메뉴 건너뛰기




Volumn 133, Issue 1-2, 2012, Pages 365-397

An optimal method for stochastic composite optimization

Author keywords

Complexity; Convex optimization; Large deviation; Optimal method; Quadratic penalty method; Stochastic approximation; Stochastic programming

Indexed keywords

COMPLEXITY; LARGE DEVIATIONS; OPTIMAL METHODS; PENALTY METHODS; STOCHASTIC APPROXIMATIONS;

EID: 84862273593     PISSN: 00255610     EISSN: 14364646     Source Type: Journal    
DOI: 10.1007/s10107-010-0434-y     Document Type: Article
Times cited : (547)

References (51)
  • 1
    • 33747151859 scopus 로고    scopus 로고
    • Interior gradient and proximal methods for convex and conic optimization
    • 2197553 1113.90118 10.1137/S1052623403427823
    • A. Auslender M. Teboulle 2006 Interior gradient and proximal methods for convex and conic optimization SIAM J. Optim. 16 697 725 2197553 1113.90118 10.1137/S1052623403427823
    • (2006) SIAM J. Optim. , vol.16 , pp. 697-725
    • Auslender, A.1    Teboulle, M.2
  • 2
    • 1942509685 scopus 로고    scopus 로고
    • Bregman monotone optimization algorithms
    • 1982285 1049.90053 10.1137/S0363012902407120
    • H.H. Bauschke J.M. Borwein P.L. Combettes 2003 Bregman monotone optimization algorithms SIAM J. Control Optim. 42 596 636 1982285 1049.90053 10.1137/S0363012902407120
    • (2003) SIAM J. Control Optim. , vol.42 , pp. 596-636
    • Bauschke, H.H.1    Borwein, J.M.2    Combettes, P.L.3
  • 4
    • 17444361978 scopus 로고    scopus 로고
    • Non-euclidean restricted memory level method for large-scale convex optimization
    • 2136222 1066.90079 10.1007/s10107-004-0553-4
    • A. Ben-Tal A. Nemirovski 2005 Non-euclidean restricted memory level method for large-scale convex optimization Math. Program. 102 407 456 2136222 1066.90079 10.1007/s10107-004-0553-4
    • (2005) Math. Program. , vol.102 , pp. 407-456
    • Ben-Tal, A.1    Nemirovski, A.2
  • 6
  • 7
    • 49949144765 scopus 로고
    • The relaxation method of finding the common point convex sets and its application to the solution of problems in convex programming
    • 10.1016/0041-5553(67)90040-7
    • L.M. Bregman 1967 The relaxation method of finding the common point convex sets and its application to the solution of problems in convex programming USSR Comput. Math. Phys. 7 200 217 10.1016/0041-5553(67)90040-7
    • (1967) USSR Comput. Math. Phys. , vol.7 , pp. 200-217
    • Bregman, L.M.1
  • 8
    • 57249107300 scopus 로고    scopus 로고
    • Smooth optimization with approximate gradient
    • 2460737 1180.90378 10.1137/060676386
    • A. d'Aspremont 2008 Smooth optimization with approximate gradient SIAM J. Optim. 19 1171 1183 2460737 1180.90378 10.1137/060676386
    • (2008) SIAM J. Optim. , vol.19 , pp. 1171-1183
    • D'Aspremont, A.1
  • 9
    • 61849097176 scopus 로고    scopus 로고
    • First-order methods for sparse covariance selection
    • 2399568 1156.90423 10.1137/060670985
    • A. d'Aspremont O. Banerjee L. El Ghaoue 2008 First-order methods for sparse covariance selection SIAM J. Matrix Anal. Appl. 30 56 66 2399568 1156.90423 10.1137/060670985
    • (2008) SIAM J. Matrix Anal. Appl. , vol.30 , pp. 56-66
    • D'Aspremont, A.1    Banerjee, O.2    El Ghaoue, L.3
  • 10
    • 0020498889 scopus 로고
    • Stochastic quasigradient methods and their application to system optimization
    • 703846 0512.90079 10.1080/17442508308833246
    • Y. Ermoliev 1983 Stochastic quasigradient methods and their application to system optimization Stochastics 9 1 36 703846 0512.90079 10.1080/ 17442508308833246
    • (1983) Stochastics , vol.9 , pp. 1-36
    • Ermoliev, Y.1
  • 11
    • 30244575435 scopus 로고
    • Nonstationary stochastic programming problems
    • A. Gaivoronski 1978 Nonstationary stochastic programming problems Kybernetika 4 89 92
    • (1978) Kybernetika , vol.4 , pp. 89-92
