June  2019, 9(2): 147-156. doi: 10.3934/naco.2019011

A Mehrotra type predictor-corrector interior-point algorithm for linear programming

Faculty of Mathematical Sciences, Shahrekord University, Shahrekord, Iran

* Corresponding author

Received  February 2017 Revised  August 2018 Published  January 2019

In this paper, we analyze a feasible predictor-corrector linear programming variant of Mehrotra's algorithm. The analysis is done in the negative infinity neighborhood of the central path. We demonstrate the theoretical efficiency of this algorithm by showing its polynomial complexity. The complexity result establishes an improvement of factor $ n^3 $ in the theoretical complexity of an earlier presented variant in [2], which is a huge improvement. We examine the performance of our algorithm by comparing its implementation results to solve some NETLIB problems with the algorithm presented in [2].

Citation: Soodabeh Asadi, Hossein Mansouri. A Mehrotra type predictor-corrector interior-point algorithm for linear programming. Numerical Algebra, Control & Optimization, 2019, 9 (2) : 147-156. doi: 10.3934/naco.2019011
References:
[1]

R. Almeida, F. Bastos and A. Teixeira, On polynomiality of a predictor-corrector variant algorithm, in International conference on numerical analysis and applied mathematica, Springer-Verlag, New York, (2010), 959–963. Google Scholar

[2]

R. Almeida and A. Teixeira, On the convergence of a predictor-corrector variant algorithm, TOP, 23 (2015), 401-418.  doi: 10.1007/s11750-014-0346-8.  Google Scholar

[3]

E. D. Andersen and K. D. Andersen, The MOSEK interior point optimizer for linear programming: an implementation of the homogeneous algorithm, in High Performance Optimization (eds. H. Frenk, K. Roos, T. Terlaky and S. Zhang), Kluwer Academic Publishers, (2000), 197–232. doi: 10.1007/978-1-4757-3216-0_8.  Google Scholar

[4]

S. Asadi, H. Mansouri, Zs. Darvay and M. Zangiabadi, On the $P_*(\kappa)$ horizontal linear complementarity problems over Cartesian product of symmetric cones, Optim. Methods Softw., 31 (2016), 233-257. doi: 10.1080/10556788.2015.1058795.  Google Scholar

[5]

S. Asadi, H. Mansouri, Zs. Darvay, G. Lesaja and M. Zangiabadi, A long-step feasible predictor-corrector interior-point algorithm for symmetric cone optimization, Optim. Methods Softw., 67 (2018), 2031–2060 doi: 10.1080/10556788.2018.1528248.  Google Scholar

[6]

S. AsadiH. MansouriG. Lesaja and M. Zangiabadi, A long-step interior-point algorithm for symmetric cone Cartesian $P_*(\kappa)$ -HLCP, Optimization, 67 (2018), 2031-2060.  doi: 10.1080/02331934.2018.1512604.  Google Scholar

[7]

S. Asadi, H. Mansouri, Zs. Darvay, M. Zangiabadi and N Mahdavi-Amiri, Large-neighborhood infeasible predictor-corrector algorithm for horizontal linear complementarity problems over cartesian product of symmetric cones, J. Optim. Theory Appl., q doi: 10.1007/s10957-018-1402-6.  Google Scholar

[8]

S. AsadiH. Mansouri and and Zs. Darvay, An infeasible full-NT step IPM for $P_*(\kappa)$ horizontal linear complementarity problem over Cartesian product of symmetric cones, Optimization, 66 (2017), 225-250.  doi: 10.1080/02331934.2016.1267732.  Google Scholar

[9]

J. CzyzykS. MehrtotraM. Wagner and S. J. Wright, PCx: an interior-point code for linear programming, Optim. Methods Softw., 11/12 (1999), 397-430.  doi: 10.1080/10556789908805757.  Google Scholar

[10]

J. JiF. Potra and R. Sheng, On a local convergence of a predictor-corrector method for semidefinite programming, SIAM J. Optim., 10 (1999), 195-210.  doi: 10.1137/S1052623497316828.  Google Scholar

[11]

N. K. Karmarkar, A new polynomial-time algorithm for linear programming, Combinatorica, 4 (1984), 373-395.  doi: 10.1007/BF02579150.  Google Scholar

