Linear equations, Inequalities, Linear Programs (LP), and a New Efficient Algorithm

The dawn of mathematical modeling and algebra occurred well over 3000 years ago in several countries (Babylonia, China, India,...). The earliest algebraic systems constructed are systems of linear equations, and soon after, the famous elimination method for solving them was discovered in China and India. This effort culminated in the writing of two books that attracted international attention by the Arabic mathematician Muhammad ibn-Musa Alkhawarizmi in the first half of 9th century. The first, Al-Maqala fi Hisab al-jabr w’almuqabilah (An essay on Algebra and equations), was translated into Latin under the title Ludus Algebrae, the name “algebra” for the subject came from this Latin title, and Alkhawarizmi is regarded as the father of algebra. Linear algebra is the branch of algebra dealing with systems of linear equations. The second book Kitab al-Jam’a wal-Tafreeq bil Hisab al-Hindi appeared in Latin translation under the title Algoritmi de Numero Indorum (meaning Alkhawarizmi Concerning the Hindu Art of Reckoning), and the word “algorithm” for procedures for solving algebraic systems originated from this Latin title. The elimination method for solving linear equations remained unknown in Europe until Gauss rediscovered it in 19th century while approximating by a quadratic formula the orbit of the asteroid Ceres based on recorded observations in tracking it earlier by the Italian astronomer Piazzi. Europeans gave the names “Gaussian elimination method”, “GJ (Gauss-Jordan) elimination method” for this method. However, there was no computationally viable method until recently to solve systems of linear constraints including inequalities. Examples of linear constraints with inequalities started appearing in published literature in mid-18th century. In the 19th and early 20th century Fourier,