메뉴 건너뛰기




Volumn 34, Issue 2, 1996, Pages 677-697

The efficiency of subgradient projection methods for convex optimization, Part II: Implementations and extensions

Author keywords

Convex programming; Linear inequalities; Nondifferentiable (nonsmooth) optimization; Parallel computing; Relaxation methods; Subgradient optimization; Successive projections

Indexed keywords

PARALLEL PROCESSING SYSTEMS; PROBLEM SOLVING;

EID: 0030104971     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0363012994261483     Document Type: Article
Times cited : (31)

References (56)
  • 1
    • 0000026159 scopus 로고
    • The relaxation method for linear inequalities
    • S. AGMON, The relaxation method for linear inequalities, Canad. J. Math., 6 (1954), pp. 382-392.
    • (1954) Canad. J. Math. , vol.6 , pp. 382-392
    • Agmon, S.1
  • 2
    • 0000198744 scopus 로고
    • Block-iterative projection methods for parallel computation of solutions to convex feasibility problems
    • R. AHARONI AND Y. CENSOR, Block-iterative projection methods for parallel computation of solutions to convex feasibility problems, Linear Algebra Appl., 120 (1989), pp. 165-175.
    • (1989) Linear Algebra Appl. , vol.120 , pp. 165-175
    • Aharoni, R.1    Censor, Y.2
  • 3
    • 0023346450 scopus 로고
    • A generalization of Polyak's convergence result for subgradient optimization
    • E. ALLEN, R. HELGASON, J. KENNINGTON, AND B. SHETTY, A generalization of Polyak's convergence result for subgradient optimization, Math. Programming, 37 (1987), pp. 309-317.
    • (1987) Math. Programming , vol.37 , pp. 309-317
    • Allen, E.1    Helgason, R.2    Kennington, J.3    Shetty, B.4
  • 4
    • 0018491453 scopus 로고
    • A survey of various tactics for generating Lagrangian multipliers in the context of Lagrangian duality
    • M. S. BAZARAA AND J. J. GOODE, A survey of various tactics for generating Lagrangian multipliers in the context of Lagrangian duality, European J. Oper. Res., 3 (1979), pp. 322-338.
    • (1979) European J. Oper. Res. , vol.3 , pp. 322-338
    • Bazaraa, M.S.1    Goode, J.J.2
  • 5
    • 0019599611 scopus 로고
    • On the choice of step size in subgradient optimization
    • M. S. BAZARAA AND H. D. SHERALI, On the choice of step size in subgradient optimization, European J. Oper. Res., 7 (1981), pp. 380-388.
    • (1981) European J. Oper. Res. , vol.7 , pp. 380-388
    • Bazaraa, M.S.1    Sherali, H.D.2
  • 7
    • 0019635237 scopus 로고
    • The ellipsoid method: A survey
    • R. B. BLAND, D. GOLDFARB, AND M. J. TODD, The ellipsoid method: A survey, Oper. Res., 29 (1981), pp. 1039-1091.
    • (1981) Oper. Res. , vol.29 , pp. 1039-1091
    • Bland, R.B.1    Goldfarb, D.2    Todd, M.J.3
  • 8
    • 70350657102 scopus 로고
    • Least squares methods
    • n (Part 1), P. G. Ciarlet and J. L. Lions, eds., North-Holland, Amsterdam
    • n (Part 1), Handbook of Numerical Analysis, vol. I, P. G. Ciarlet and J. L. Lions, eds., North-Holland, Amsterdam, 1990, pp. 465-652.
    • (1990) Handbook of Numerical Analysis , vol.1 , pp. 465-652
    • Björck, A.1
  • 10
  • 11
    • 0016601948 scopus 로고
    • On improving relaxation methods by modified gradient techniques
    • P. M. CAMERINI, L. FRATTA, AND F. MAFFIOLI, On improving relaxation methods by modified gradient techniques, Math. Programming Stud., 3 (1975), pp. 26-34.
    • (1975) Math. Programming Stud. , vol.3 , pp. 26-34
