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A New Technique for Determining Approximate Center of a Polytope

A New Technique for Determining Approximate Center of a Polytope Hindawi Advances in Operations Research Volume 2019, Article ID 8218329, 7 pages https://doi.org/10.1155/2019/8218329 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. inside the polytope. )erefore, the center of a polytope 1. Introduction depends on the definition we are using. But, fortunately, all Linear programming (LP) is a mathematical technique for those definitions are equivalent in the sense that, as shown optimizing a linear function subject to a set of linear con- in [1], if we get a polynomial time algorithm for a center of a straints and nonnegativity restrictions. Linear programs polytope, then that algorithm could also be used to con- frequently show up in various areas of applied sciences struct a polynomial time algorithm for solving linear today. )e prime reason for this is their manageable, program. enormous impact in various disciplines; it has become a core Many techniques [2–4] are used for finding the center of research area of many mathematicians, economists, decision a polytope, but they are taking so many iterations, slow in scientists, etc. Linear programming was developed during convergence, and require mostly complex computations. World War II, when a system with which to maximize the Most of the interior point methods for solving LPs efficiency of resources was of utmost importance. Since then, depend on the computations of a center finding method, many researchers have strived to advance their ideas and either explicitly or implicitly [3, 5]. made centering of the polytope as a core step in the major )e analytic center [6–12] is no doubt the most used optimization techniques (named as interior point methods) notion of center of a polytope in linear optimization because in science and industry. 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 2. Definitions of the Center of a Polytope spatial positions of the half-spaces that define the polytope. So in that case, analytic center may not look like located at a )ere are several ways to define the center of a polytope, it may be the center of gravity i.e., centroid, mean position of good central position (see also Section 3). )e P-center [3], also to be discussed in Section 4, provides a much better all vertices i.e., vertex centroid, point at the location where product of distances from all boundary lines is maximized center than the analytic center, but it is found in practice that it takes much longer time to obtain a good approximation of i.e., analytic center, center of the least volume ellipsoid that contains the polytope, or the center of the biggest ball the P-center. 2 Advances in Operations Research by taking the average of all m midpoints 􏼈x : i � 1, . . . , m􏼉. 3. Analytic Center k+1 m k )at is x � 􏽐 x /m. i�1 i k+1 k Let S be a polytope described by a (normalized) system of If max􏼈abs(x − x )􏼉< ε (where ε is the tolerance linear inequalities: value), then stop with the result that “P-center is obtained up to tolerance level of ε.” Otherwise, perform the (k + 1)th k+1 􏽘 a x ≤ b , i � 1, 2, . . . , m. ij j i (1) iteration taking x as the initial interior point. j�1 Carlos defined it as an approximation to the vertex centroid because in every iteration, he has taken 2 -m distinct points on )e analytic center of S is the point the boundary in distinct directions. Method looks to be very ξ � (ξ , ξ , . . . , ξ ) ∈ R which satisfies the following 1 2 n effective and he has also shown that quality of the center is maximization problem: also very good, even most of the times the centrality of m n P-center is way better than analytic center, but practically, ⎛ ⎝ ⎞ ⎠ Maximize 􏽙 b − 􏽘 a x i ij j it is found that it become very slow to converge and to (2) i�1 j�1 attain a particular tolerance. Sometimes, it requires a huge subject to x ∈ S. number of iterations just to reach near about centroid. Computational results are shown in the Section 6. When S is bounded, this maximization problem always has a solution. 