Concise algorithm for linear programs in matlab: monotonic convergence, basic variables, boundedness

A physically concise polynomial-time iterative algorithm due to Barnes – a variation of Karmarkar projective transformation algorithm – is presented in Matlab for linear programs 0 ,   x b Ax to subject x c Min t . The concerned monotonic convergence of the solution vector and the consequent detection of basic variables are stated. The boundedness of the solution, multiple solutions, and no solution (inconsistency) cases are discussed. The possibility of applying Aitken’s 2  -process to accelerate convergence of the solution vector has been studied taking advantage of monotonic convergence. The effect of degeneracy of the primal linear program and/or its dual on the uniqueness of the optimal solution is mentioned. The foregoing algorithm is implemented in another way based on detection of basic variables and then solving the resulting linear system involving only the basic variables mathematically non-iteratively. The second way of implementation also includes optimality test and is coded in Matlab. It results in less number of iterations and usually more accurate optimal solution. Numerical experiments are carried out on these algorithms considering several typical linear programs and the Matlab implementation of these algorithms is found to be useful for solving many real-world problems.