    • Camerini, P.M.1    Fratta, L.2    Maffioli, F.3
  • 12
    • 84966217951 scopus 로고
    • Reorthogonalization and stable algorithms for updating the Gram-Schmidt QR factorization
    • J. W. DANIEL, W. B. GRAGG, L. C. KAUFMAN, AND G. W. STEWART, Reorthogonalization and stable algorithms for updating the Gram-Schmidt QR factorization, Math. Comp., 30 (1976), pp. 772-795.
    • (1976) Math. Comp. , vol.30 , pp. 772-795
    • Daniel, J.W.1    Gragg, W.B.2    Kaufman, L.C.3    Stewart, G.W.4
  • 15
    • 0041413679 scopus 로고
    • Nondifferentiable optimization and the relaxation method
    • C. Lemaréchal and R. Mifflin, eds., Pergamon Press, Oxford
    • J.-L. GOFFIN, Nondifferentiable optimization and the relaxation method, in Nonsmooth Optimization, C. Lemaréchal and R. Mifflin, eds., Pergamon Press, Oxford, 1978, pp. 31-49.
    • (1978) Nonsmooth Optimization , pp. 31-49
    • Goffin, J.-L.1
  • 16
    • 0004886259 scopus 로고
    • Convergence results in a class of variable metric subgradient methods
    • L. Mangasarian, R. R. Meyer, and S. M. Robinson, eds., Academic Press, New York
    • Convergence results in a class of variable metric subgradient methods, in Non-linear Programming 4, O. L. Mangasarian, R. R. Meyer, and S. M. Robinson, eds., Academic Press, New York, 1981, pp. 283-326.
    • (1981) Non-linear Programming 4, O , pp. 283-326
  • 17
    • 0020132663 scopus 로고
    • Modifications and implementation of the ellipsoid algorithm for linear programming
    • D. GOLDFARB AND M. J. TODD, Modifications and implementation of the ellipsoid algorithm for linear programming, Math. Programming, 23 (1982), pp. 1-19.
    • (1982) Math. Programming , vol.23 , pp. 1-19
    • Goldfarb, D.1    Todd, M.J.2
  • 20
    • 0016025814 scopus 로고
    • Validation of subgradient optimization
    • M. HELD, P. WOLFE, AND H. P. CROWDER, Validation of subgradient optimization, Math. Programming, 6 (1974), pp. 62-88.
    • (1974) Math. Programming , vol.6 , pp. 62-88
    • Held, M.1    Wolfe, P.2    Crowder, H.P.3
  • 21
    • 34249920356 scopus 로고
    • On the convergence of Han's method for convex programming with quadratic objective
    • A. N. IUSEM AND A. R. DE PIERRO, On the convergence of Han's method for convex programming with quadratic objective, Math. Programming, 52 (1991), pp. 265-284.
    • (1991) Math. Programming , vol.52 , pp. 265-284
    • Iusem, A.N.1    De Pierro, A.R.2
  • 22
    • 0025745988 scopus 로고
    • Variable target value subgradient method
    • S. KIM, H. AHN, AND S.-C. CHO, Variable target value subgradient method, Math. Programming, 49 (1991), pp. 359-369.
    • (1991) Math. Programming , vol.49 , pp. 359-369
    • Kim, S.1    Ahn, H.2    Cho, S.-C.3
  • 23
    • 84989682249 scopus 로고
    • Convergence properties of the modified subgradient method of Camerini et al
    • S. KIM AND H. AHN, Convergence properties of the modified subgradient method of Camerini et al., Naval Res. Logist. Quart., 37 (1990), pp. 961-966.
    • (1990) Naval Res. Logist. Quart. , vol.37 , pp. 961-966
    • Kim, S.1    Ahn, H.2
  • 24
    • 0001570373 scopus 로고
    • Polyak's subgradient method with simplified projection for nondifferentiable optimization with linear constraints
    • S. KIM AND B.-S. UM, Polyak's subgradient method with simplified projection for nondifferentiable optimization with linear constraints, Optimization, 20 (1989), pp. 451-456.
    • (1989) Optimization , vol.20 , pp. 451-456
