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

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 Tucker's 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, which 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. At each iteration 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. 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. The worst case computational complexity of spdspds is shown to be O(L^1.5).

A comparative study of pivot selection strategies for unsupervised cross-domain sentiment classification

Selecting pivot features that connect a source domain to a target domain is an important first step in unsupervised domain adaptation (UDA). Although different strategies such as the frequency of a feature in a domain, mutual (or pointwise mutual) information have been proposed in prior work in domain adaptation (DA) for selecting pivots, a comparative study into (a) how the pivots selected using existing strategies differ, and (b) how the pivot selection strategy affects the performance of a target DA task remain unknown. In this paper, we perform a comparative study covering different strategies that use both labelled (available for the source domain only) as well as unlabelled (available for both the source and target domains) data for selecting pivots for UDA. Our experiments show that in most cases pivot selection strategies that use labelled data outperform their unlabelled counterparts, emphasising the importance of the source domain labelled data for UDA. Moreover, pointwise mutual information and frequency-based pivot selection strategies obtain the best performances in two state-of-the-art UDA methods.