    • Gaivoronski, A.1
  • 12
    • 31344435933 scopus 로고    scopus 로고
    • Recursive aggregation of estimators via the mirror descent algorithm with average
    • 10.1007/s11122-006-0005-2
    • A. Juditsky A. Nazin A.B. Tsybakov N. Vayatis 2005 Recursive aggregation of estimators via the mirror descent algorithm with average Probl. Inf. Transm. 41 n.4 10.1007/s11122-006-0005-2
    • (2005) Probl. Inf. Transm. , vol.41 , pp. 4
    • Juditsky, A.1    Nazin, A.2    Tsybakov, A.B.3    Vayatis, N.4
  • 14
    • 54349121305 scopus 로고    scopus 로고
    • Learning by mirror averaging
    • 2458184 05368488 10.1214/07-AOS546
    • A. Juditsky P. Rigollet A.B. Tsybakov 2008 Learning by mirror averaging Ann. Stat. 36 2183 2206 2458184 05368488 10.1214/07-AOS546
    • (2008) Ann. Stat. , vol.36 , pp. 2183-2206
    • Juditsky, A.1    Rigollet, P.2    Tsybakov, A.B.3
  • 15
    • 0031190440 scopus 로고    scopus 로고
    • Proximal minimization methods with generalized bregman functions
    • 1453294 0890.65061 10.1137/S0363012995281742
    • K.C. Kiwiel 1997 Proximal minimization methods with generalized bregman functions SIAM J. Control Optim. 35 1142 1168 1453294 0890.65061 10.1137/S0363012995281742
    • (1997) SIAM J. Control Optim. , vol.35 , pp. 1142-1168
    • Kiwiel, K.C.1
  • 16
    • 0036013019 scopus 로고    scopus 로고
    • The sample average approximation method for stochastic discrete optimization
    • 1885572 0991.90090 10.1137/S1052623499363220
    • A.J. Kleywegt A. Shapiro T. Homem de Mello 2001 The sample average approximation method for stochastic discrete optimization SIAM J. Optim. 12 479 502 1885572 0991.90090 10.1137/S1052623499363220
    • (2001) SIAM J. Optim. , vol.12 , pp. 479-502
    • Kleywegt, A.J.1    Shapiro, A.2    Homem De Mello, T.3
  • 18
    • 78649411310 scopus 로고    scopus 로고
    • Primal-dual first-order methods with O(1/ε)iteration-complexity for cone programming
    • to appear
    • Lan, G,; Lu, Z,; Monteiro, R.D.C.: Primal-dual first-order methods with O(1/ε)iteration-complexity for cone programming. Math. Program. (2009, to appear)
    • (2009) Math. Program.
    • Lan, G.1    Lu, Z.2    Monteiro, R.D.C.3
  • 21
    • 78649429189 scopus 로고    scopus 로고
    • Validation analysis of robust stochastic approximation method
    • submitted to
    • Lan, G,; Nemirovski, A,; Shapiro, A.: Validation analysis of robust stochastic approximation method. submitted to Math. Program. (2008). http://www.optimization-online.org
    • (2008) Math. Program.
    • Lan, G.1    Nemirovski, A.2    Shapiro, A.3
  • 23
    • 33644697284 scopus 로고    scopus 로고
    • The empirical behavior of sampling methods for stochastic programming
    • 2222918 1122.90391 10.1007/s10479-006-6169-8
    • J. Linderoth A. Shapiro S. Wright 2006 The empirical behavior of sampling methods for stochastic programming Ann. Oper. Res. 142 215 241 2222918 1122.90391 10.1007/s10479-006-6169-8
    • (2006) Ann. Oper. Res. , vol.142 , pp. 215-241
    • Linderoth, J.1    Shapiro, A.2    Wright, S.3
  • 24
    • 70450200096 scopus 로고    scopus 로고
    • Smooth optimization approach for sparse covariance selection
    • 1179.90257 10.1137/070695915
    • Z. Lu 2009 Smooth optimization approach for sparse covariance selection SIAM J. Optim. 19 1807 1827 1179.90257 10.1137/070695915
    • (2009) SIAM J. Optim. , vol.19 , pp. 1807-1827
    • Lu, Z.1
  • 26
    • 33846655826 scopus 로고    scopus 로고
    • Large-scale semidefinite programming via saddle point mirror-prox algorithm
    • 2295141 1148.90009 10.1007/s10107-006-0031-2
    • Z. Lu A. Nemirovski R.D.C. Monteiro 2007 Large-scale semidefinite programming via saddle point mirror-prox algorithm Math. Program. 109 211 237 2295141 1148.90009 10.1007/s10107-006-0031-2
    • (2007) Math. Program. , vol.109 , pp. 211-237