5. A New Approximation of Central Location: In general, the analytic center depends on how the set of CN-Center particular inequalities is defined. )e addition of redundant In this section, we describe a recursive version of the method inequalities could push analytic center towards the boundary. In the Sections 4 and 5, we discuss two new described in Section 4. Because of the recursive nature of this efficient and alternative methods, which could be used to method, it uses the most updated value of center for next find a good approximation of the central location of a computation so it has a quick movement towards the central polytope. location. We would call the central location obtained by this method as CN-center. Overall, for a problem of m constraints, each iteration of 4. P-Center [3] this procedure holds m steps. Here onward, x denotes the Consider a linear programming polytope described by a set coordinates of center obtained in ith intermediate step of kth n m×n T S � {x ∈ R : Ax≤ b}, where A ∈ R . Let H � 􏼈x : a iteration, x denotes the center obtained after kth iteration, th x≤ b } be the half-space corresponding to the i row of A and x denotes the coordinates of the initial feasible point. k+1 and G � 􏼈x : a x � b 􏼉 be the corresponding hyperplane. ∧ i i Here, x would be used to represent the midpoint of )e method assumes that the polytope is full dimensional. + k − k + k − k P (x ) and P (x ), where P (x ) and P (x ) on the i i i i )e overall technique is based on the following definitions of + k k − k k boundary defined by P (x ) � x + θ a and P (x ) � x − i i i i projections. k k λ a , where θ � max t : x + ta ∈ S and λ � max t : x − 􏼈 􏼉 􏼈 i i i i i ta ∈ S}. 4.1. Orthogonal Feasible Projections. Let x be some feasible For any kth iteration, the method needs an interior k− 1 interior point; for each hyperplane G , we can easily generate feasible point x : + k − k k two distinct points (projections) P (x ) and P (x ) on the + k− 1 − k− 1 i i Step k : compute x ≔ 1/2(P (x ) + P (x )). Set + k k − k k 1 1 1 1 boundary, defined by P (x ) � x + θ a and P (x ) � x − i i i i k k k λ a , where θ � max􏼈t : x + ta ∈ S􏼉 and λ � max􏼈t : x − x ≔ x . i i i i i 1 1 + k − k ta ∈ S}. Step k : compute x ≔ 1/2(P (x ) + P (x )). Set i i i− 1 i i− 1 i− 1 k k x ≔ 1/i(􏽐 x + x ), ∀i � 2, . . . , m. i j�1 j i 4.2. Central Location Using Orthogonal Projections. A vertex Now, since all hyperplanes have contributed, we can set centroid of a polytope could be defined as the average of all k k x ≔ x . We can terminate the procedure when points on the boundary and [3] defined the P-center as an k k− 1 max abs(x − x ) < ε, where ε is the required tolerance. If 􏼈 􏼉 approximation of vertex centroid, which could be obtained the tolerance level is not achieved, then the process could by taking average of some finite number of points on the proceed to (k + 1)th iteration, taking x as an initial point. boundary of a polytope. )e main task of the method is to )e main difference between computation of P-center generate points on the boundary as much as possible. For and CN-center could simply be illustrated by performing this purpose, an initial feasible interior point is needed and initial two steps of the first iteration. then the method generates the iterates by taking a convex combination of the orthogonal projections into the hyper- Iteration 1. Step 1: both the methods identically take an planes associated with the inequalities that define the interior feasible point, say x , as starting point and find + k − k polytope. two points P (x ) and P (x ) on the boundary of the 1 1 + k − k Each G could generate two points P (x ) and P (x ) on feasible region using the direction of normal of first i i i k + k − k the boundary of S and a midpoint x � (P (x ) + P (x ))/2 constraint. )en, both methods take the average of i i i k+1 0 on the chord joining them. )e new iterate x is obtained boundary points to get a new point, say x . 1 Advances in Operations Research 3 6 6 5.5 5.5 5 5 4.5 4.5 4 4 3.5 3.5 2.5 2.5 2 2 1.5 1.5 1 1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 P-center CN-center Figure 1: Polytope 1, P-center is obtained in 56 iterations but CN-center is obtained in just 11 iterations. Centrality wise both looked equivalent. 9 9 8 8 7 7 6 6 5 5 4 4 3 3 2 2 0 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 P-center CN-center Figure 2: Polytope 2, convergence towards center. 1.6 1.6 1.4 1.4 1.2 1.2 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 P-center CN-center Figure 3: Polytope 3, convergence towards center. 4 Advances in Operations Research 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 P-center CN-center Figure 4: Polytope 4, convergence towards center. Table 1: A comparison of number of iterations and quality of P-center and CN-center. P-center CN-center Polytope number C(x) X Iterations C(x) X Iterations 1 0.7597 (1.4453, 3.4772) 56 0.6903 (1.3980, 3.6719) 11 2 0.9272 (0.9790, 3.0236) 38 0.9262 (0.9843, 3.0236) 7 3 0.3332 (0.3602, 0.7120) 16 0.3311 (0.3585, 0.7503) 2 4 0.1594 (0.1947, 0.3126) 12 0.1656 (0.2014, 0.3064) 3 Table 2: A numerical comparison between coordinates of analytic center, P-center, CN-center, and centroid. Polytope number Analytic center P-center CN-center Centroid 1 (2.3932, 2.8696) (1.4453, 3.4772) (1.3980, 3.6719) (1.4364, 0.6318) 2 (0.6340, 3.5490) (0.9790, 3.0236) (0.984, 3.0236) (1.0000, 