# zbMATH — the first resource for mathematics

An $$O(n^ 3L)$$ potential reduction algorithm for linear programming. (English) Zbl 0734.90057
Summary: We describe a primal-dual potential function for linear programming: $\phi (x,s)=\rho \ln (x^ Ts)-\sum^{n}_{j=1}\ln (x_ js_ j),$ where $$\rho\geq n$$, x is the primal variable, and s is the dual-slack variable. As a result, we develop an interior point algorithm seeking reductions in the potential function with $$\rho =n+\sqrt{n}$$. Neither tracing the central path nor using the projective transformation, the algorithm converges to the optimal solution set in O($$\sqrt{n}L)$$ iterations and uses $$O(n^ 3L)$$ total arithmetic operations. We also suggest a practical approach to implementing the algorithm.

##### MSC:
 90C05 Linear programming 90C60 Abstract computational complexity for mathematical programming problems 90-08 Computational methods for problems pertaining to operations research and mathematical programming
Full Text:
##### References:
 [1] I. Adler, N. Karmarkar, M.G.C. Resende and G. Veiga, ”An implementation of Karmarkar’s algorithm for linear programming,”Mathematical Programming 44 (1989) 297–335. · Zbl 0682.90061 · doi:10.1007/BF01587095 [2] K.M. Anstreicher, ”A monotonic projective algorithm for fractional linear programming,”Algorithmica 1 (1986) 483–498. · Zbl 0625.90088 · doi:10.1007/BF01840458 [3] E.R. Barnes, ”A variation on Karmarkar’s algorithm for solving linear programming problems,”Mathematical Programming 36 (1986) 174–182. · Zbl 0626.90052 · doi:10.1007/BF02592024 [4] D.A. Bayer and J.C. Lagarias, ”The nonlinear geometry of linear programming: I. Affine and projective scaling trajectories, II. Legendre transform coordinates and central trajectories,”Transactions of the American Mathematical Society 314 (1989) 499–581. · Zbl 0671.90046 [5] M. Ben Daya and C.M. Shetty, ”Polynomial barrier function algorithms for convex quadratic programming,” Report J 88-5, School of ISE, Georgia Institute of Technology (Atlanta, GA, 1988). · Zbl 0641.90058 [6] G.B. Dantzig,Linear Programming and Extensions (Princeton University Press, Princeton, NJ, 1963). [7] I.I. Dikin, ”Iterative solution of problems of linear and quadratic programming,”Soviet Mathematics Doklady 8 (1967) 674–675. · Zbl 0189.19504 [8] A.V. Fiacco and G.P. McCormick,Nonlinear Programming: Sequential Unconstrained Minimization Techniques (Wiley, New York, NY, 1968). · Zbl 0193.18805 [9] K.R. Frisch, ”The logarithmic potential method of convex programming,” Technical Report, University Institute of Economics (Oslo, Norway, 1955). [10] D.M. Gay, ”A variant of Karmarkar’s linear programming algorithm for problems in standard form,”Mathematical Programming 37 (1987) 81–90. · Zbl 0629.90056 · doi:10.1007/BF02591685 [11] G. de Ghellinck and J.-P. Vial, ”A polynomial Newton method for linear programming,”Algorithmica 1 (1986) 425–454. · Zbl 0629.90058 · doi:10.1007/BF01840456 [12] P.E. Gill, W. Murray, M.A. Saunders, J.A. Tomlin and M.H. Wright, ”On projected Newton barrier methods for linear programming and an equivalence to Karmarkar’s projective method,”Mathematical Programming 36 (1986) 183–209. · Zbl 0624.90062 · doi:10.1007/BF02592025 [13] D. Goldfarb and S. Liu, ”An O(n 3 L) primal interior point algorithm for convex quadratic programming,” Technical Report, Department of IEOR, Columbia University (New York, NY, 1988). [14] C.C. Gonzaga, ”Polynomial affine algorithms for linear programming,”Mathematical Programming 49 (1991) 7–21. · Zbl 0777.90027 · doi:10.1007/BF01588776 [15] C.C. Gonzaga, ”An algorithm for solving linear programming problems in O(n 3 L) operations,” in: N. Megiddo, ed.,Progress in Mathematical Programming, Interior Point and Related Methods (Springer, New York, 1988) pp. 1–28. [16] M. Iri and H. Imai, ”A multiplicative barrier function method for linear programming,”Algorithmica 1 (1986) 455–482. · Zbl 0641.90048 · doi:10.1007/BF01840457 [17] N. Karmarkar, ”A new polynomial-time algorithm for linear programming,”Combinatorica 4 (1984) 373–395. · Zbl 0557.90065 · doi:10.1007/BF02579150 [18] M. Kojima, S. Mizuno and A. Yoshise, ”A polynomial-time algorithm for a class of linear complementarity problems,”Mathematical Programming 44 (1989) 1–26. · Zbl 0676.90087 · doi:10.1007/BF01587074 [19] K.O. Kortanek and M. Shi, ”Convergence results and numerical experiments on a linear programming hybrid algorithm,”European Journal of Operations Research 32 (1987) 47–61. · Zbl 0631.90038 · doi:10.1016/0377-2217(87)90270-0 [20] K.A. McShane, C.L. Monma and D.F. Shanno, ”An implementation of a primal-dual interior point method for linear programming,”ORSA Journal on Computing 1 (1989) 70–83. · Zbl 0752.90047 [21] N. Megiddo, ”Pathways to the optimal set in linear programming,” in: N. Megiddo, ed.,Progress in Mathematical Programming, Interior Point and Related Methods (Springer, New York, 1988) 131–158. [22] N. Megiddo and M. Shub, ”Boundary behavior of interior point algorithms in linear programming,”Mathematics of Operations Research 14 (1989) 97–146. · Zbl 0675.90050 · doi:10.1287/moor.14.1.97 [23] S. Mehrotra and J. Sun, ”An algorithm for convex quadratic programming that requires O(n 3.5 L) arithmetic operations,”Mathematics of Operations Research 15 (1990) 342–363. · Zbl 0714.90075 · doi:10.1287/moor.15.2.342 [24] C.L. Monma and A.J. Morton, ”Computational experimental with a dual affine variant of Karmarkar’s method for linear programming,” Technical Report, Bell Communications Research (Morristown, NJ, 1987). · Zbl 0627.90065 [25] R.C. Monteiro, I. Adler and M.C. Resende, ”A polynomial-time primal-dual affine scaling algorithm for linear and convex quadratic programming and its power series extension,”Mathematics of Operations Research 15 (1990) 191–214. · Zbl 0714.90060 · doi:10.1287/moor.15.2.191 [26] R.C. Monteiro and I. Adler, ”Interior path following primal-dual algorithms. Part I: Linear programming,”Mathematical Programming 44 (1989) 27–42. · Zbl 0676.90038 · doi:10.1007/BF01587075 [27] J. Renegar, ”A polynomial-time algorithm, based on Newton’s method, for linear programming,”Mathematical Programming 40 (1988) 59–93. · Zbl 0654.90050 · doi:10.1007/BF01580724 [28] D.F. Shanno, ”Computing Karmarkar projection quickly,”Mathematical Programming 41 (1988) 61–71. · Zbl 0652.90069 · doi:10.1007/BF01580753 [29] G. Sonnevend, ”An analytical center for polyhedrons and new classes of global algorithms for linear (smooth, convex) programming,” in:Lecture Notes in Control and Information Sciences No. 84 (Springer, New York, 1985) pp. 866–876. [30] M.J. Todd and B.P. Burrell, ”An extension of Karmarkar’s algorithm for linear programming using dual variables,”Algorithmica 1 (1986) 409–424. · Zbl 0621.90048 · doi:10.1007/BF01840455 [31] M.J. Todd and Y. Ye, ”A centered projective algorithm for linear programming,”Mathematics of Operations Research 15 (1990) 508–529. · Zbl 0722.90044 · doi:10.1287/moor.15.3.508 [32] P.M. Vaidya, ”An algorithm for linear programming which requires O((m+n)n 2+(m+n) 1.5 n)L) arithmetic operations,”Mathematical Programming 47 (1990) 175–201. · Zbl 0708.90047 · doi:10.1007/BF01580859 [33] R.J. Vanderbei and J.C. Lagarias, ”I.I. Dikin’s convergence result for the affine-scaling algorithm,” manuscript, AT&T Bell Laboratories (Murray Hill, NJ, 1988). · Zbl 0727.90050 [34] R.J. Vanderbei, M.S. Meketon and B.A. Freedman, ”On a modification of Karmarkar’s linear programming algorithm,”Algorithmica 1 (1986) 395–407. · Zbl 0626.90056 · doi:10.1007/BF01840454 [35] Y. Ye, ”Interior algorithms for linear, quadratic, and linearly constrained convex programming,” Ph.D. Thesis, Department of Engineering-Economic Systems, Stanford Universty (Stanford, CA, 1987). [36] Y. Ye and M. Kojima, ”Recovering optimal dual solutions in Karmarkar’s polynomial algorithm for linear programming,”Mathematical Programming 39 (1987) 305–317. · Zbl 0639.90062 · doi:10.1007/BF02592079 [37] Y. Ye and M.J. Todd, ”Containing and shrinking ellipsoids in the path-following algorithm,”Mathematical Programming 47 (1990) 1–9. · Zbl 0746.90049 · doi:10.1007/BF01580848
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.