    • Lu, Z.1    Nemirovski, A.2    Monteiro, R.D.C.3
  • 27
    • 0032632474 scopus 로고    scopus 로고
    • Monte carlo bounding techniques for determining solution quality in stochastic programs
    • 1683170 0956.90022 10.1016/S0167-6377(98)00054-6
    • W.K. Mak D.P. Morton R.K. Wood 1999 Monte carlo bounding techniques for determining solution quality in stochastic programs Oper. Res. Lett. 24 47 56 1683170 0956.90022 10.1016/S0167-6377(98)00054-6
    • (1999) Oper. Res. Lett. , vol.24 , pp. 47-56
    • Mak, W.K.1    Morton, D.P.2    Wood, R.K.3
  • 29
    • 14944353419 scopus 로고    scopus 로고
    • Prox-method with rate of convergence o(1/t) for variational inequalities with lipschitz continuous monotone operators and smooth convex-concave saddle point problems
    • 2112984 1106.90059 10.1137/S1052623403425629
    • A. Nemirovski 2004 Prox-method with rate of convergence o(1/t) for variational inequalities with lipschitz continuous monotone operators and smooth convex-concave saddle point problems SIAM J. Optim. 15 229 251 2112984 1106.90059 10.1137/S1052623403425629
    • (2004) SIAM J. Optim. , vol.15 , pp. 229-251
    • Nemirovski, A.1
  • 30
    • 70450197241 scopus 로고    scopus 로고
    • Robust stochastic approximation approach to stochastic programming
    • 2486041 1189.90109 10.1137/070704277
    • A. Nemirovski A. Juditsky G. Lan A. Shapiro 2009 Robust stochastic approximation approach to stochastic programming SIAM J. Optim. 19 1574 1609 2486041 1189.90109 10.1137/070704277
    • (2009) SIAM J. Optim. , vol.19 , pp. 1574-1609
    • Nemirovski, A.1    Juditsky, A.2    Lan, G.3    Shapiro, A.4
  • 32
    • 17444382102 scopus 로고
    • 2)
    • Translated as Soviet Math. Docl
    • 2). Doklady AN SSSR 269:543-547 (1983). Translated as Soviet Math. Docl
    • (1983) Doklady AN SSSR , vol.269 , pp. 543-547
    • Nesterov, Y.E.1
  • 34
    • 17444406259 scopus 로고    scopus 로고
    • Smooth minimization of nonsmooth functions
    • 2166537 1079.90102 10.1007/s10107-004-0552-5
    • Y.E. Nesterov 2005 Smooth minimization of nonsmooth functions Math. Program. 103 127 152 2166537 1079.90102 10.1007/s10107-004-0552-5
    • (2005) Math. Program. , vol.103 , pp. 127-152
    • Nesterov, Y.E.1
  • 35
    • 65249121279 scopus 로고    scopus 로고
    • Primal-dual subgradient methods for convex problems
    • 2496434 10.1007/s10107-007-0149-x
    • Y.E. Nesterov 2006 Primal-dual subgradient methods for convex problems Math. Program. 120 221 259 2496434 10.1007/s10107-007-0149-x
    • (2006) Math. Program. , vol.120 , pp. 221-259
    • Nesterov, Y.E.1
  • 36
    • 67651063011 scopus 로고    scopus 로고
    • Technical report, Center for Operations Research and Econometrics (CORE), Catholic University of Louvain (September)
    • Nesterov, Y.E.: Gradient Methods for Minimizing Composite Objective Functions. Technical report, Center for Operations Research and Econometrics (CORE), Catholic University of Louvain (2007, September)
    • (2007) Gradient Methods for Minimizing Composite Objective Functions
    • Nesterov, Y.E.1
  • 37
    • 33947732413 scopus 로고    scopus 로고
    • Smoothing technique and its applications in semidefinite optimization
    • 2313779 1126.90058 10.1007/s10107-006-0001-8
    • Y.E. Nesterov 2007 Smoothing technique and its applications in semidefinite optimization Math. Program. 110 245 259 2313779 1126.90058 10.1007/s10107-006-0001-8
    • (2007) Math. Program. , vol.110 , pp. 245-259
    • Nesterov, Y.E.1
  • 38
    • 84862283032 scopus 로고    scopus 로고
    • Nash equilibria computation via smoothing techniques
    • J. Peña 2008 Nash equilibria computation via smoothing techniques Optima 78 12 13
    • (2008) Optima , vol.78 , pp. 12-13
    • Peña, J.1
  • 40
    • 0000828406 scopus 로고
    • New stochastic approximation type procedures
    • 1071220