    • Kim, S.1    Um, B.-S.2
  • 25
    • 38249002574 scopus 로고
    • An improved subgradient method for constrained nondifferentiable optimization
    • An improved subgradient method for constrained nondifferentiable optimization, Oper. Res. Lett., 14 (1993), pp. 61-64.
    • (1993) Oper. Res. Lett. , vol.14 , pp. 61-64
  • 26
    • 0003970531 scopus 로고
    • Methods of Descent for Nondifferentiable Optimization
    • Springer-Verlag, Berlin
    • K. C. KIWIEL, Methods of Descent for Nondifferentiable Optimization, Lecture Notes in Mathematics 1133, Springer-Verlag, Berlin, 1985.
    • (1985) Lecture Notes in Mathematics 1133
    • Kiwiel, K.C.1
  • 27
    • 0004835363 scopus 로고
    • A survey of bundle methods for nondifferentiable optimization
    • M. Iri and K. Tanabe, eds., KTT/Kluwer, Tokyo
    • A survey of bundle methods for nondifferentiable optimization, in Mathematical Programming: Recent Developments and Applications, M. Iri and K. Tanabe, eds., KTT/Kluwer, Tokyo, 1989, pp. 263-282.
    • (1989) Mathematical Programming: Recent Developments and Applications , pp. 263-282
  • 28
    • 0025208765 scopus 로고
    • Proximity control in bundle methods for convex nondifferentiable minimization
    • Proximity control in bundle methods for convex nondifferentiable minimization, Math. Programming, 46 (1990), pp. 105-122.
    • (1990) Math. Programming , vol.46 , pp. 105-122
  • 30
    • 0345295872 scopus 로고
    • Block-iterative surrogate projection methods for convex feasibility problems
    • Block-iterative surrogate projection methods for convex feasibility problems, Linear Algebra Appl., 215 (1995), pp. 225-259.
    • (1995) Linear Algebra Appl. , vol.215 , pp. 225-259
  • 31
    • 0030110299 scopus 로고    scopus 로고
    • The efficiency of subgradient projection methods for convex optimization, part I: General level methods
    • The efficiency of subgradient projection methods for convex optimization, part I: General level methods, SIAM J. Control Optim., 34 (1996), pp. 660-676.
    • (1996) SIAM J. Control Optim. , vol.34 , pp. 660-676
  • 32
    • 0023311153 scopus 로고
    • Two-direction subgradient method for non-differentiate optimization problems
    • S. KIM, S. KOH, AND H. AHN, Two-direction subgradient method for non-differentiate optimization problems, Oper. Res. Lett., 6 (1987), pp. 43-46.
    • (1987) Oper. Res. Lett. , vol.6 , pp. 43-46
    • Kim, S.1    Koh, S.2    Ahn, H.3
  • 33
    • 0343855915 scopus 로고
    • Convex optimization with prescribed accuracy
    • In Russian
    • A. N. KULIKOV AND V. R. FAZILOV, Convex optimization with prescribed accuracy, Zh. Vychisl. Mat. i Mat. Fiz., 30 (1990), pp. 663-671. (In Russian.)
    • (1990) Zh. Vychisl. Mat. i Mat. Fiz. , vol.30 , pp. 663-671
    • Kulikov, A.N.1    Fazilov, V.R.2
  • 34
    • 77957820648 scopus 로고
    • Nondifferentiable optimization
    • G. L. Nemhauser, A. H. G. Rinnooy-Kan, and M. J. Todd, eds., North-Holland, Amsterdam
    • C. LEMARÉCHAL, Nondifferentiable optimization, in Optimization, Handbooks in Operations Research and Management Science, vol. 1, G. L. Nemhauser, A. H. G. Rinnooy-Kan, and M. J. Todd, eds., North-Holland, Amsterdam, 1989, pp. 529-572.
    • (1989) Optimization, Handbooks in Operations Research and Management Science , vol.1 , pp. 529-572
    • Lemaréchal, C.1
  • 36
    • 4243131247 scopus 로고
