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CurveLP-A MATLAB implementation of an infeasible interior-point algorithm for linear programming. (English) Zbl 1378.65127
The author develops a competitive arc-feasible infeasible interior-point algorithm. He shows that based on the results on Netlib problems, the comparison of Mehrotra’s algorithm and the arc-feasible infeasible interior-point algorithm yields that the proposed arc-feasible infeasible interior-point algorithm is a more reliable and efficient algorithm than Mehrotra’s algorithm.
This article is well written, structured and explained, it contains six sections: Section 1 on Introduction, Section 2 on Problem descriptions, Section 3 on Arc-search algorithm for linear programming, Section 4 on Implementation details, Section 5 on Numerical tests, and, Section 6 on Conclusions.

65K05 Numerical mathematical programming methods
90C05 Linear programming
90C51 Interior-point methods
Full Text: DOI arXiv
[1] Wright, S.: Primal-Dual Interior-Point Methods. SIAM, Philadelphia (1997) · Zbl 0863.65031
[2] Megiddo, N; Megiddo, N (ed.), Pathway to the optimal set in linear propramming, 131-158, (1989), New York
[3] Kojima, M; Mizuno, S; Yoshise, A, A polynomial-time algorithm for a class of linear complementarity problem, Math. Program., 44, 1-26, (1989) · Zbl 0676.90087
[4] Kojima, M; Mizuno, S; Yoshise, A; Megiddo, N (ed.), A primal-dual interior point algorithm for linear programming, (1989), New York · Zbl 0708.90049
[5] Mehrotra, S, On the implementation of a primal-dual interior point method, SIAM J. Optim., 2, 575-601, (1992) · Zbl 0773.90047
[6] Lustig, I; Marsten, R; Shannon, D, Computational experience with a primal-dual interior-point method for linear programming, Linear Algebra Appl., 152, 191-222, (1991) · Zbl 0731.65049
[7] Lustig, I; Marsten, R; Shannon, D, On implementing mehrotra’s predictor-corrector interior-point method for linear programming, SIAM J. Optim., 2, 432-449, (1992) · Zbl 0771.90066
[8] Mizuno, S, Polynomiality of the kojima-megiddo-mizuno infeasible interior point algorithm for linear programming, Math. Program., 67, 109-119, (1994) · Zbl 0828.90086
[9] Zhang, Y, On the convergence of a class of infeasible-interior-point methods for the horizontal linear complementarity problem, SIAM J. Optim., 4, 208-227, (1994) · Zbl 0803.90092
[10] Mizuno, S; Todd, M; Ye, Y, On adaptive step primal-dual interior-point algorithms for linear programming, Math. Oper. Res., 18, 964-981, (1993) · Zbl 0810.90091
[11] Miao, J, Two infeasible interior-point predictor-corrector algorithms for linear programming, SIAM J. Optim., 6, 587-599, (1996) · Zbl 0856.90075
[12] Roos, C, A full-Newton step O(n) infeasible interior-point algorithm for linear optimization, SIAM J. Optim., 16, 1110-1136, (2006) · Zbl 1131.90029
[13] Salahi, M; Peng, J; Terlaky, T, On mehrotra-type predictor-corrector algorithms, SIAM J. Optim., 18, 1377-1397, (2007) · Zbl 1165.90569
[14] Kheirfam, B; Ahmadi, K; Hasani, F, A modified full-Newton step infeasible interior-point algoirhm for linear optimization, Asia-Pacific J. Operational Research, 30, 11-23, (2013) · Zbl 1401.90267
[15] Yang, Y., Yamashita, M. An \(\mathcal{O},(nL))\) Infeasible-Interior-Point Algorithm for Linear Programming (2015). arXiv:1506.06365 · Zbl 1165.90569
[16] Winternitz, LB; Tits, AL; Absil, P-A, Addressing rank degeneracy in constraint-reduced interior-point methods for linear optimization, J Optim Theory Appl, 160, 127-157, (2014) · Zbl 1311.90180
[17] Vanderbei, RJ, LOQO: an interior point code for quadratic programming, Optim. Methods Soft., 12, 451-484, (1999) · Zbl 0973.90518
[18] Czyzyk, J., Mehrotra, S., Wagner, M., Wright, S.J.: PCX User Guide (version 1.1), Technical Report OTC 96/01 Optimization Technology Center (1997) · Zbl 0973.90518
[19] Zhang, Y.: Solving large-scale linear programs by interior-point methods under the matlab environment, Technical Report TR96-01, Department of Mathematics and Statistics University of Maryland (1996)