3.0000) 3 (0.3732, 0.8454) (0.3602, 0.7120) (0.358, 0.7503) (0.3524, 0.7403) 4 (0.2152, 0.3704) (0.1947, 0.3126) (0.2014, 0.3064) (0.2644, 0.2738) Iteration 1. Step 2: difference of strategy starts from represent four polytopes with convergences to their here; the method of P-center now again finds the av- P-center and CN-center, respectively. nd erage of two other boundary points using normal of 2 Table 1 shows the measure of coordinates of center x, constraint and the same starting point x , and calls it number of iterations, and measure of centrality C(x) for x . P-center and CN-center for each polytope illustrated in nd Figures 1–4 In contrast, method of CN-center takes normal of 2 As we see in Table 1 as well as in Figures 1–4, centrality constraint and a new point x to obtain the average of C(x) of CN-center and P-center is almost equal but con- boundary points, denoting it by x . Now, x is obtained 2 2 ∧ vergence towards CN-center is multiplex faster than by taking the average of x and x . 1 2 P-center. Now, we are taking other four different polytopes and presenting the comparison between observable quality of 6. Computational Experiences analytic center, P-center, centroid, and CN-center. Table 2 and Figure 5 represent numerical and graphical results. We performed numerical experiences to compare the CN- center against P-center, analytic center, and centroid in Based on Figures 1–5, it is easy to see that location-wise CN- center is almost equal to P-center but with a less number of several polytopes in MATLAB and convergence with a iterations. suitable tolerance level. First, we show pictures of polytopes in 2D space to illustrate the convergence of the CN-center, Our calculation shows that if we take the initial point near a narrow corner of the region, then there is a huge P-center, analytic center, and centroid. Second, we present the numerical results in tables for randomly generated polytope. difference in the number of iterations for P-center and CN- center as shown in Figure 6, and if we take the initial point To visualize the convergence of P-center and CN-center, we have taken some examples from [13]. Figures 1–4 near a wide corner of the region (see Figure 7), difference in Advances in Operations Research 5 5.5 4.5 3.5 2.5 1.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 1.6 0.8 1.4 0.7 1.2 0.6 0.5 0.8 0.4 0.6 0.3 0.4 0.2 0.2 0.1 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Figure 5: Graphical comparison of quality between analytic center (denoted by yellow square), P-center (denoted by blue asterisk), CN- center (denoted by black dot), and centroid (denoted by green diamond). 6 6 5.5 5.5 5 5 4.5 4.5 3.5 3.5 3 3 2.5 2.5 2 2 1.5 1.5 1 1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 P-center CN-center Figure 6: When initial point lies near a narrow corner, convergence of CN-center is much quicker than P-center. 6 Advances in Operations Research 5.5 5.5 4.5 4.5 3.5 3.5 2.5 2.5 1.5 1.5 1 1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 P-center CN-center Figure 7: When initial point lies near a wide corner, both P-center and CN-center bear almost equal number of iterations. Table 3: Computational results of randomly generated LPs using MATLAB. P-center CN-center Number of constraint Coordinates of center Number of iteration Coordinates of center Number of iteration (0.0965, 0.0752) 6 (0.1122, 0.0793) 1 (0.0465, 0.0746) 8 (0.0469, 0.1064) 1 (0.3132, 0.7748) 12 (0.3187, 0.8938) 1 (0.4704, 0.0892) 24 (0.4837, 0.0898) 2 (0.5839, 0.1610) 21 (0.5442, 0.1541) 2 25 (0.0686, 0.2428) 32 (0.0731, 0.3027) 5 (0.1244, 0.3280) 12 (0.1294, 0.3847) 1 (0.3476, 0.0421) 24 (0.3151, 0.0403) 4 (0.2793, 0.1585) 12 (0.2714, 0.1569) 2 (0.2134, 0.0223) 2 (0.2255, 0.0221) 3 (0.1856, 0.0341) 23 (0.2183, 0.03257) 3 (0.0346, 0.0922) 13 (0.0328, 0.1205) 1 (0.0761, 0.0101) 19 (0.0943, 0.0098) 2 (0.1607, 0.2225) 6 (0.1714, 0.2511) 1 (0.1092, 0.0408) 17 (0.1538, 0.0368) 2 (0.1669, 0.0229) 19 (0.1592, 0.0229) 1 50 (0.2277, 0.0482) 28 (0.2159, 0.0479) 2 (0.4047, 0.03670) 31 (0.3385, 0.0342) 3 (0.1925, 0.3076) 14 (0.1881, 0.3085) 1 (0.1546, 0.1301) 12 (0.1678, 0.1517) 1 (0.2142, 0.0126) 89 (0.3065, 0.0111) 10 (0.3657, 0.2198) 16 (0.370951, 0.2122) 1 Table 4: Comparison of average number of iterations for higher dimensional random LPs. Order P-center CN-center 3 × 5 62.71 16.57 5 × 3 35.85 8.23 5 × 5 85.5 55.33 10 × 5 136.8 34.4 10 × 10 476.44 77.22 15 × 15 650.71 249.85 15 × 10 348.6 96.34 20 × 20 331.44 258.88 30 × 20 213.2 162.7 20 × 30 508.2 236 30 × 30 206.75 193.125 Note: here, ε � 0.001 is more than enough. If we observe the convergence pattern of CN-center, we can see that convergence is initially fast and gets slower in later iterations. Generally, we do not need exact central point; for