Improving an interior-point algorithm for multicommodity flows by quadratic regularizations
One of the best approaches for some classes of multicommodity flow problems is a specialized interior‐point method that solves the normal equations by a combination of Cholesky factorizations and preconditioned conjugate gradient. Its efficiency depends on the spectral radius—in [0,1)—of a certain m...
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| Vydané v: | Networks Ročník 59; číslo 1; s. 117 - 131 |
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| Abstract | One of the best approaches for some classes of multicommodity flow problems is a specialized interior‐point method that solves the normal equations by a combination of Cholesky factorizations and preconditioned conjugate gradient. Its efficiency depends on the spectral radius—in [0,1)—of a certain matrix in the definition of the preconditioner. In a recent work, the authors improved this algorithm (i.e., reduced the spectral radius) for general block‐angular problems by adding a quadratic regularization to the logarithmic barrier. This barrier was shown to be self‐concordant, which guarantees the convergence and polynomial complexity of the algorithm. In this work, we focus on linear multicommodity problems, a particular case of primal block‐angular ones. General results are tailored for multicommodity flows, allowing a local sensitivity analysis on the effect of the regularization. Extensive computational results on some standard and some difficult instances, testing several regularization strategies, are also provided. These results show that the regularized interior‐point algorithm is more efficient than the nonregularized one. From this work it can be concluded that, if interior‐point methods based on conjugate gradients are used, linear multicommodity flow problems are most efficiently solved as a sequence of quadratic ones. © 2011 Wiley Periodicals, Inc. NETWORKS, 2012 |
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| AbstractList | One of the best approaches for some classes of multicommodity flow problems is a specialized interior‐point method that solves the normal equations by a combination of Cholesky factorizations and preconditioned conjugate gradient. Its efficiency depends on the spectral radius—in [0,1)—of a certain matrix in the definition of the preconditioner. In a recent work, the authors improved this algorithm (i.e., reduced the spectral radius) for general block‐angular problems by adding a quadratic regularization to the logarithmic barrier. This barrier was shown to be self‐concordant, which guarantees the convergence and polynomial complexity of the algorithm. In this work, we focus on linear multicommodity problems, a particular case of primal block‐angular ones. General results are tailored for multicommodity flows, allowing a local sensitivity analysis on the effect of the regularization. Extensive computational results on some standard and some difficult instances, testing several regularization strategies, are also provided. These results show that the regularized interior‐point algorithm is more efficient than the nonregularized one. From this work it can be concluded that, if interior‐point methods based on conjugate gradients are used, linear multicommodity flow problems are most efficiently solved as a sequence of quadratic ones. © 2011 Wiley Periodicals, Inc. NETWORKS, 2012 |
| Author | Castro, Jordi Cuesta, Jordi |
| Author_xml | – sequence: 1 givenname: Jordi surname: Castro fullname: Castro, Jordi email: jordi.castro@upc.edu organization: Department of Statistics and Operations Research, Universitat Politècnica de Catalunya, c. Jordi Girona 1-3, 08034 Barcelona, Catalonia, Spain – sequence: 2 givenname: Jordi surname: Cuesta fullname: Cuesta, Jordi organization: Unit of Statistics and Operations Research, Department of Chemical Engineering, Universitat Rovira i Virgili, c. de l'Escorxador s/n, 43003 Tarragona, Catalonia, Spain |
