메뉴 건너뛰기




Volumn 35, Issue 3, 2010, Pages 641-654

Randomized methods for linear constraints: Convergence rates and conditioning

Author keywords

Averaged projections; Condition number; Coordinate descent; Distance to ill posedness; Error bound; Iterated projections; Linear constraint; Metric regularity; Randomization

Indexed keywords

AVERAGED PROJECTIONS; CONDITION NUMBERS; COORDINATE DESCENT; DISTANCE TO ILL-POSEDNESS; ERROR BOUND; ITERATED PROJECTIONS; LINEAR CONSTRAINTS; METRIC REGULARITY; RANDOMIZATION;

EID: 77956625417     PISSN: 0364765X     EISSN: 15265471     Source Type: Journal    
DOI: 10.1287/moor.1100.0456     Document Type: Article
Times cited : (327)

References (45)
  • 1
    • 0002662980 scopus 로고
    • On a successive transformation of probability distribution and its application to the analysis of the optimum gradient method
    • Akaike, H. 1959. On a successive transformation of probability distribution and its application to the analysis of the optimum gradient method. Ann. Inst. Statist. Math. 11 1-16.
    • (1959) Ann. Inst. Statist. Math. , vol.11 , pp. 1-16
    • Akaike, H.1
  • 3
    • 0013467971 scopus 로고
    • Convergence of random products of contractions in Hilbert space
    • Amemiya, I., T. Ando. 1965. Convergence of random products of contractions in Hilbert space. Acta. Sci. Math. 26 239-244.
    • (1965) Acta. Sci. Math. , vol.26 , pp. 239-244
    • Amemiya, I.1    Ando, T.2
  • 4
    • 0000746478 scopus 로고
    • A norm convergence result on random products of relaxed projections in Hilbert space
    • Bauschke, H. H. 1995. A norm convergence result on random products of relaxed projections in Hilbert space. Trans. Amer. Math. Soc. 347 1365-1373.
    • (1995) Trans. Amer. Math. Soc. , vol.347 , pp. 1365-1373
    • Bauschke, H.H.1
  • 6
    • 0001448913 scopus 로고
    • On the convergence of von Neumann's alternating projection algorithm for two sets
    • Bauschke, H. H., J. M. Borwein. 1993. On the convergence of von Neumann's alternating projection algorithm for two sets. Set-Valued Anal. 1 185-212.
    • (1993) Set-Valued Anal. , vol.1 , pp. 185-212
    • Bauschke, H.H.1    Borwein, J.M.2
  • 7
    • 0030246542 scopus 로고    scopus 로고
    • On projection algorithms for solving convex feasibility problems
    • Bauschke, H. H., J. M. Borwein. 1996. On projection algorithms for solving convex feasibility problems. SIAM Rev. 38 367-426.
    • (1996) SIAM Rev. , vol.38 , pp. 367-426
    • Bauschke, H.H.1    Borwein, J.M.2
  • 9
    • 5244239809 scopus 로고
    • Random products of contractions in metric and Banach spaces
    • Bruck, R. E. 1982. Random products of contractions in metric and Banach spaces. J. Math. Anal. Appl. 88 319-332.
    • (1982) J. Math. Anal. Appl. , vol.88 , pp. 319-332
    • Bruck, R.E.1
  • 10
    • 0001345156 scopus 로고
    • On condition numbers and the distance to the nearest ill-posed problem
    • Demmel, J. W. 1987. On condition numbers and the distance to the nearest ill-posed problem. Numerische Mathematik 51 251-289.
    • (1987) Numerische Mathematik , vol.51 , pp. 251-289
    • Demmel, J.W.1
  • 11
    • 84966237760 scopus 로고
    • The probability that a numerical analysis problem is difficult
    • Demmel, J. W. 1988. The probability that a numerical analysis problem is difficult. Math. Comput. 50 449-480.