[20] Monteiro, R; Adler, I; Resende, M, A polynomial-time primal-dual affine scaling algorithm for linear and convex quadratic programming and its power series extension, Math. Oper. Res., 15, 191-214, (1990) · Zbl 0714.90060
[21] Cartis, C, Some disadvantages of a mehrotra-type primal-dual corrector interior point algorithm for linear programming, Appl. Numer. Math., 59, 1110-1119, (2009) · Zbl 1163.90042
[22] Yang, Y, A polynomial arc-search interior-point algorithm for linear programming, J. Optim. Theory Appl., 158, 859-873, (2013) · Zbl 1274.90494
[23] Yang, Y, A polynomial arc-search interior-point algorithm for convex quadratic programming, Eur. J. Oper. Res., 215, 25-38, (2011) · Zbl 1252.90059
[24] Cartis, C., Gould, N.I.M.: Finding a point in the relative interior of a polyhedron, Technical Report NA-07/01, Computing Laboratory Oxford University (2007)
[25] Do Carmo, M.P.: Differential Geometry of Curves and Surfaces. Prentice-Hall, New Jersey (1976) · Zbl 0326.53001
[26] Brearley, AL; Mitra, G; Williams, HP, Analysis of methematical programming problems prior to applying the simplex algorithm, Math. Program., 8, 54-83, (1975) · Zbl 0317.90037
[27] Andersen, ED; Andersen, KD, Presolving in linear programming, Math. Program., 71, 221-245, (1993) · Zbl 0846.90068
[28] Mahajan, A.: Presolving mixed-integer linear programs, Preprint ANL/MCS-p1752-0510 argonne national laboratory (2010)
[29] Ghidini, CTLS; Oliveira, ARL; Silvab, J; Velazco, MI, Combining a hybrid preconditioner and a optimal adjustment algorithm to accelerate the convergence of interior point methods, Linear Algebra Appl., 436, 1267-1284, (2012) · Zbl 1236.65065
[30] Tits, AL; Yang, Y, Globally convergent algorithms for robust pole assignment by state feedback, IEEE Trans. Autom. Control, 41, 1432-1452, (1996) · Zbl 0884.93030
[31] Dolan, ED; More, JJ, Benchmarking optimization software with performance profiles, Math. Program., 91, 201-213, (2002) · Zbl 1049.90004
[32] Curtis, AR; Reid, JK, On the automatic scaling of matrices for Gaussian elimination, J. Inst. Maths Applics, 10, 118-124, (1972) · Zbl 0249.65026
[33] Andersen, ED, Finding all linearly dependent rows in large-scale linear programming, Optim. Methods Soft., 6, 219-227, (1995)
[34] Duff, J.F., Erisman, A.M., Reid, J.K.: Direct method for sparse matrices. Oxford University Press, New York (1989) · Zbl 0666.65024
[35] Dobes, J.: A modified Markowitz criterion for the fast modes of the LU factorization. In: Proceedings of 48th Midwest Symposium on Circuits and Systems, (2005), pp. 955-959 · Zbl 1049.90004
[36] Ng, E; Peyton, BW, Block sparse Cholesky algorithm on advanced uniprocessor computers, SIAM J. Sci. Comput., 14, 1034-1056, (1993) · Zbl 0785.65015
[37] Liu, JW, Modification of the minimum degree algorithm by multiple elimination, ACM Trans. Math. Softw., 11, 141-153, (1985) · Zbl 0568.65015
[38] Goldman, A.J., Tucker, A.W.: Theory of Linear Programming. In: Kuhn, H.W., Tucker (eds.) Linear Equalities and Related Systems, pp. 53-97. Princeton University Press, Princeton, N.J (1956) · Zbl 0072.37601
[39] Guler, O., den Hertog, D., Roos, C., Terlaky, T., Tsuchiya T.: Degeneracy in interior-point methods for linear programming: a survey. Ann. Oper. Res. 46, 107-138 (1993) · Zbl 0785.90067
[40] Gill, PE; Murray, W; Saunders, MA; Tomlin, JA; Wright, MH, On projected Newton barrier methods for linear programming and an equivalence of karmarkar’s projective method, Math. Program., 36, 183-209, (1986) · Zbl 0624.90062
[41] Vavasis, SA; Ye, Y, A primal dual interior-point method whose running time depends on the constraint matrix, Math. Program. A, 74, 79-120, (1996) · Zbl 0868.90081
[42] Yang, Y.: Arc-Search Infeasible Interior-Point Algorithm for Linear Programming. arXiv:1406.4539 (2014) · Zbl 1252.90059
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