    • A parallel projection method for solving generalized linear least-squares problems
    • G. LOU AND S.-P. HAN, A parallel projection method for solving generalized linear least-squares problems, Numer. Math., 53 (1988), pp. 255-264.
    • (1988) Numer. Math. , vol.53 , pp. 255-264
    • Lou, G.1    Han, S.-P.2
  • 37
    • 0000026160 scopus 로고
    • The relaxation method for linear inequalities
    • T. MOTZKIN AND I. J. SCHOENBERG, The relaxation method for linear inequalities, Canad. J. Math., 6 (1954), pp. 393-404.
    • (1954) Canad. J. Math. , vol.6 , pp. 393-404
    • Motzkin, T.1    Schoenberg, I.J.2
  • 38
    • 0026822012 scopus 로고
    • Surrogate methods for linear inequalities
    • S. O. OKO, Surrogate methods for linear inequalities, J. Optim. Theory Appl., 72 (1992), pp. 247-268.
    • (1992) J. Optim. Theory Appl. , vol.72 , pp. 247-268
    • Oko, S.O.1
  • 39
    • 49849127729 scopus 로고
    • Minimization of unsmooth functional
    • In Russian
    • B. T. POLYAK, Minimization of unsmooth functional, Zh. Vychisl. Mat. i Mat. Fiz., 9 (1969), pp. 509-521. (In Russian.) English transl. in U.S.S.R. Comput. Math. and Math. Phys. 9 (1969), pp. 14-29.
    • (1969) Zh. Vychisl. Mat. i Mat. Fiz. , vol.9 , pp. 509-521
    • Polyak, B.T.1
  • 40
    • 49849127729 scopus 로고
    • English transl.
    • B. T. POLYAK, Minimization of unsmooth functional, Zh. Vychisl. Mat. i Mat. Fiz., 9 (1969), pp. 509-521. (In Russian.) English transl. in U.S.S.R. Comput. Math. and Math. Phys. 9 (1969), pp. 14-29.
    • (1969) U.S.S.R. Comput. Math. and Math. Phys. , vol.9 , pp. 14-29
  • 41
    • 0023234881 scopus 로고
    • Computational evaluation of two subgradient search methods
    • S. SARIN AND M. H. KARWAN, Computational evaluation of two subgradient search methods, Comput. Oper. Res., 14 (1987), pp. 241-247.
    • (1987) Comput. Oper. Res. , vol.14 , pp. 241-247
    • Sarin, S.1    Karwan, M.H.2
  • 42
    • 0000091855 scopus 로고
    • A version of the bundle idea for minimizing a nonsmooth function: Conceptual idea, convergence analysis, numerical results
    • H. SCHRAMM AND J. ZOWE, A version of the bundle idea for minimizing a nonsmooth function: Conceptual idea, convergence analysis, numerical results, SIAM J. Optim., 2 (1992), pp. 121-152.
    • (1992) SIAM J. Optim. , vol.2 , pp. 121-152
    • Schramm, H.1    Zowe, J.2
  • 43
    • 0022751757 scopus 로고
    • A class of convergent primal-dual subgradient algorithms for decomposable convex programs
    • S. SEN AND H. D. SHERALI, A class of convergent primal-dual subgradient algorithms for decomposable convex programs, Math. Programming, 35 (1986), pp. 279-297.
    • (1986) Math. Programming , vol.35 , pp. 279-297
    • Sen, S.1    Sherali, H.D.2
  • 44
    • 4243119331 scopus 로고
    • On a modification of a class of algorithms for mathematical programming
    • In Russian
    • M. B. SHCHEPAKIN, On a modification of a class of algorithms for mathematical programming, Zh. Vychisl. Mat. i Mat. Fiz., 19 (1979), pp. 1387-1395. (In Russian.)
    • (1979) Zh. Vychisl. Mat. i Mat. Fiz. , vol.19 , pp. 1387-1395
    • Shchepakin, M.B.1
  • 45
    • 0040761628 scopus 로고
    • On the method of orthogonal descent
    • In Russian
    • On the method of orthogonal descent, Kibernetika, (1987), pp. 58-62. (In Russian.)
    • (1987) Kibernetika , pp. 58-62
  • 46
    • 85033833607 scopus 로고