practical purposes, any good central location is enough for working. So a good central point is obtainable by CN-center within just a few iterations for a 100 × 100 or even a very high dimensional problem. Advances in Operations Research 7 [5] E. R. Barnes and A. C. Moretti, “Some results on centers of the number of iterations for P-center and CN-center would polytopes,” Optimization Methods and Software, vol. 20, no. 1, not be so significant. pp. 9–24, 2005. Now, Table 3 presents some computational results on [6] J. Renegar, “A polynomial-time algorithm, based on Newton’s number of iterations needed for obtaining P-center and CN- method,” For Linear Programming, vol. 40, no. 1–3, pp. 59–93, center of randomly generated 2D LPs with higher number of constraints. [7] P. T. Boggs, P. D. Domich, J. R. Donaldson, and C. Witzgall, Finally, results for higher dimensional random LPs are “Algorithmic enhancements to the method of centers for presented in Table 4. Here, we have taken the average of linear programming problems,” ORSA Journal on Computing, number of iterations of 20 random LPs of each order with vol. 1, no. 3, pp. 159–171, 1989. ε � 0.001. Results showed that the new approach is still [8] R. Freund, “Projective transformation for interior point superior in efficiency even in higher dimensional problems. method and superlinear convergent algorithum for the w-center problem,” Mathematical Programming, vol. 58, no. 1–3, pp. 385–414, 1993. 7. Applications [9] P. Huard, “Resolution of mathematical programming with nonlinear constraints by the method of centers,” in Nonlinear )ere are a lot of areas where approximate center finding Programming, J. Abadie, Ed., pp. 207–219, North-Holland methods for LPs could be used, for example, solving both Publishing Company, Amsterdam, Netherlands, 1967. linear and general convex programming [13], the support [10] F. Jarre, G. Sonnevend, and J. Stoer, “An implementation of vector machine (SVM) solution that corresponds to the the method of analytic center,” in Lecture Notes in Control and center of the largest sphere inscribed in version space [9, 14], Information Sciences, pp. 297–307, Springer-Verlag, Berlin, computing cubature formulae [15], and sphere method for Germany, 1998. linear programming [16]. [11] S. Boyd and L. Vandenberghe, Convex Optimization, Cam- bridge University Press, Cambridge, UK, 2004. 8. Conclusion [12] D. G. Luenberger and Y. Yinyu, Linear and Non Linear Programming, Springer, New York, NY, USA, third edition, In this paper, we have presented a modified form of P-center [3] and called it as CN-center. Our experimental results [13] A. Carlos Moretti, A Technique for Finding the Center of a Polytope, Georgia Institute of Technology, Atlanta, GA, USA, show that quality of centrality of P-center and CN-center is almost same, but in terms of number of iterations,CN-center [14] F. Maire, “An algorithm for the exact computation of the is much faster in computation of a good central location in centroid of higher dimensional polyhedra and its application the feasible region in lower and as well as in higher di- to kernel machines,” in Proceedings of the 4ird IEEE In- mensional problems. ternational Conference on Data Mining, Melbourne, FL, USA, Generally, finding the central location of an LP is the November 2003. main crucial step for most of the interior point methods. [15] G. Sonnevend, “Applications of the notion of analytic center Usually, we do not need the exact center instead a good in approximation (estimation) problems,” Journal of Com- central location would be enough if it is obtained in less putational and Applied Mathematics, vol. 28, pp. 349–358, number of computations. So, in this sense, CN-center is a better option to use instead of P-center or analytic center. [16] K. G. Murty, Ball Center of Special Polytopes, Department of Industrial and Operation Engineering, University of Michi- gan, Ann Arbor, MI, USA, 2009. Data Availability Data were randomly generated using MATLAB software. )e seed of the random numbers and associated MATLAB files could be provided on request. Conflicts of Interest )e authors declare that they have no conflicts of interest. References [1] V. Klee, “Convexity,” in Proceedings of the Seventh Symposium in Pure Mathematics of American Mathematical Society, Seattle, WA, USA, June 1961. [2] N. D. Botkin and V. L. Turvova, “An algorithm for finding the Chebyshev center of a convex polyheron,” Applied Mathe- matics & Optimization, vol. 29, no. 2, 1995. [3] A. Carlos Moretti, “A weighted projection centering method,” Computational and Applied Mathematics, vol. 22, no. 1, pp. 19–36, 2003. [4] D. P. Bertsekas, Convex Optimization 4eory, Athena Sci- entific, Belmont, MA, USA, 1st edition, 2009. 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A New Technique for Determining Approximate Center of a Polytope

