Developing a Boundary Point Method Algorithm for Solving Linear Programming Problems with Feasible Initial Solution

Abstract

In this research, SALCHOW algorithm has been developed to solve linear programming problems. In each step SLACHOW moves towards the constrained gradient of the objective function, so that it always remains within the feasible region. This type of generating sequence of feasible solutions on the boundary of the feasible region differs from the behavior of the simplex. Simplex moves on the corners of the feasible region. On the other hand, SALCHOW is also different from interior point methods; because interior point methods generate solutions that are not on the corner points or even borders of feasible region. SALCHOW assigns a set of coefficients to some active constraints for appending to objective function and updating constrained gradient of objective function. Finally at the optimal point, the Lagrange coefficients of the active constraints are found. Computational results are generated by using a set of randomly generated instance problems and a few standard ones from OR-Library. These results show the superiority of SALCHOW over the simplex in these small instances. In other words, the mean time of solving an instance with SALCHOW is a function of the number of decision variables in contrast with Simplex. Runtime of simplex in the average is a function of the number of constraints. The computational errors caused by round off errors in developed code in MATLAB exhibits that our developed code for SALCHOW suffers from cumulative errors; and it obstructs the possibility of judging the definite superiority of SALCHOW over the simplex in solving small instance problems.

Keywords

• linear programming
• active set methods
• polynomial time order
• feasible direction method