    • An algorithm of orthogonal descent for searching for the zero of a convex function, identification of unsolvability of the problem and the questions of the speed of convergence
    • In Russian
    • An algorithm of orthogonal descent for searching for the zero of a convex function, identification of unsolvability of the problem and the questions of the speed of convergence, Kibernetika, (1992), pp. 87-96. (In Russian.)
    • (1992) Kibernetika , pp. 87-96
  • 47
    • 0024014124 scopus 로고
    • Dual formulations and subgradient optimization strategies for linear programming relaxations of mixed-integer programs
    • H. D. SHERALI AND D, C. MYERS, Dual formulations and subgradient optimization strategies for linear programming relaxations of mixed-integer programs, Discrete Appl. Math., 20 (1988), pp. 51-68.
    • (1988) Discrete Appl. Math. , vol.20 , pp. 51-68
    • Sherali, H.D.1    Myers, D.C.2
  • 48
  • 49
    • 4243070520 scopus 로고
    • A primal-dual conjugate subgradient algorithm for specially structured linear and convex programming problems
    • H. D. SHERALI AND O. ULULAR, A primal-dual conjugate subgradient algorithm for specially structured linear and convex programming problems, Appl. Math. Optim., 20 (1989), pp. 193-221.
    • (1989) Appl. Math. Optim. , vol.20 , pp. 193-221
    • Sherali, H.D.1    Ulular, O.2
  • 50
    • 0023366344 scopus 로고
    • A new surrogate-dual multiplier search procedure
    • S. SARIN, M. H. KARWAN, AND R. L. RADIN, A new surrogate-dual multiplier search procedure, Naval Res. Logist. Quart., 34 (1987), pp. 431-450.
    • (1987) Naval Res. Logist. Quart. , vol.34 , pp. 431-450
    • Sarin, S.1    Karwan, M.H.2    Radin, R.L.3
  • 51
    • 2942677299 scopus 로고
    • A projection method for least-squares solutions to overdetermined systems of linear inequalities
    • J. E. SPINGARN, A projection method for least-squares solutions to overdetermined systems of linear inequalities, Linear Algebra Appl., 86 (1987), pp. 211-236.
    • (1987) Linear Algebra Appl. , vol.86 , pp. 211-236
    • Spingarn, J.E.1
  • 52
    • 0020087678 scopus 로고
    • On relaxation methods for systems of linear inequalities
    • J. TELGEN, On relaxation methods for systems of linear inequalities, European J. Oper. Res., 9 (1982), pp. 184-189.
    • (1982) European J. Oper. Res. , vol.9 , pp. 184-189
    • Telgen, J.1
  • 54
    • 0025233021 scopus 로고
    • Dual ascent methods for problems with strictly convex costs and linear constraints: A unified approach
    • P. TSENG, Dual ascent methods for problems with strictly convex costs and linear constraints: A unified approach, SIAM J. Control Optim., 28 (1990), pp. 214-242.
    • (1990) SIAM J. Control Optim. , vol.28 , pp. 214-242
    • Tseng, P.1
  • 55
    • 0000626995 scopus 로고
    • Further applications of a splitting algorithm to decomposition in variational inequalities and convex programming
    • further applications of a splitting algorithm to decomposition in variational inequalities and convex programming, Math. Programming, 48 (1990), pp. 249-263.
    • (1990) Math. Programming , vol.48 , pp. 249-263
  • 56
    • 0026634445 scopus 로고
    • New iterative methods for linear inequalities
    • K. YANG AND K. G. MURTY, New iterative methods for linear inequalities, J. Optim. Theory Appl., 72 (1992), pp. 163-185.
    • (1992) J. Optim. Theory Appl. , vol.72 , pp. 163-185
    • Yang, K.1    Murty, K.G.2


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.