Unbelievable $O(L^{1.5})$ worst case computational complexity achieved by $spdspds$ algorithm for linear programming problem

TL;DR

The paper introduces the spdspds algorithm, a novel iterative method for linear programming with a worst-case complexity of O(L^{1.5}), improving efficiency over traditional approaches.

Contribution

It presents a new symmetric primal-dual pivot strategy that achieves sub-quadratic worst-case complexity for linear programming problems.

Findings

Achieves worst-case complexity of O(L^{1.5})

Defines a global effectiveness measure based on infeasibility index

Allows exploration of alternative solutions beyond optimality

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 simply an exchange between a basic variable and a non-basic 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 is the problem-size expressed in terms of the size(length) of the input data.