UNBELIEVABLE O(L^1.5) WORST CASE COMPUTATIONAL COMPLEXITY ACHIEVED BY spdspds ALGORITHM FOR LINEAR PROGRAMMING PROBLEM

UNBELIEVABLE  O(L^1.5)  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.  Here, 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 governed by the anticipated decrease in the infeasibility index - seeking the best possible 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 in a given context, including a comprehensive parametric analysis, etc.  The worst-case computational complexity of spdspds algorithm is shown to be O(L^1.5).