메뉴 건너뛰기




Volumn 37, Issue 1, 2007, Pages 1-34

Inner solvers for interior point methods for large scale nonlinear programming

Author keywords

Interior point method; Large scale nonlinear programming; Method of multipliers; Preconditioned conjugate gradient method

Indexed keywords

CODES (SYMBOLS); CONJUGATE GRADIENT METHOD; CONVERGENCE OF NUMERICAL METHODS; ITERATIVE METHODS; LARGE SCALE SYSTEMS; PROBLEM SOLVING;

EID: 34047126461     PISSN: 09266003     EISSN: 15732894     Source Type: Journal    
DOI: 10.1007/s10589-007-9012-5     Document Type: Article
Times cited : (6)

References (58)
  • 1
    • 0033296703 scopus 로고    scopus 로고
    • Regularized symmetric indefinite systems in interior point methods for linear and quadratic optimization
    • Altman, A., Godson, J.: Regularized symmetric indefinite systems in interior point methods for linear and quadratic optimization. Optim. Methods Softw. 11-12, 275-302 (1999)
    • (1999) Optim. Methods Softw , vol.11-12 , pp. 275-302
    • Altman, A.1    Godson, J.2
  • 3
    • 9744233649 scopus 로고    scopus 로고
    • Numerical comparisons of path-following strategies for a primal-dual interior-point method for nonlinear programming
    • Argáez, M., Tapia, R., Velázquez, L.: Numerical comparisons of path-following strategies for a primal-dual interior-point method for nonlinear programming. J. Optim. Theory Appl. 114, 255-272 (2002)
    • (2002) J. Optim. Theory Appl , vol.114 , pp. 255-272
    • Argáez, M.1    Tapia, R.2    Velázquez, L.3
  • 4
    • 3543086890 scopus 로고    scopus 로고
    • Preconditioning indefinite systems in interior point methods for optimization
    • Bergamaschi, L., Gondzio, J., Zilli, G.: Preconditioning indefinite systems in interior point methods for optimization. Comput. Optim. Appl. 28, 149-171 (2004)
    • (2004) Comput. Optim. Appl , vol.28 , pp. 149-171
    • Bergamaschi, L.1    Gondzio, J.2    Zilli, G.3
  • 5
    • 27744552861 scopus 로고    scopus 로고
    • A nonmonotone inexact Newton method
    • Bonettini, S.: A nonmonotone inexact Newton method. Optim. Methods Softw. 20, 475-491 (2005)
    • (2005) Optim. Methods Softw , vol.20 , pp. 475-491
    • Bonettini, S.1
  • 6
    • 25644459151 scopus 로고    scopus 로고
    • An inexact Newton method combined with Hestenes multipliers' scheme for the solution of Karush-Kuhn-Tucker systems
    • Bonettini, S., Galligani, E., Ruggiero, V.: An inexact Newton method combined with Hestenes multipliers' scheme for the solution of Karush-Kuhn-Tucker systems. Appl. Math. Comput. 168, 651-676 (2005)
    • (2005) Appl. Math. Comput , vol.168 , pp. 651-676
    • Bonettini, S.1    Galligani, E.2    Ruggiero, V.3
  • 7
    • 34547570237 scopus 로고    scopus 로고
    • Some iterative methods for the solution of a symmetric indefinite KKT system
    • to appear
    • Bonettini, S., Ruggiero, V.: Some iterative methods for the solution of a symmetric indefinite KKT system. Comput. Optim. Appl. (2006, to appear)
    • (2006) Comput. Optim. Appl
    • Bonettini, S.1    Ruggiero, V.2
  • 8
    • 0015200767 scopus 로고
    • Direct methods for solving symmetric indefinite systems of linear equations
    • Bunch, J.R., Parlert, B.N.: Direct methods for solving symmetric indefinite systems of linear equations. SIAM J. Numer. Anal. 8, 639-655 (1971)
    • (1971) SIAM J. Numer. Anal , vol.8 , pp. 639-655
    • Bunch, J.R.1    Parlert, B.N.2
  • 9
    • 0001130234 scopus 로고    scopus 로고
    • A trust region method based on interior-point techniques for nonlinear programming
    • Byrd, R.H., Gilbert, J.C., Nocedal, J.: A trust region method based on interior-point techniques for nonlinear programming. Math. Program. 89, 149-185 (2000)
    • (2000) Math. Program , vol.89 , pp. 149-185
    • Byrd, R.H.1    Gilbert, J.C.2    Nocedal, J.3
  • 10
    • 0003917112 scopus 로고    scopus 로고
