jarvis, and M. S. Bazaraa):
A linear program is said to be in standard format if all restrictions are equalities and all variables are non-negative. The simplex method is designed to be applied only after the problem is put in standard form.