    • (1988) Math. Comput. , vol.50 , pp. 449-480
    • Demmel, J.W.1
  • 14
    • 0013499061 scopus 로고
    • Random products of contractions in Banach spaces
    • Dye, J., M. A. Khamsi, S. Reich. 1991. Random products of contractions in Banach spaces. Trans. Amer. Math. Soc. 325 87-99.
    • (1991) Trans. Amer. Math. Soc. , vol.325 , pp. 87-99
    • Dye, J.1    Khamsi, M.A.2    Reich, S.3
  • 15
    • 0000802374 scopus 로고
    • The approximation of one matrix by another of low rank
    • Eckart, C., G. Young. 1936. The approximation of one matrix by another of low rank. Psychometrika 1 211-218.
    • (1936) Psychometrika , vol.1 , pp. 211-218
    • Eckart, C.1    Young, G.2
  • 17
  • 18
    • 21844502875 scopus 로고
    • Approximations to solutions to systems of linear inequalities
    • Güler, O., A. J. Hoffman, U. Rothblum. 1995. Approximations to solutions to systems of linear inequalities. SIAM J. Matrix Anal. Appl. 16 688-696.
    • (1995) SIAM J. Matrix Anal. Appl. , vol.16 , pp. 688-696
    • Güler, O.1    Hoffman, A.J.2    Rothblum, U.3
  • 19
    • 31144464193 scopus 로고    scopus 로고
    • Reconciliation of various complexity and condition measures for linear programming problems and a generalization of Tardos' theorem
    • World Scientific, Singapore
    • Ho, J. C. K., L. Tunçel. 2002. Reconciliation of various complexity and condition measures for linear programming problems and a generalization of Tardos' theorem. Foundations Comput. Math.: Proc. Smalefest 2000, World Scientific, Singapore, 93-148.
    • (2002) Foundations Comput. Math.: Proc. Smalefest 2000 , pp. 93-148
    • Ho, J.C.K.1    Tunçel, L.2
  • 20
    • 0000699491 scopus 로고
    • On approximate solutions of systems of linear inequalities
    • Hoffman, A. J. 1952. On approximate solutions of systems of linear inequalities. J. Res. National Bureau Standards 49 263-265.
    • (1952) J. Res. National Bureau Standards , vol.49 , pp. 263-265
    • Hoffman, A.J.1
  • 21
    • 0004151494 scopus 로고    scopus 로고
    • Cambridge University Press, Cambridge, UK
    • Horn, R., C. Johnson. 1999. Matrix Analysis. Cambridge University Press, Cambridge, UK.
    • (1999) Matrix Analysis
    • Horn, R.1    Johnson, C.2
  • 22
    • 70349376403 scopus 로고    scopus 로고
    • Metric subregularity and the proximal point method
    • Leventhal, D. 2009. Metric subregularity and the proximal point method. J. Math. Anal. Appl. 360 681-688.
    • (2009) J. Math. Anal. Appl. , vol.360 , pp. 681-688
    • Leventhal, D.1
  • 23
    • 70349387487 scopus 로고    scopus 로고
    • Local convergence for alternating and averaged nonconvex projections
    • Lewis, A. S., D. R. Luke, J. Malick. 2009. Local convergence for alternating and averaged nonconvex projections. Foundations Comput. Math. 9 485-513.
    • (2009) Foundations Comput. Math. , vol.9 , pp. 485-513
    • Lewis, A.S.1    Luke, D.R.2    Malick, J.3
  • 24
    • 38248998880 scopus 로고
    • The sharp Lipschitz constants for feasible and optimal solutions of a perturbed linear program
    • Li, W. 1993. The sharp Lipschitz constants for feasible and optimal solutions of a perturbed linear program. Linear Algebra Appl. 187 15-40.
    • (1993) Linear Algebra Appl. , vol.187 , pp. 15-40
    • Li, W.1
  • 25
    • 0001518167 scopus 로고
    • On the convergence of the LMS algorithm with adaptive learning rate for linear feedback networks
