Research Article A New Technique for Determining Approximate Center of a Polytope Syed Inayatullah , Maria Aman, Asma Rani, Hina Zaheer, and Tanveer Ahmed Siddiqi Department of Mathematics, University of Karachi, Karachi 75270, Pakistan Correspondence should be addressed to Syed Inayatullah; inayat@uok.edu.pk Received 5 March 2019; Revised 19 June 2019; Accepted 23 June 2019; Published 15 November 2019 Academic Editor: Imed Kacem Copyright © 2019 Syed Inayatullah et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this article, we have presented a method for finding the approximate center of a linear programming polytope. is method provides a point near the center of a polytope in few simple and easy steps. Geometrical interpretation and some numerical examples have also been presented to demonstrate the proposed approach and comparison of quality of the center obtained by using the new method with existing methods of finding exact and approximate centers. At the end, we also presented com- putational results on the randomly generated polytopes to compare the quality of the center obtained by using the new method. 1.Introduction Linear programming (LP) is a mathematical technique for optimizing a linear function subject to a set of linear con- straints and nonnegativity restrictions. Linear programs frequently show up in various areas of applied sciences today. e prime reason for this is their manageable, enormous impact in various disciplines; it has become a core research area of many mathematicians, economists, decision scientists, etc. Linear programming was developed during World War II, when a system with which to maximize the efficiency of resources was of utmost importance. Since then, many researchers have strived to advance their ideas and made centering of the polytope as a core step in the major optimization techniques (named as interior point methods) in science and industry. 2.DefinitionsoftheCenterofaPolytope ere are several ways to define the center of a polytope, it may be the center of gravity i.e., centroid, mean position of all vertices i.e., vertex centroid, point at the location where product of distances from all boundary lines is maximized i.e., analytic center, center of the least volume ellipsoid that contains the polytope, or the center of the biggest ball inside the polytope. erefore, the center of a polytope depends on the definition we are using. But, fortunately, all those definitions are equivalent in the sense that, as shown in [1], if we get a polynomial time algorithm for a center of a polytope, then that algorithm could also be used to con- struct a polynomial time algorithm for solving linear program. Many techniques [2–4] are used for finding the center of a polytope, but they are taking so many iterations, slow in convergence, and require mostly complex computations. Most of the interior point methods for solving LPs depend on the computations of a center finding method, either explicitly or implicitly [3, 5]. e analytic center [6–12] is no doubt the most used notion of center of a polytope in linear optimization because of its easy computation, but its disadvantage is that it can be pushed near the boundary of the polytope by using re- dundant constraints because its position depends on the spatial positions of the half-spaces that define the polytope. So in that case, analytic center may not look like located at a good central position (see also Section 3). e P-center [3], also to be discussed in Section 4, provides a much better center than the analytic center, but it is found in practice that it takes much longer time to obtain a good approximation of the P-center. Hindawi Advances in Operations Research Volume 2019, Article ID 8218329, 7 pages https://doi.org/10.1155/2019/8218329