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Hindawi Advances in Operations Research Volume 2019, Article ID 8218329, 7 pages https://doi.org/10.1155/2019/8218329 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. inside the polytope. )erefore, the center of a polytope 1. Introduction depends on the definition we are using. But, fortunately, all Linear programming (LP) is a mathematical technique for those definitions are equivalent in the sense that, as shown optimizing a linear function subject to a set of linear con- in [1], if we get a polynomial time algorithm for a center of a straints and nonnegativity restrictions. Linear programs polytope, then that algorithm could also be used to con- frequently show up in various areas of applied sciences struct a polynomial time algorithm for solving linear today. )e prime reason for this is their manageable, program. enormous impact in various disciplines; it has become a core Many techniques [2–4] are used for finding the center of research area of many mathematicians, economists, decision a polytope, but they are taking so many iterations, slow in scientists, etc. Linear programming was developed during convergence, and require mostly complex computations. World War II, when a system with which to maximize the Most of the interior point methods for solving LPs efficiency of resources was of utmost importance. Since then, depend on the computations of a center finding method, many researchers have strived to advance their ideas and either explicitly or implicitly [3, 5]. made centering of the polytope as a core step in the major )e analytic center [6–12] is no doubt the most used optimization techniques (named as interior point methods) notion of center of a polytope in linear optimization because in science and industry. 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 2. Definitions of the Center of a Polytope spatial positions of the half-spaces that define the polytope. So in that case, analytic center may not look like located at a )ere are several ways to define the center of a polytope, it may be the center of gravity i.e., centroid, mean position of good central position (see also Section 3). )e P-center [3], also to be discussed in Section 4, provides a much better all vertices i.e., vertex centroid, point at the location where product of distances from all boundary lines is maximized center than the analytic center, but it is found in practice that it takes much longer time to obtain a good approximation of i.e., analytic center, center of the least volume ellipsoid that contains the polytope, or the center of the biggest ball the P-center. 2 Advances in Operations Research by taking the average of all m midpoints 􏼈x : i � 1, . . . , m􏼉. 3. Analytic Center k+1 m k )at is x � 􏽐 x /m. i�1 i k+1 k Let S be a polytope described by a (normalized) system of If max􏼈abs(x − x )􏼉< ε (where ε is the tolerance linear inequalities: value), then stop with the result that “P-center is obtained up to tolerance level of ε.” Otherwise, perform the (k + 1)th k+1 􏽘 a x ≤ b , i � 1, 2, . . . , m. ij j i (1) iteration taking x as the initial interior point. j�1 Carlos defined it as an approximation to the vertex centroid because in every iteration, he has taken 2 -m distinct points on )e analytic center of S is the point the boundary in distinct directions. Method looks to be very ξ � (ξ , ξ , . . . , ξ ) ∈ R which satisfies the following 1 2 n effective and he has also shown that quality of the center is maximization problem: also very good, even most of the times the centrality of m n P-center is way better than analytic center, but practically, ⎛ ⎝ ⎞ ⎠ Maximize 􏽙 b − 􏽘 a x i ij j it is found that it become very slow to converge and to (2) i�1 j�1 attain a particular tolerance. Sometimes, it requires a huge subject to x ∈ S. number of iterations just to reach near about centroid. Computational results are shown in the Section 6. When S is bounded, this maximization problem always has a solution. 