    • Luo, Z.-Q. 1991. On the convergence of the LMS algorithm with adaptive learning rate for linear feedback networks. Neural Comput. 3 226-245.
    • (1991) Neural Comput. , vol.3 , pp. 226-245
    • Luo, Z.-Q.1
  • 26
    • 0026678659 scopus 로고
    • On the convergence of the coordinate descent method for convex differentiable minimization
    • Luo, Z.-Q., P. Tseng. 1992a. On the convergence of the coordinate descent method for convex differentiable minimization. J. Optim. Theory Appl. 72 7-35.
    • (1992) J. Optim. Theory Appl. , vol.72 , pp. 7-35
    • Luo, Z.-Q.1    Tseng, P.2
  • 27
    • 0026839499 scopus 로고
    • On the linear convergence of descent methods for convex essentially smooth minimization
    • Luo, Z.-Q., P. Tseng. 1992b. On the linear convergence of descent methods for convex essentially smooth minimization. SIAM J. Control Optim. 30 408-425.
    • (1992) SIAM J. Control Optim. , vol.30 , pp. 408-425
    • Luo, Z.-Q.1    Tseng, P.2
  • 28
    • 0003095772 scopus 로고
    • Error bound and convergence analysis of matrix splitting algorithms for the affine variational inequality problem
    • Luo, Z.-Q., P. Tseng. 1992c. Error bound and convergence analysis of matrix splitting algorithms for the affine variational inequality problem. SIAM J. Optim. 2 43-54.
    • (1992) SIAM J. Optim. , vol.2 , pp. 43-54
    • Luo, Z.-Q.1    Tseng, P.2
  • 29
    • 6244272639 scopus 로고
    • Error bound and reduced gradient projection algorithms for convex minimization over a polyhedral set
    • Luo, Z.-Q., P. Tseng. 1993a. Error bound and reduced gradient projection algorithms for convex minimization over a polyhedral set. SIAM J. Optim. 3 43-59.
    • (1993) SIAM J. Optim. , vol.3 , pp. 43-59
    • Luo, Z.-Q.1    Tseng, P.2
  • 30
    • 0001357819 scopus 로고
    • On the convergence rate of dual ascent methods for strictly convex minimization
    • Luo, Z.-Q., P. Tseng. 1993b. On the convergence rate of dual ascent methods for strictly convex minimization. Math. Oper. Res. 18 846-867.
    • (1993) Math. Oper. Res. , vol.18 , pp. 846-867
    • Luo, Z.-Q.1    Tseng, P.2
  • 31
    • 21344480786 scopus 로고
    • Error bounds and convergence analysis of feasible descent methods: A general approachs
    • Luo, Z.-Q., P. Tseng. 1993c. Error bounds and convergence analysis of feasible descent methods: A general approach. Ann. Oper. Res. 46 157-178.
    • (1993) Ann. Oper. Res. , vol.46 , pp. 157-178
    • Luo, Z.-Q.1    Tseng, P.2
  • 32
    • 84973041797 scopus 로고
    • Analysis of an approximate gradient projection method with applications to the back propagation algorithm
    • Luo, Z.-Q., P. Tseng. 1994. Analysis of an approximate gradient projection method with applications to the back propagation algorithm. Optim. Methods Software 4 85-101.
    • (1994) Optim. Methods Software , vol.4 , pp. 85-101
    • Luo, Z.-Q.1    Tseng, P.2
  • 33
    • 77956642913 scopus 로고    scopus 로고
    • CORE Discussion Paper 2010/2, Université Catholique de Louvain, Center for Operations Research and Economics, Louvain-la-Neuve, Belgium
    • Nesterov, Y. 2010. Efficiency of coordinate descent methods on huge-scale optimization problems. CORE Discussion Paper 2010/2, Université Catholique de Louvain, Center for Operations Research and Economics, Louvain-la-Neuve, Belgium.