[12]

M. Kojima, N. Megiddo, T. Noma and A. Yoshise, A Unified Approach to Interior Point Algorithms for Linear Complementarity Problems, Springer, Berlin, 1991. doi: 10.1007/3-540-54509-3.  Google Scholar

[13]

M. KojimaN. Megiddo and S. Mizuno, A primal-dual infeasible-interior-point algorithm for linear programming, Math. Program., 61 (1993), 263-280.  doi: 10.1007/BF01582151.  Google Scholar

[14]

S. Mehrotra, On finding a vertex solution using interior-point methods, Linear Algebra Appl., 152 (1991), 233-253.  doi: 10.1016/0024-3795(91)90277-4.  Google Scholar

[15]

S. Mehrotra, On the implementation of a primal-dual interior point method, SIAM J. Optim., 2 (1992), 575-601.  doi: 10.1137/0802028.  Google Scholar

[16]

N. Megiddo, Pathways to the optimal set in linear programming, in Progress in Mathematical Programming, (1989), 135–158.  Google Scholar

[17]

S. MizunoM. J. Todd and Y. Ye, On adaptive-step primal-dual interior-point algorithms for linear programming, Math. Oper. Res., 18 (1993), 964-981.  doi: 10.1287/moor.18.4.964.  Google Scholar

[18]

R. D. C. Monteiro, Primal-dual path-following algorithm for semidefinite programming, SIAM J. Optim., 7 (1997), 663-678.  doi: 10.1137/S1052623495293056.  Google Scholar

[19]

J. Peng, C. Roos and T. Terlaky, Self-Regularity: A New Paradigm for Primal-Dual Interior-Point Algorithms. Princeton University Press, Princeton, New Jersey, 2002.  Google Scholar

[20]

M. SalahiJ. Peng and T. Terlaky, On mehrotra-type predictor-corrector algorithms, SIAM J. Optim., 18 (2007), 1377-1397.  doi: 10.1137/050628787.  Google Scholar

[21]

M. Salahi, A finite termination mehrotra type predictor-corrector algorithm, Appl. Math. Comput., 190 (2007), 1740-1746.  doi: 10.1016/j.amc.2007.02.061.  Google Scholar

[22]

Gy. Sonnevend, An analytic center for polyhedrons and new classes of global algorithms for linear (smooth, convex) programming, in Lecture Notes in Control and Information Sciences, Springer, Berlin, (1985), 866–876. doi: 10.1007/BFb0043914.  Google Scholar

[23]

J. Stoer and M. Wechs, Infeasible-interior-point paths for sufficient linear complementarity problems and their analyticity, Math. Program. Ser. A., 83 (1998), 407-423.  doi: 10.1016/S0025-5610(98)00011-2.  Google Scholar

[24]

G. Q. Wang and Y. Q. Bai, Polynomial interior-point algorithms for $P_*(\kappa)$ horizontal linear complementarity problem, J. Comput. Appl. Math., 233 (2009), 248-263.  doi: 10.1016/j.cam.2009.07.014.  Google Scholar

[25]

G. Q. Wang and G. Lesaja, Full Nesterov-Todd step feasible interior-point method for the Cartesian $P_*(\kappa)$-SCLCP, Optim. Methods Softw., 28 (2013), 600-618.  doi: 10.1080/10556788.2013.781600.  Google Scholar

[26]

Y. Zhang and D. Zhang, Superlinear convergence of infeasible-interior-point methods for linear programming, Math. Program., 66 (1994), 361-377.  doi: 10.1007/BF01581155.  Google Scholar

[27]

Y. Zhang, On the convergence of a class of infeasible interior-point methods for the horizontal linear complementarity problem, SIAM J. Optim., 4 (1994), 208-227.  doi: 10.1137/0804012.  Google Scholar

[28]

Y. Zhang, Solving large scale linear programmes by interior point methods under the Matlab environment, Optim. Methods Softw., 10 (1999), 1-31.  doi: 10.1080/10556789808805699.  Google Scholar

show all references

References:
[1]

R. Almeida, F. Bastos and A. Teixeira, On polynomiality of a predictor-corrector variant algorithm, in International conference on numerical analysis and applied mathematica, Springer-Verlag, New York, (2010), 959–963. Google Scholar