5. A New Approximation of Central Location: In general, the analytic center depends on how the set of CN-Center particular inequalities is defined. )e addition of redundant In this section, we describe a recursive version of the method inequalities could push analytic center towards the boundary. In the Sections 4 and 5, we discuss two new described in Section 4. Because of the recursive nature of this efficient and alternative methods, which could be used to method, it uses the most updated value of center for next find a good approximation of the central location of a computation so it has a quick movement towards the central polytope. location. We would call the central location obtained by this method as CN-center. Overall, for a problem of m constraints, each iteration of 4. P-Center [3] this procedure holds m steps. Here onward, x denotes the Consider a linear programming polytope described by a set coordinates of center obtained in ith intermediate step of kth n m×n T S � {x ∈ R : Ax≤ b}, where A ∈ R . Let H � 􏼈x : a iteration, x denotes the center obtained after kth iteration, th x≤ b } be the half-space corresponding to the i row of A and x denotes the coordinates of the initial feasible point. k+1 and G � 􏼈x : a x � b 􏼉 be the corresponding hyperplane. ∧ i i Here, x would be used to represent the midpoint of )e method assumes that the polytope is full dimensional. + k − k + k − k P (x ) and P (x ), where P (x ) and P (x ) on the i i i i )e overall technique is based on the following definitions of + k k − k k boundary defined by P (x ) � x + θ a and P (x ) � x − i i i i projections. k k λ a , where θ � max t : x + ta ∈ S and λ � max t : x − 􏼈 􏼉 􏼈 i i i i i ta ∈ S}. 4.1. Orthogonal Feasible Projections. Let x be some feasible For any kth iteration, the method needs an interior k− 1 interior point; for each hyperplane G , we can easily generate feasible point x : + k − k k two distinct points (projections) P (x ) and P (x ) on the + k− 1 − k− 1 i i Step k : compute x ≔ 1/2(P (x ) + P (x )). Set + k k − k k 1 1 1 1 boundary, defined by P (x ) � x + θ a and P (x ) � x − i i i i k k k λ a , where θ � max􏼈t : x + ta ∈ S􏼉 and λ � max􏼈t : x − x ≔ x . i i i i i 1 1 + k − k ta ∈ S}. Step k : compute x ≔ 1/2(P (x ) + P (x )). Set i i i− 1 i i− 1 i− 1 k k x ≔ 1/i(􏽐 x + x ), ∀i � 2, . . . , m. i j�1 j i 4.2. Central Location Using Orthogonal Projections. A vertex Now, since all hyperplanes have contributed, we can set centroid of a polytope could be defined as the average of all k k x ≔ x . We can terminate the procedure when points on the boundary and [3] defined the P-center as an k k− 1 max abs(x − x ) < ε, where ε is the required tolerance. If 􏼈 􏼉 approximation of vertex centroid, which could be obtained the tolerance level is not achieved, then the process could by taking average of some finite number of points on the proceed to (k + 1)th iteration, taking x as an initial point. boundary of a polytope. )e main task of the method is to )e main difference between computation of P-center generate points on the boundary as much as possible. For and CN-center could simply be illustrated by performing this purpose, an initial feasible interior point is needed and initial two steps of the first iteration. then the method generates the iterates by taking a convex combination of the orthogonal projections into the hyper- Iteration 1. Step 1: both the methods identically take an planes associated with the inequalities that define the interior feasible point, say x , as starting point and find + k − k polytope. two points P (x ) and P (x ) on the boundary of the 1 1 + k − k Each G could generate two points P (x ) and P (x ) on feasible region using the direction of normal of first i i i k + k − k the boundary of S and a midpoint x � (P (x ) + P (x ))/2 constraint. )en, both methods take the average of i i i k+1 0 on the chord joining them. )e new iterate x is obtained boundary points to get a new point, say x . 1 Advances in Operations Research 3 6 6 5.5 5.5 5 5 4.5 4.5 4 4 3.5 3.5 2.5 2.5 2 2 1.5 1.5 1 1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 P-center CN-center Figure 1: Polytope 1, P-center is obtained in 56 iterations but CN-center is obtained in just 11 iterations. Centrality wise both looked equivalent. 9 9 8 8 7 7 6 6 5 5 4 4 3 3 2 2 0 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 P-center CN-center Figure 2: Polytope 2, convergence towards center. 1.6 1.6 1.4 1.4 1.2 1.2 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 P-center CN-center Figure 3: Polytope 3, convergence towards center. 4 Advances in Operations Research 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 P-center CN-center Figure 4: Polytope 4, convergence towards center. Table 1: A comparison of number of iterations and quality of P-center and CN-center. P-center CN-center Polytope number C(x) X Iterations C(x) X Iterations 1 0.7597 (1.4453, 3.4772) 56 0.6903 (1.3980, 3.6719) 11 2 0.9272 (0.9790, 3.0236) 38 0.9262 (0.9843, 3.0236) 7 3 0.3332 (0.3602, 0.7120) 16 0.3311 (0.3585, 0.7503) 2 4 0.1594 (0.1947, 0.3126) 12 0.1656 (0.2014, 0.3064) 3 Table 2: A numerical comparison between coordinates of analytic center, P-center, CN-center, and centroid. Polytope number Analytic center P-center CN-center Centroid 1 (2.3932, 2.8696) (1.4453, 3.4772) (1.3980, 3.6719) (1.4364, 0.6318) 2 (0.6340, 3.5490) (0.9790, 3.0236) (0.984, 3.0236) (1.0000, 3.0000) 3 (0.3732, 0.8454) (0.3602, 