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| References | R. D. McBride, Progress made in solving the multicommodity flow problem, SIAM J Optim 8 ( 1998), 947-955. A. Ouorou, Implementing a proximal point algorithm to some nonlinear multicommodity flow problems, Networks 18 ( 2007), 18-27. A. Altman and J. Gondzio, Regularized symmetric indefinite systems in interior point methods for linear and quadratic optimization, Optim Methods Software 11 ( 1999), 275-302. J.-L. Goffin, J. Gondzio, R. Sarkissian, and J.-P. Vial, Solving nonlinear multicommodity flow problems by the analytic center cutting plane method, Math Prog 76 ( 1996), 131-154. W. J. Carolan, J. E. Hill, J. L. Kennington, S. Niemi, and S. J. Wichmann, An empirical evaluation of the KORBX algorithms for military airlift applications, Oper Res 38 ( 1990), 240-248. A. Ouorou, P. Mahey, and J.-P. Vial, A survey of algorithms for convex multicommodity flow problems, Manage Sci 46 ( 2000), 126-147. J. Castro and J. Cuesta, Quadratic regularizations in an interior-point method for primal block-angular problems, Math Prog, doi:10.1007/s10107-010-0341-2. J. Castro, Solving difficult multicommodity problems through a specialized interior-point algorithm, Ann Oper Res 124 ( 2003), 35-48. G. H. Golub and C. F. Van Loan, Matrix computations, 3rd edition, Johns Hopkins University Press, Baltimore, 1996. R. Setiono, Interior proximal point algorithm for linear programs, J Optim Theory Appl 74 ( 1992), 425-444. F. Babonneau, O. du Merle, and J.-P. Vial, Solving large-scale linear multicommodity flow problems with an active set strategy and proximal-ACCPM, Oper Res 54 ( 2006), 184-197. A. Frangioni and C. Gentile, New preconditioners for KKT systems of network flow problems, SIAM J Optim 14 ( 2004), 894-913. D. Bienstock, Potential function methods for approximately solving linear programming problems. Theory and Practice, Kluwer, Boston, 2002. A. Frangioni and G. Gallo, A bundle type dual-ascent approach to linear multicommodity min cost flow problems, INFORMS J Comp 11 ( 1999), 370-393. Y. Nesterov, Introductory lectures on convex optimization: A basic course, Kluwer, Boston, 2004. M. G. C. Resende and G. Veiga, An implementation of the dual affine scaling algorithm for minimum-cost flow on bipartite uncapacitated networks, SIAM J Optim 3 ( 1993), 516-537. J. Castro and N. Nabona, An implementation of linear and nonlinear multicommodity network flows, Eur J Oper Res 92 ( 1996), 37-53. C. Lemaréchal, A. Ouorou, and G. Petrou, A bundle-type algorithm for routing in telecommunication data networks, Comput Optim Appl 44 ( 2009), 385-409. J. Castro and J. Cuesta, Existence, uniqueness and convergence of the regularized primal-dual central path, Oper Res Lett 38 ( 2010), 366-371. E. Ng and B. W. Peyton, Block sparse Cholesky algorithms on advanced uniprocessor computers, SIAM J Sci Comput 14 ( 1993), 1034-1056. P. Chardaire and A. Lisser, Simplex and interior point specialized algorithms for solving nonoriented multicommodity flow problems, Oper Res 50 ( 2002), 260-276. F. Babonneau and J.-P. Vial, ACCPM with a nonlinear constraint and an active set strategy to solve nonlinear multicommodity flow problems, Math Prog 120 ( 2009), 179-210. J. Castro, A specialized interior-point algorithm for multicommodity network flows, SIAM J Optim 10 ( 2000), 852-877. S. J. Wright, Primal-dual interior-point methods, SIAM, Philadelphia, 1996. J. Castro, An interior-point approach for primal block-angular problems, Comput Optim Appl 36 ( 2007), 195-219. 2009; 44 1993; 14 2007; 18 2010; 38 1990; 38 2006; 54 2002; 50 2000; 46 2000; 10 2004; 14 1999; 11 1996 2004 2003 1996; 92 2009; 120 2002 2003; 124 1996; 76 1993; 3 2007; 36 1977 1998; 8 1992; 74 e_1_2_9_10_2 e_1_2_9_11_2 Wright S. J. (e_1_2_9_28_2) 1996 Castro J. (e_1_2_9_12_2) e_1_2_9_14_2 e_1_2_9_13_2 e_1_2_9_16_2 e_1_2_9_15_2 e_1_2_9_18_2 e_1_2_9_17_2 e_1_2_9_21_2 e_1_2_9_20_2 e_1_2_9_23_2 e_1_2_9_22_2 e_1_2_9_7_2 e_1_2_9_5_2 Golub G. H. (e_1_2_9_19_2) 1996 e_1_2_9_4_2 e_1_2_9_3_2 e_1_2_9_2_2 e_1_2_9_9_2 e_1_2_9_8_2 e_1_2_9_25_2 e_1_2_9_24_2 e_1_2_9_27_2 e_1_2_9_26_2 Bienstock D. (e_1_2_9_6_2) 2002 |