    • B.T. Polyak 1990 New stochastic approximation type procedures Automat. i Telemekh 7 98 107 1071220
    • (1990) Automat. i Telemekh , vol.7 , pp. 98-107
    • Polyak, B.T.1
  • 41
    • 0026899240 scopus 로고
    • Acceleration of stochastic approximation by averaging
    • 1167814 0762.62022 10.1137/0330046
    • B.T. Polyak A.B. Juditsky 1992 Acceleration of stochastic approximation by averaging SIAM J. Control Optim. 30 838 855 1167814 0762.62022 10.1137/0330046
    • (1992) SIAM J. Control Optim. , vol.30 , pp. 838-855
    • Polyak, B.T.1    Juditsky, A.B.2
  • 42
    • 0000016172 scopus 로고
    • A stochastic approximation method
    • 42668 0054.05901 10.1214/aoms/1177729586
    • H. Robbins S. Monro 1951 A stochastic approximation method Ann. Math. Stat. 22 400 407 42668 0054.05901 10.1214/aoms/1177729586
    • (1951) Ann. Math. Stat. , vol.22 , pp. 400-407
    • Robbins, H.1    Monro, S.2
  • 43
    • 0004267646 scopus 로고
    • Princeton University Press Princeton 0193.18401
    • Rockafellar R.T.: Convex Analysis. Princeton University Press, Princeton (1970)
    • (1970) Convex Analysis
    • Rockafellar, R.T.1
  • 44
    • 0022693667 scopus 로고
    • A method of aggregate stochastic subgradients with on-line stepsize rules for convex stochastic programming problems
    • 0597.90064 10.1007/BFb0121128
    • A. Ruszczyński W. Sysk 1986 A method of aggregate stochastic subgradients with on-line stepsize rules for convex stochastic programming problems Math. Program. Study 28 113 131 0597.90064 10.1007/BFb0121128
    • (1986) Math. Program. Study , vol.28 , pp. 113-131
    • Ruszczyński, A.1    Sysk, W.2
  • 45
    • 28044445224 scopus 로고    scopus 로고
    • Monte carlo sampling methods
    • A. Ruszczyński A. Shapiro (eds). North-Holland Amsterdam
    • Shapiro A.: Monte carlo sampling methods. In: Ruszczyński, A,; Shapiro, A. (eds) Stochastic Programming, North-Holland, Amsterdam (2003)
    • (2003) Stochastic Programming
    • Shapiro, A.1
  • 46
    • 33750315463 scopus 로고    scopus 로고
    • On complexity of stochastic programming problems
    • V. Jeyakumar A.M. Rubinov (eds). Springer Berlin
    • Shapiro A,; Nemirovski A.: On complexity of stochastic programming problems. In: Jeyakumar, V,; Rubinov, A.M. (eds) Continuous Optimization: Current Trends and Applications, pp. 111-144. Springer, Berlin (2005)
    • (2005) Continuous Optimization: Current Trends and Applications , pp. 111-144
    • Shapiro, A.1    Nemirovski, A.2
  • 48
    • 0000649228 scopus 로고
    • The existence of probability measures with given marginals
    • 177430 10.1214/aoms/1177700153
    • V. Strassen 1965 The existence of probability measures with given marginals Ann. Math. Stat. 30 423 439 177430 10.1214/aoms/1177700153
    • (1965) Ann. Math. Stat. , vol.30 , pp. 423-439
    • Strassen, V.1
  • 49
    • 0031285685 scopus 로고    scopus 로고
    • Convergence of proximal-like algorithms
    • 1479615 0890.90151 10.1137/S1052623495292130
    • M. Teboulle 1997 Convergence of proximal-like algorithms SIAM J. Optim. 7 1069 1083 1479615 0890.90151 10.1137/S1052623495292130
    • (1997) SIAM J. Optim. , vol.7 , pp. 1069-1083
    • Teboulle, M.1
  • 51
    • 0037322974 scopus 로고    scopus 로고
    • The sample average approximation method applied to stochastic routing problems: A computational study
    • 1969156 1094.90029 10.1023/A:1021814225969
    • B. Verweij S. Ahmed J.A. Kleywegt G. Nemhauser A. Shapiro 2003 The sample average approximation method applied to stochastic routing problems: a computational study Comput. Optim. Appl. 24 289 333 1969156 1094.90029 10.1023/A:1021814225969
    • (2003) Comput. Optim. Appl. , vol.24 , pp. 289-333
    • Verweij, B.1    Ahmed, S.2    Kleywegt, J.A.3    Nemhauser, G.4    Shapiro, A.5


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.