UNBELIEVABLE O(L1.5) WORST CASE COMPUTATIONAL COMPLEXITY ACHIEVED BY spdspds ALGORITHM FOR LINEAR PROGRAMMING PROBLEM ©Dr.(Prof.) Keshava Prasad Halemane, Professor - retired from Department of Mathematical And Computational Sciences National Institute of Technology Karnataka, Surathkal Srinivasnagar, Mangaluru - 575025, India. Residence: 8-129/12 SASHESHA, Sowjanya Road, Naigara Hills, Bikarnakatte, Kulshekar Post, Mangaluru - 575005. Karnataka State, India. k.prasad.h@gmail.com [+919481022946] https://arxiv.org/abs/1405.6902 https://archive.org/details/s-p-d-s-p-d-s https://doi.org/10.5281/zenodo.4553754 https://hal.archives-ouvertes.fr/hal-03087745 https://www.linkedin.com/in/keshavaprasadahalemane/ ABSTRACT The Symmetric Primal-Dual Symplex Pivot Decision Strategy (spdspds) is a novel iterative algorithm to solve linear programming problems. A symplex pivoting operation is considered simply as an exchange between a basic (dependent) variable and a non-basic (independent) variable, in the Goldman-Tucker Compact-Symmetric-Tableau (CST) which is a unique symmetric representation common to both the primal as well as the dual of a linear programming problem in its standard canonical form. From this viewpoint, the classical simplex pivoting operation of Dantzig may be considered as a restricted special case. The infeasibility index associated with a symplex tableau is defined as the sum of the number of primal variables and the number of dual variables that are infeasible. A measure of goodness as a global effectiveness measure of a pivot selection is defined/determined as/by the decrease in the infeasibility index associated with such a pivot selection. The selection of the symplex pivot element is made by seeking the best possible anticipated decrease in the infeasibility index from among a wide range of candidate choices with non-zero values - limited only by considerations of potential numerical instability. After passing through a non-repeating sequence of CST tableaus, the algorithm terminates when further reduction in the infeasibility index is not possible; then the tableau is checked for the terminal tableau type to facilitate the problem classification - a termination with an infeasibility index of zero indicates optimum solution. Even in the absence of an optimum solution, the versatility of the spdspds algorithm allows one to explore/determine the most suitable alternative solutions, including possibly a comprehensive parametric analysis, etc. The worst-case computational complexity of the spdspds algorithm is shown to be O(L1.5).

AMS MSC Mathematics Subject Classification: 90C05 ACM CCS Computing Classification System: F.2.1, G.1.6