| References_xml | – reference: G. H. Golub and C. F. Van Loan, Matrix computations, 3rd edition, Johns Hopkins University Press, Baltimore, 1996. – reference: W. J. Carolan, J. E. Hill, J. L. Kennington, S. Niemi, and S. J. Wichmann, An empirical evaluation of the KORBX algorithms for military airlift applications, Oper Res 38 ( 1990), 240-248. – reference: P. Chardaire and A. Lisser, Simplex and interior point specialized algorithms for solving nonoriented multicommodity flow problems, Oper Res 50 ( 2002), 260-276. – reference: A. Ouorou, Implementing a proximal point algorithm to some nonlinear multicommodity flow problems, Networks 18 ( 2007), 18-27. – reference: A. Ouorou, P. Mahey, and J.-P. Vial, A survey of algorithms for convex multicommodity flow problems, Manage Sci 46 ( 2000), 126-147. – reference: R. Setiono, Interior proximal point algorithm for linear programs, J Optim Theory Appl 74 ( 1992), 425-444. – reference: A. Frangioni and G. Gallo, A bundle type dual-ascent approach to linear multicommodity min cost flow problems, INFORMS J Comp 11 ( 1999), 370-393. – reference: E. Ng and B. W. Peyton, Block sparse Cholesky algorithms on advanced uniprocessor computers, SIAM J Sci Comput 14 ( 1993), 1034-1056. – reference: J.-L. Goffin, J. Gondzio, R. Sarkissian, and J.-P. Vial, Solving nonlinear multicommodity flow problems by the analytic center cutting plane method, Math Prog 76 ( 1996), 131-154. – reference: F. Babonneau, O. du Merle, and J.-P. Vial, Solving large-scale linear multicommodity flow problems with an active set strategy and proximal-ACCPM, Oper Res 54 ( 2006), 184-197. – reference: R. D. McBride, Progress made in solving the multicommodity flow problem, SIAM J Optim 8 ( 1998), 947-955. – reference: J. Castro, Solving difficult multicommodity problems through a specialized interior-point algorithm, Ann Oper Res 124 ( 2003), 35-48. – reference: M. G. C. Resende and G. Veiga, An implementation of the dual affine scaling algorithm for minimum-cost flow on bipartite uncapacitated networks, SIAM J Optim 3 ( 1993), 516-537. – reference: S. J. Wright, Primal-dual interior-point methods, SIAM, Philadelphia, 1996. – reference: Y. Nesterov, Introductory lectures on convex optimization: A basic course, Kluwer, Boston, 2004. – reference: J. Castro and J. Cuesta, Quadratic regularizations in an interior-point method for primal block-angular problems, Math Prog, doi:10.1007/s10107-010-0341-2. – reference: F. Babonneau and J.-P. Vial, ACCPM with a nonlinear constraint and an active set strategy to solve nonlinear multicommodity flow problems, Math Prog 120 ( 2009), 179-210. – reference: J. Castro, An interior-point approach for primal block-angular problems, Comput Optim Appl 36 ( 2007), 195-219. – reference: J. Castro, A specialized interior-point algorithm for multicommodity network flows, SIAM J Optim 10 ( 2000), 852-877. – reference: D. Bienstock, Potential function methods for approximately solving linear programming problems. Theory and Practice, Kluwer, Boston, 2002. – reference: J. Castro and J. Cuesta, Existence, uniqueness and convergence of the regularized primal-dual central path, Oper Res Lett 38 ( 2010), 366-371. – reference: J. Castro and N. Nabona, An implementation of linear and nonlinear multicommodity network flows, Eur J Oper Res 92 ( 1996), 37-53. – reference: A. Altman and J. Gondzio, Regularized symmetric indefinite systems in interior point methods for linear and quadratic optimization, Optim Methods Software 11 ( 1999), 275-302. – reference: A. Frangioni and C. Gentile, New preconditioners for KKT systems of network flow problems, SIAM J Optim 14 ( 2004), 894-913. – reference: C. Lemaréchal, A. Ouorou, and G. Petrou, A bundle-type algorithm for routing in telecommunication data networks, Comput Optim Appl 44 ( 2009), 385-409. – volume: 50 start-page: 260 year: 2002 end-page: 276 article-title: Simplex and interior point specialized algorithms for solving nonoriented multicommodity flow problems publication-title: Oper Res – volume: 38 start-page: 366 year: 2010 end-page: 371 article-title: Existence, uniqueness and convergence of the regularized primal‐dual central path publication-title: Oper Res Lett – volume: 18 start-page: 18 year: 2007 end-page: 27 article-title: Implementing a proximal point algorithm to some nonlinear multicommodity flow problems publication-title: Networks – volume: 38 start-page: 240 year: 1990 end-page: 248 article-title: An empirical evaluation of the KORBX algorithms for military airlift applications publication-title: Oper Res – volume: 3 start-page: 516 year: 1993 end-page: 537 article-title: An implementation of the dual affine scaling algorithm for minimum‐cost flow on bipartite uncapacitated networks publication-title: SIAM J Optim – volume: 92 start-page: 37 year: 1996 end-page: 53 article-title: An implementation of linear and nonlinear multicommodity network flows publication-title: Eur J Oper Res – volume: 14 start-page: 