0.7120) (0.358, 0.7503) (0.3524, 0.7403) 4 (0.2152, 0.3704) (0.1947, 0.3126) (0.2014, 0.3064) (0.2644, 0.2738) Iteration 1. Step 2: difference of strategy starts from represent four polytopes with convergences to their here; the method of P-center now again finds the av- P-center and CN-center, respectively. nd erage of two other boundary points using normal of 2 Table 1 shows the measure of coordinates of center x, constraint and the same starting point x , and calls it number of iterations, and measure of centrality C(x) for x . P-center and CN-center for each polytope illustrated in nd Figures 1–4 In contrast, method of CN-center takes normal of 2 As we see in Table 1 as well as in Figures 1–4, centrality constraint and a new point x to obtain the average of C(x) of CN-center and P-center is almost equal but con- boundary points, denoting it by x . Now, x is obtained 2 2 ∧ vergence towards CN-center is multiplex faster than by taking the average of x and x . 1 2 P-center. Now, we are taking other four different polytopes and presenting the comparison between observable quality of 6. Computational Experiences analytic center, P-center, centroid, and CN-center. Table 2 and Figure 5 represent numerical and graphical results. We performed numerical experiences to compare the CN- center against P-center, analytic center, and centroid in Based on Figures 1–5, it is easy to see that location-wise CN- center is almost equal to P-center but with a less number of several polytopes in MATLAB and convergence with a iterations. suitable tolerance level. First, we show pictures of polytopes in 2D space to illustrate the convergence of the CN-center, Our calculation shows that if we take the initial point near a narrow corner of the region, then there is a huge P-center, analytic center, and centroid. Second, we present the numerical results in tables for randomly generated polytope. difference in the number of iterations for P-center and CN- center as shown in Figure 6, and if we take the initial point To visualize the convergence of P-center and CN-center, we have taken some examples from [13]. Figures 1–4 near a wide corner of the region (see Figure 7), difference in Advances in Operations Research 5 5.5 4.5 3.5 2.5 1.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 1.6 0.8 1.4 0.7 1.2 0.6 0.5 0.8 0.4 0.6 0.3 0.4 0.2 0.2 0.1 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Figure 5: Graphical comparison of quality between analytic center (denoted by yellow square), P-center (denoted by blue asterisk), CN- center (denoted by black dot), and centroid (denoted by green diamond). 6 6 5.5 5.5 5 5 4.5 4.5 3.5 3.5 3 3 2.5 2.5 2 2 1.5 1.5 1 1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 P-center CN-center Figure 6: When initial point lies near a narrow corner, convergence of CN-center is much quicker than P-center. 6 Advances in Operations Research 5.5 5.5 4.5 4.5 3.5 3.5 2.5 2.5 1.5 1.5 1 1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 P-center CN-center Figure 7: When initial point lies near a wide corner, both P-center and CN-center bear almost equal number of iterations. Table 3: Computational results of randomly generated LPs using MATLAB. P-center CN-center Number of constraint Coordinates of center Number of iteration Coordinates of center Number of iteration (0.0965, 0.0752) 6 (0.1122, 0.0793) 1 (0.0465, 0.0746) 8 (0.0469, 0.1064) 1 (0.3132, 0.7748) 12 (0.3187, 0.8938) 1 (0.4704, 0.0892) 24 (0.4837, 0.0898) 2 (0.5839, 0.1610) 21 (0.5442, 0.1541) 2 25 (0.0686, 0.2428) 32 (0.0731, 0.3027) 5 (0.1244, 0.3280) 12 (0.1294, 0.3847) 1 (0.3476, 0.0421) 24 (0.3151, 0.0403) 4 (0.2793, 0.1585) 12 (0.2714, 0.1569) 2 (0.2134, 0.0223) 2 (0.2255, 0.0221) 3 (0.1856, 0.0341) 23 (0.2183, 0.03257) 3 (0.0346, 0.0922) 13 (0.0328, 0.1205) 1 (0.0761, 0.0101) 19 (0.0943, 0.0098) 2 (0.1607, 0.2225) 6 (0.1714, 0.2511) 1 (0.1092, 0.0408) 17 (0.1538, 0.0368) 2 (0.1669, 0.0229) 19 (0.1592, 0.0229) 1 50 (0.2277, 0.0482) 28 (0.2159, 0.0479) 2 (0.4047, 0.03670) 31 (0.3385, 0.0342) 3 (0.1925, 0.3076) 14 (0.1881, 0.3085) 1 (0.1546, 0.1301) 12 (0.1678, 0.1517) 1 (0.2142, 0.0126) 89 (0.3065, 0.0111) 10 (0.3657, 0.2198) 16 (0.370951, 0.2122) 1 Table 4: Comparison of average number of iterations for higher dimensional random LPs. Order P-center CN-center 3 × 5 62.71 16.57 5 × 3 35.85 8.23 5 × 5 85.5 55.33 10 × 5 136.8 34.4 10 × 10 476.44 77.22 15 × 15 650.71 249.85 15 × 10 348.6 96.34 20 × 20 331.44 258.88 30 × 20 213.2 162.7 20 × 30 508.2 236 30 × 30 206.75 193.125 Note: here, ε � 0.001 is more than enough. If we observe the convergence pattern of CN-center, we can see that convergence is initially fast and gets slower in later iterations. Generally, we do not need exact central point; for practical purposes, any good