Topological structure and a polynomial-time solution of linear programming over the real numbers

TL;DR

This paper introduces a polynomial-time algorithm for linear programming over real numbers that leverages topological structures and squeeze mappings to efficiently identify optimal support sets.

Contribution

It presents a novel O(mn^2) algorithm utilizing topological insights and squeeze mappings to solve linear programming problems in polynomial time.

Findings

Algorithm runs in O(mn^2) time.

Supports identification of optimal solutions via topological projection.

Uses geometric properties of hypercubes and spheres for solution detection.

Abstract

We present an O(mn^2) algorithm for linear programming over the real numbers with n primal and m dual variables through deciding the support set a of an optimal solution. Let z and e be two 2(n+m)-tuples with z representing the primal, dual and slack variables of linear programming, and e the all-one vector. Let Z denote the region including all (tz, t) with z meeting the zero duality gap constraint, all primal and dual constraints except for the non-negativity constraints, and without limit on the real number t. Let L be the projection of Z on the hyperplane defined by t = 0. Consider a squeeze mapping involving the two variables of each complementary pair of z. The projection of e on the image of L of the mapping lies in an (n+m-1)-sphere Q centered at e/2 of a diameter whose square equals 2(n+m). The sum of the two components of a complementary pair of z in Q equals one, and Q is the circumsphere of the hypercube where each component of its vertices takes value in {0, 1}. One vertex v* called the solution vertex is the indicator vector of a. The algorithm uses squeeze mapping to move the aforementioned projection around v* along Q so that a is identified at certain position. It consists of O(n) unidimensional squeeze mappings, each of which uses O(mn) arithmetic operations.