[2]

R. Almeida and A. Teixeira, On the convergence of a predictor-corrector variant algorithm, TOP, 23 (2015), 401-418.  doi: 10.1007/s11750-014-0346-8.  Google Scholar

[3]

E. D. Andersen and K. D. Andersen, The MOSEK interior point optimizer for linear programming: an implementation of the homogeneous algorithm, in High Performance Optimization (eds. H. Frenk, K. Roos, T. Terlaky and S. Zhang), Kluwer Academic Publishers, (2000), 197–232. doi: 10.1007/978-1-4757-3216-0_8.  Google Scholar

[4]

S. Asadi, H. Mansouri, Zs. Darvay and M. Zangiabadi, On the $P_*(\kappa)$ horizontal linear complementarity problems over Cartesian product of symmetric cones, Optim. Methods Softw., 31 (2016), 233-257. doi: 10.1080/10556788.2015.1058795.  Google Scholar

[5]

S. Asadi, H. Mansouri, Zs. Darvay, G. Lesaja and M. Zangiabadi, A long-step feasible predictor-corrector interior-point algorithm for symmetric cone optimization, Optim. Methods Softw., 67 (2018), 2031–2060 doi: 10.1080/10556788.2018.1528248.  Google Scholar

[6]

S. AsadiH. MansouriG. Lesaja and M. Zangiabadi, A long-step interior-point algorithm for symmetric cone Cartesian $P_*(\kappa)$ -HLCP, Optimization, 67 (2018), 2031-2060.  doi: 10.1080/02331934.2018.1512604.  Google Scholar

[7]

S. Asadi, H. Mansouri, Zs. Darvay, M. Zangiabadi and N Mahdavi-Amiri, Large-neighborhood infeasible predictor-corrector algorithm for horizontal linear complementarity problems over cartesian product of symmetric cones, J. Optim. Theory Appl., q doi: 10.1007/s10957-018-1402-6.  Google Scholar

[8]

S. AsadiH. Mansouri and and Zs. Darvay, An infeasible full-NT step IPM for $P_*(\kappa)$ horizontal linear complementarity problem over Cartesian product of symmetric cones, Optimization, 66 (2017), 225-250.  doi: 10.1080/02331934.2016.1267732.  Google Scholar

[9]

J. CzyzykS. MehrtotraM. Wagner and S. J. Wright, PCx: an interior-point code for linear programming, Optim. Methods Softw., 11/12 (1999), 397-430.  doi: 10.1080/10556789908805757.  Google Scholar

[10]

J. JiF. Potra and R. Sheng, On a local convergence of a predictor-corrector method for semidefinite programming, SIAM J. Optim., 10 (1999), 195-210.  doi: 10.1137/S1052623497316828.  Google Scholar

[11]

N. K. Karmarkar, A new polynomial-time algorithm for linear programming, Combinatorica, 4 (1984), 373-395.  doi: 10.1007/BF02579150.  Google Scholar

[12]

M. Kojima, N. Megiddo, T. Noma and A. Yoshise, A Unified Approach to Interior Point Algorithms for Linear Complementarity Problems, Springer, Berlin, 1991. doi: 10.1007/3-540-54509-3.  Google Scholar

[13]

M. KojimaN. Megiddo and S. Mizuno, A primal-dual infeasible-interior-point algorithm for linear programming, Math. Program., 61 (1993), 263-280.  doi: 10.1007/BF01582151.  Google Scholar

[14]

S. Mehrotra, On finding a vertex solution using interior-point methods, Linear Algebra Appl., 152 (1991), 233-253.  doi: 10.1016/0024-3795(91)90277-4.  Google Scholar

[15]

S. Mehrotra, On the implementation of a primal-dual interior point method, SIAM J. Optim., 2 (1992), 575-601.  doi: 10.1137/0802028.  Google Scholar

[16]

N. Megiddo, Pathways to the optimal set in linear programming, in Progress in Mathematical Programming, (1989), 135–158.  Google Scholar

[17]

S. MizunoM. J. Todd and Y. Ye, On adaptive-step primal-dual interior-point algorithms for linear programming, Math. Oper. Res., 18 (1993), 964-981.  doi: 10.1287/moor.18.4.964.  Google Scholar

