A gravitational interior point method for LP

Katta G. Murty ، In [4, 1] gravitational methods for linear programming (LP) have been introduced. Several versions exist, the three main versions discussed there use a ball of (a): 0 radius, (b): small positive radius, and (c): the ball of largest possible radius with the given center that will completely fit within the polytope, with the option of changing its radius as the algorithm progresses. In versions (a), (b), after the first move, the center of the ball always remains very close to the boundary (because the ball hugs the boundary), and hence these versions behave like other boundary algorithms such as the simplex algorithm in terms of exponential complexity in the worst case [3]. Here we discuss a gravitational method of type (c) that behaves like an interior point method. To guarantee that the ball used has the largest possible radius, it uses a new centering strategy that moves any interior feasible solution x 0 to the center of the intersection of the feasible region with the objective hyperplane through x 0 before beginning each gravi-tational descent move. Also, using this centering strategy we discuss a method that can obtain an approximate optimum solution for an LP by a very efficient method without using any matrix inversions. Key words: Linear programming (LP), gravitational method, in-terior point method, avoiding zigzagging, approximate optimum without matrix inversions