Title: Classes of ABS methods for solving Diophantine equations, certain inequalities and LP problems

ABS (Abaffy-Broyden-Spedicato) algorithms have been used extensively for solving systems of equations and certain optimization problems during the past two decades. We have recently used the ABS approach to solve Diophantine systems of linear equations, and full row rank inequalities. We show the conditions for the existence of the solution of the Diophantine equations and we present algorithms for the computation of the general solution. We also explain how the general solution of full row rank linear inequality systems can be computed over both the real and the integer spaces. We apply the results to certain class of Linear Programming (LP) Problems, obtaining optimality conditions and computing the general solutions.