Standard Form linear programming

Question

What are the advantages of using the Standard Form in Linear Programming, over the general form of a linear programming problem ?

Answer 1

Standard form is important for historical algorithmic reasons. The simplex method assumes an LP in standard form (see for example this report from 1956 or any modern textbook). Nowadays we almost never directly interface with a simplex solver. Most linear optimization software accepts inequality constraints and internally reformulates them if the simplex method is used. Algebraic modeling languages such as AMPL also do not require a specific input format. Despite not having practical value to the average student, standard form remains a useful tool in teaching some basic reformulations of linear optimization problems.

Please allow me to dwell a bit in history. The term 'standard form' was already known in 1956, but very few papers use that term. This paper from 1969 mentions 'standard form' but has inequality constraints, whereas this paper from 1971 uses the form as we know it today:

After providing slack variables for the inequality equations, the linear program in standard form is solved with a modified version of the MFOR-360 code, which uses the revised simplex method with the product form of the inverse.

The MFOR-360 solver was written by Joel Shwimer, who was a PhD student at MIT. This document from 1977 has a description and lists the system requirements of this solver:

MFOR 360 IS AN INDEPENDENT ROUTINE WHICH USES THE REVISED SIMPLEX METHOD WITH THE PRODUCT FORM OF THE INVERSE TO SOLVE THE LINEAR PROGRAMMING PROBLEM IN STANDARD FORM. MFOR 360 HAS BEEN COMPILED AND TESTED USING OS VERSION 11 ON A S/360 MODEL 65.

THE ROUTINE IS AN ALL-IN-CORE ROUTINE, THEREFORE NO SECONDARY STORAGE IS NEEDED. SYMBOLIC CONTROL CARDS DIRECT THE OPERATION OF MFOR 360.

PROGRAMMING SYSTEMS - WRITTEN IN FORTRAN IV (G LEVEL) WITH ONE SUBROUTINE IN 360 BASIC ASSEMBLER lANGUAGE.

MINIMUM SYSTEM REQUIREMENTS - THOSE NEEDED TO RUN OS/360.

MFOR-360 was not the only solver at athe time. This 1974 NASA report mentions LINPROG (1969) as an alternative, but I also found a solver called LPI (1974). Those solvers also required standard form as input.