894 year: 2004 end-page: 913 article-title: New preconditioners for KKT systems of network flow problems publication-title: SIAM J Optim – year: 1996 – volume: 14 start-page: 1034 year: 1993 end-page: 1056 article-title: Block sparse Cholesky algorithms on advanced uniprocessor computers publication-title: SIAM J Sci Comput – volume: 8 start-page: 947 year: 1998 end-page: 955 article-title: Progress made in solving the multicommodity flow problem publication-title: SIAM J Optim – year: 1977 – volume: 10 start-page: 852 year: 2000 end-page: 877 article-title: A specialized interior‐point algorithm for multicommodity network flows publication-title: SIAM J Optim – start-page: 199 year: 2003 end-page: 212 – volume: 54 start-page: 184 year: 2006 end-page: 197 article-title: Solving large‐scale linear multicommodity flow problems with an active set strategy and proximal‐ACCPM publication-title: Oper Res – year: 2002 – volume: 76 start-page: 131 year: 1996 end-page: 154 article-title: Solving nonlinear multicommodity flow problems by the analytic center cutting plane method publication-title: Math Prog – volume: 74 start-page: 425 year: 1992 end-page: 444 article-title: Interior proximal point algorithm for linear programs publication-title: J Optim Theory Appl – volume: 36 start-page: 195 year: 2007 end-page: 219 article-title: An interior‐point approach for primal block‐angular problems publication-title: Comput Optim Appl – year: 2004 – volume: 120 start-page: 179 year: 2009 end-page: 210 article-title: ACCPM with a nonlinear constraint and an active set strategy to solve nonlinear multicommodity flow problems publication-title: Math Prog – volume: 46 start-page: 126 year: 2000 end-page: 147 article-title: A survey of algorithms for convex multicommodity flow problems publication-title: Manage Sci – volume: 11 start-page: 275 year: 1999 end-page: 302 article-title: Regularized symmetric indefinite systems in interior point methods for linear and quadratic optimization publication-title: Optim Methods Software – article-title: Quadratic regularizations in an interior‐point method for primal block‐angular problems publication-title: Math Prog – volume: 11 start-page: 370 year: 1999 end-page: 393 article-title: A bundle type dual‐ascent approach to linear multicommodity min cost flow problems publication-title: INFORMS J Comp – volume: 124 start-page: 35 year: 2003 end-page: 48 article-title: Solving difficult multicommodity problems through a specialized interior‐point algorithm publication-title: Ann Oper Res – volume: 44 start-page: 385 year: 2009 end-page: 409 article-title: A bundle‐type algorithm for routing in telecommunication data networks publication-title: Comput Optim Appl – ident: e_1_2_9_17_2 doi: 10.1137/S105262340240519X – ident: e_1_2_9_22_2 doi: 10.1007/978-1-4419-8853-9 – ident: e_1_2_9_2_2 – ident: e_1_2_9_7_2 doi: 10.1287/opre.38.2.240 – volume-title: Potential function methods for approximately solving linear programming problems. Theory and Practice year: 2002 ident: e_1_2_9_6_2 – ident: e_1_2_9_10_2 doi: 10.1007/978-0-387-35699-0_10 – ident: e_1_2_9_3_2 doi: 10.1080/10556789908805754 – ident: e_1_2_9_20_2 doi: 10.1007/s10589-007-9160-7 – ident: e_1_2_9_8_2 doi: 10.1137/S1052623498341879 – ident: e_1_2_9_4_2 doi: 10.1007/s10107-007-0151-3 – ident: e_1_2_9_5_2 doi: 10.1287/opre.1050.0262 – ident: e_1_2_9_9_2 doi: 10.1023/B:ANOR.0000004761.99649.a5 – ident: e_1_2_9_21_2 doi: 10.1137/S1052623496304542 – ident: e_1_2_9_23_2 doi: 10.1137/0914063 – ident: e_1_2_9_16_2 doi: 10.1287/ijoc.11.4.370 – ident: e_1_2_9_14_2 doi: 10.1016/0377-2217(95)00137-9 – ident: e_1_2_9_13_2 doi: 10.1016/j.orl.2010.07.010 – ident: e_1_2_9_15_2 doi: 10.1287/opre.50.2.260.436 – ident: e_1_2_9_18_2 doi: 10.1007/BF02614381 – ident: e_1_2_9_27_2 doi: 10.1007/BF00940319 – volume-title: Matrix computations year: 1996 ident: e_1_2_9_19_2 – ident: e_1_2_9_26_2 doi: 10.1137/0803025 – ident: e_1_2_9_25_2 doi: 10.1287/mnsc.46.1.126.15132 – ident: e_1_2_9_24_2 doi: 10.1002/net.20138 – volume-title: Primal‐dual interior‐point methods year: 1996 ident: e_1_2_9_28_2 – ident: e_1_2_9_12_2 article-title: Quadratic regularizations in an interior‐point method for primal block‐angular problems publication-title: Math Prog – ident: e_1_2_9_11_2 doi: 10.1007/s10589-006-9000-1 |
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