The cosine simplex algorithm

Abstract

An extension of the simplex algorithm is presented. For a given linear programming problem, a sequence of relaxed linear programming problems is solved until a solution to the original problem is reached. Each successive relaxed problem is obtained from the previous one by adding a single constraint chosen from the constraints violated by the solution to the previous relaxed problem. This added constraint maximizes the cosine of the angle that the gradient of any violated constraint forms with the gradient of the objective function. In other words, each successive relaxed problem is obtained by adding the violated constraint most parallel to the objective function. The proposed algorithm terminates when no constraints are violated. Preliminary results indicate that this cosine simplex algorithm reduces both the number of simplex iterations and the number of computations at each iteration.