    • (2010) Efficiency of Coordinate Descent Methods on Huge-Scale Optimization Problems
    • Nesterov, Y.1
  • 34
    • 14644391456 scopus 로고    scopus 로고
    • Hoffman's least error bounds for linear inequalities
    • Ng, K. F., X. Y. Zheng. 2004. Hoffman's least error bounds for linear inequalities. J. Global Optim. 30 391-403.
    • (2004) J. Global Optim. , vol.30 , pp. 391-403
    • Ng, K.F.1    Zheng, X.Y.2
  • 35
    • 0000150038 scopus 로고    scopus 로고
    • Error bounds in mathematical programming
    • Pang, J.-S. 1997. Error bounds in mathematical programming. Math. Programming 79 299-332.
    • (1997) Math. Programming , vol.79 , pp. 299-332
    • Pang, J.-S.1
  • 36
    • 0021201713 scopus 로고
    • Decomposition through formalization in a product space
    • Pierra, G. 1984. Decomposition through formalization in a product space. Math. Programming 28 96-115.
    • (1984) Math. Programming , vol.28 , pp. 96-115
    • Pierra, G.1
  • 37
    • 0242294539 scopus 로고    scopus 로고
    • Random algorithms for solving convex inequalities
    • D. Butnariu, Y. Censor, S. Reich, eds. Elsevier, Amsterdam
    • Polyak, B. T. 2001. Random algorithms for solving convex inequalities. D. Butnariu, Y. Censor, S. Reich, eds. Inherently Parallel Algorithms in Feasibility and Other Applications. Elsevier, Amsterdam.
    • (2001) Inherently Parallel Algorithms in Feasibility and Other Applications
    • Polyak, B.T.1
  • 39
    • 0028426653 scopus 로고
    • Perturbation theory for linear programming
    • Renegar, J. 1994. Perturbation theory for linear programming. Math. Programming 65 73-91.
    • (1994) Math. Programming , vol.65 , pp. 73-91
    • Renegar, J.1
  • 40
    • 0000208672 scopus 로고
    • Incorporating conditions measures into the complexity theory of linear programming
    • Renegar, J. 1995a. Incorporating conditions measures into the complexity theory of linear programming. SIAM J. Optim. 5 506-524.
    • (1995) SIAM J. Optim. , vol.5 , pp. 506-524
    • Renegar, J.1
  • 41
    • 58149364311 scopus 로고
    • Linear programming, complexity theory and elementary functional analysis
    • Renegar, J. 1995b. Linear programming, complexity theory and elementary functional analysis. Math. Programming 70 279-351.
    • (1995) Math. Programming , vol.70 , pp. 279-351
    • Renegar, J.1
  • 43
    • 67349206945 scopus 로고    scopus 로고
    • A randomized Kaczmarz algorithm with exponential convergence
    • Strohmer, T., R. Vershynin. 2009. A randomized Kaczmarz algorithm with exponential convergence. J. Fourier Anal. Appl. 15 262-278.
    • (2009) J. Fourier Anal. Appl. , vol.15 , pp. 262-278
    • Strohmer, T.1    Vershynin, R.2
  • 44
    • 0000488063 scopus 로고
    • On linear convergence of iterative methods for the variational inequality problem
    • Tseng, P. 1995. On linear convergence of iterative methods for the variational inequality problem. J. Comput. Appl. Math. 60 237-252.
    • (1995) J. Comput. Appl. Math. , vol.60 , pp. 237-252
    • Tseng, P.1
  • 45
    • 0034391880 scopus 로고    scopus 로고
    • Global error bounds for convex conic problems
    • Zhang, S. 2000. Global error bounds for convex conic problems. SIAM J. Optim. 10 836-851.
    • (2000) SIAM J. Optim. , vol.10 , pp. 836-851
    • Zhang, S.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.