A family of search directions for Karmarkar’s algorithm

The authors consider a new family of search directions for the standard form variant of Karmarkar’s projective linear programming algorithm. The family includes the usual projected gradient direction, and also a direction first proposed by Mike Todd. The authors prove that any choice from the family preserves the algorithm’s polynomial-time complexity. They then examine the computational behavior of the algorithm using different choices of directions. Although the theoretical complexity is the same for the different directions, in practice the authors find wide variations in algorithm performance. One particular choice consistently requires about 20% fewer iterations than the usual direction, while another requires a number of iterations which grows rapidly with problem size. The present computational results also demonstrate that a small number of monotonic steps on early iterations may considerably improve the performance of the algorithm.