central location is enough for working. So a good central point is obtainable by CN-center within just a few iterations for a 100 × 100 or even a very high dimensional problem. Advances in Operations Research 7 [5] E. R. Barnes and A. C. Moretti, “Some results on centers of the number of iterations for P-center and CN-center would polytopes,” Optimization Methods and Software, vol. 20, no. 1, not be so significant. pp. 9–24, 2005. Now, Table 3 presents some computational results on [6] J. Renegar, “A polynomial-time algorithm, based on Newton’s number of iterations needed for obtaining P-center and CN- method,” For Linear Programming, vol. 40, no. 1–3, pp. 59–93, center of randomly generated 2D LPs with higher number of constraints. [7] P. T. Boggs, P. D. Domich, J. R. Donaldson, and C. Witzgall, Finally, results for higher dimensional random LPs are “Algorithmic enhancements to the method of centers for presented in Table 4. Here, we have taken the average of linear programming problems,” ORSA Journal on Computing, number of iterations of 20 random LPs of each order with vol. 1, no. 3, pp. 159–171, 1989. ε � 0.001. Results showed that the new approach is still [8] R. Freund, “Projective transformation for interior point superior in efficiency even in higher dimensional problems. method and superlinear convergent algorithum for the w-center problem,” Mathematical Programming, vol. 58, no. 1–3, pp. 385–414, 1993. 7. Applications [9] P. Huard, “Resolution of mathematical programming with nonlinear constraints by the method of centers,” in Nonlinear )ere are a lot of areas where approximate center finding Programming, J. Abadie, Ed., pp. 207–219, North-Holland methods for LPs could be used, for example, solving both Publishing Company, Amsterdam, Netherlands, 1967. linear and general convex programming [13], the support [10] F. Jarre, G. Sonnevend, and J. Stoer, “An implementation of vector machine (SVM) solution that corresponds to the the method of analytic center,” in Lecture Notes in Control and center of the largest sphere inscribed in version space [9, 14], Information Sciences, pp. 297–307, Springer-Verlag, Berlin, computing cubature formulae [15], and sphere method for Germany, 1998. linear programming [16]. [11] S. Boyd and L. Vandenberghe, Convex Optimization, Cam- bridge University Press, Cambridge, UK, 2004. 8. Conclusion [12] D. G. Luenberger and Y. Yinyu, Linear and Non Linear Programming, Springer, New York, NY, USA, third edition, In this paper, we have presented a modified form of P-center [3] and called it as CN-center. Our experimental results [13] A. Carlos Moretti, A Technique for Finding the Center of a Polytope, Georgia Institute of Technology, Atlanta, GA, USA, show that quality of centrality of P-center and CN-center is almost same, but in terms of number of iterations,CN-center [14] F. Maire, “An algorithm for the exact computation of the is much faster in computation of a good central location in centroid of higher dimensional polyhedra and its application the feasible region in lower and as well as in higher di- to kernel machines,” in Proceedings of the 4ird IEEE In- mensional problems. ternational Conference on Data Mining, Melbourne, FL, USA, Generally, finding the central location of an LP is the November 2003. main crucial step for most of the interior point methods. [15] G. Sonnevend, “Applications of the notion of analytic center Usually, we do not need the exact center instead a good in approximation (estimation) problems,” Journal of Com- central location would be enough if it is obtained in less putational and Applied Mathematics, vol. 28, pp. 349–358, number of computations. So, in this sense, CN-center is a better option to use instead of P-center or analytic center. [16] K. G. Murty, Ball Center of Special Polytopes, Department of Industrial and Operation Engineering, University of Michi- gan, Ann Arbor, MI, USA, 2009. Data Availability Data were randomly generated using MATLAB software. )e seed of the random numbers and associated MATLAB files could be provided on request. Conflicts of Interest )e authors declare that they have no conflicts of interest. References [1] V. Klee, “Convexity,” in Proceedings of the Seventh Symposium in Pure Mathematics of American Mathematical Society, Seattle, WA, USA, June 1961. [2] N. D. Botkin and V. L. Turvova, “An algorithm for finding the Chebyshev center of a convex polyheron,” Applied Mathe- matics & Optimization, vol. 29, no. 2, 1995. [3] A. Carlos Moretti, “A weighted projection centering method,” Computational and Applied Mathematics, vol. 22, no. 1, pp. 19–36, 2003. [4] D. P. Bertsekas, Convex Optimization 4eory, Athena Sci- entific, Belmont, MA, USA, 1st edition, 2009. 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