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

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(L^1.5) where L refers to the problem-size expressed in terms of the size(length) of the input data.