A hybrid gradient and feasible direction pivotal solution algorithm for general linear programs

摘要

The simplex solution algorithm for linear programs (LP) can be considered as a sub-gradient direction method therefore a full gradient solution algorithm might be more efficient. We have developed a full gradient method which consists of three phases. The initialization phase provides the initial tableau which may not have a full set of basis. The push phase uses a full gradient vector of the objective function to obtain a feasible vertex. This is then followed by a series of pivotal steps using the sub-gradient, which leads to an optimal solution (if exists) in the final iteration phase. At each of these iterations, the sub-gradient provides the desired direction of motion within the feasible region. The algorithm hits and/or moves on the constraint hyper-planes and their intersections to reach an optimal vertex (if exists). The algorithm works in the original decision variables space, therefore, there is no need to introduce any new extra variables such as artificial variables. The proposed algorithm is practical, easy for the user to understand, and provides useful information for the decision makers.