[18]

R. D. C. Monteiro, Primal-dual path-following algorithm for semidefinite programming, SIAM J. Optim., 7 (1997), 663-678.  doi: 10.1137/S1052623495293056.  Google Scholar

[19]

J. Peng, C. Roos and T. Terlaky, Self-Regularity: A New Paradigm for Primal-Dual Interior-Point Algorithms. Princeton University Press, Princeton, New Jersey, 2002.  Google Scholar

[20]

M. SalahiJ. Peng and T. Terlaky, On mehrotra-type predictor-corrector algorithms, SIAM J. Optim., 18 (2007), 1377-1397.  doi: 10.1137/050628787.  Google Scholar

[21]

M. Salahi, A finite termination mehrotra type predictor-corrector algorithm, Appl. Math. Comput., 190 (2007), 1740-1746.  doi: 10.1016/j.amc.2007.02.061.  Google Scholar

[22]

Gy. Sonnevend, An analytic center for polyhedrons and new classes of global algorithms for linear (smooth, convex) programming, in Lecture Notes in Control and Information Sciences, Springer, Berlin, (1985), 866–876. doi: 10.1007/BFb0043914.  Google Scholar

[23]

J. Stoer and M. Wechs, Infeasible-interior-point paths for sufficient linear complementarity problems and their analyticity, Math. Program. Ser. A., 83 (1998), 407-423.  doi: 10.1016/S0025-5610(98)00011-2.  Google Scholar

[24]

G. Q. Wang and Y. Q. Bai, Polynomial interior-point algorithms for $P_*(\kappa)$ horizontal linear complementarity problem, J. Comput. Appl. Math., 233 (2009), 248-263.  doi: 10.1016/j.cam.2009.07.014.  Google Scholar

[25]

G. Q. Wang and G. Lesaja, Full Nesterov-Todd step feasible interior-point method for the Cartesian $P_*(\kappa)$-SCLCP, Optim. Methods Softw., 28 (2013), 600-618.  doi: 10.1080/10556788.2013.781600.  Google Scholar

[26]

Y. Zhang and D. Zhang, Superlinear convergence of infeasible-interior-point methods for linear programming, Math. Program., 66 (1994), 361-377.  doi: 10.1007/BF01581155.  Google Scholar

[27]

Y. Zhang, On the convergence of a class of infeasible interior-point methods for the horizontal linear complementarity problem, SIAM J. Optim., 4 (1994), 208-227.  doi: 10.1137/0804012.  Google Scholar

[28]

Y. Zhang, Solving large scale linear programmes by interior point methods under the Matlab environment, Optim. Methods Softw., 10 (1999), 1-31.  doi: 10.1080/10556789808805699.  Google Scholar

Table 1.  The number of iterations
Problem $ m $ $ n $ Alg 1 (It.) Alg 1 ($ x^Tv $) Alg 2 (It.) Alg 2 ($ x^Tv $)
blend 74 114 242 8.3720e-4 280 8.4720e-4
adlittle 56 138 61 3.8658e-4 376 3.7909e-4
scagr7 129 185 308 4.2065e-4 217 7.5233e-4
share1b 117 253 51 1.3551e-4 344 1.0125e-4
share2b 96 162 191 3.8527e-4 296 3.9533e-4
scsd1 77 760 75 1.1725e-4 112 1.0346e-4
sc105 105 163 238 5.0063e-4 266 1.6058e-4
agg 488 615 31 1.0920e-4 199 1.0088e-4
Problem $ m $ $ n $ Alg 1 (It.) Alg 1 ($ x^Tv $) Alg 2 (It.) Alg 2 ($ x^Tv $)
blend 74 114 242 8.3720e-4 280 8.4720e-4
adlittle 56 138 61 3.8658e-4 376 3.7909e-4
scagr7 129 185 308 4.2065e-4 217 7.5233e-4
share1b 117 253 51 1.3551e-4 344 1.0125e-4
share2b 96 162 191 3.8527e-4 296 3.9533e-4
scsd1 77 760 75 1.1725e-4 112 1.0346e-4
sc105 105 163 238 5.0063e-4 266 1.6058e-4
agg 488 615 31 1.0920e-4 199 1.0088e-4
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