    • On the convergence of Newton iterations to non-stationary points
    • Technical report OTC 2001/01, Optimization Technology Center, Northwestern University, Evanston
    • Byrd, R.H., Marazzi, M., Nocedal, J.: On the convergence of Newton iterations to non-stationary points. Technical report OTC 2001/01, Optimization Technology Center, Northwestern University, Evanston (2001)
    • (2001)
    • Byrd, R.H.1    Marazzi, M.2    Nocedal, J.3
  • 11
    • 0037378185 scopus 로고    scopus 로고
    • Parallel computational issues of an interior point method for solving large bound-constrained quadratic programming problems
    • D'Apuzzo, M., Marino, M.: Parallel computational issues of an interior point method for solving large bound-constrained quadratic programming problems. Parallel Comput. 29, 467-483 (2003)
    • (2003) Parallel Comput , vol.29 , pp. 467-483
    • D'Apuzzo, M.1    Marino, M.2
  • 13
    • 34047152533 scopus 로고    scopus 로고
    • Numerical solution of discrete quadratic optimal control problems by Hestenes' method
    • Durazzi, C.: Numerical solution of discrete quadratic optimal control problems by Hestenes' method. Rend. Circ. Mat. Palermo Suppl. 58(2), 133-154 (1999)
    • (1999) Rend. Circ. Mat. Palermo Suppl , vol.58 , Issue.2 , pp. 133-154
    • Durazzi, C.1
  • 14
    • 0034349445 scopus 로고    scopus 로고
    • On the Newton interior-point method for nonlinear programming problems
    • Durazzi, C.: On the Newton interior-point method for nonlinear programming problems. J. Optim. Theory Appl. 104, 73-90 (2000)
    • (2000) J. Optim. Theory Appl , vol.104 , pp. 73-90
    • Durazzi, C.1
  • 16
    • 0345547686 scopus 로고    scopus 로고
    • Indefinitely preconditioned conjugate gradient method for large sparse equality and inequality constrained quadratic problems
    • Durazzi, C., Ruggiero, V.: Indefinitely preconditioned conjugate gradient method for large sparse equality and inequality constrained quadratic problems. Numer. Linear Algebra Appl. 10, 673-688 (2003)
    • (2003) Numer. Linear Algebra Appl , vol.10 , pp. 673-688
    • Durazzi, C.1    Ruggiero, V.2
  • 17
    • 0037377464 scopus 로고    scopus 로고
    • Numerical solution of special linear and quadratic programs via a parallel interior-point method
    • Durazzi, C., Ruggiero, V.: Numerical solution of special linear and quadratic programs via a parallel interior-point method. Parallel Comput. 29, 485-503 (2003)
    • (2003) Parallel Comput , vol.29 , pp. 485-503
    • Durazzi, C.1    Ruggiero, V.2
  • 18
    • 0347723949 scopus 로고    scopus 로고
    • Global convergence of the Newton interior-point method for nonlinear programming
    • Durazzi, C., Ruggiero, V.: Global convergence of the Newton interior-point method for nonlinear programming. J. Optim. Theory Appl. 120, 199-208 (2004)
    • (2004) J. Optim. Theory Appl , vol.120 , pp. 199-208
    • Durazzi, C.1    Ruggiero, V.2
  • 19
    • 0035536764 scopus 로고    scopus 로고
    • Parallel interior-point method for linear and quadratic programs with special structure
    • Durazzi, C., Ruggiero, V., Zanghirati, G.: Parallel interior-point method for linear and quadratic programs with special structure. J. Optim. Theory Appl. 110, 289-313 (2001)
    • (2001) J. Optim. Theory Appl , vol.110 , pp. 289-313
    • Durazzi, C.1    Ruggiero, V.2    Zanghirati, G.3
  • 20
    • 0001330894 scopus 로고
    • Globally convergent inexact Newton methods
    • Eisenstat, S.C., Walker, H.E: Globally convergent inexact Newton methods. SIAM J. Optim. 4, 393-422 (1994)
    • (1994) SIAM J. Optim , vol.4 , pp. 393-422
    • Eisenstat, S.C.1    Walker, H.E.2
  • 21
    • 0030541525 scopus 로고    scopus 로고
    • On the formulation and theory of Newton interior-point method for nonlinear programming
    • El-Bakry, A.S., Tapia, R.A., Tsuchiya, T., Zhang, Y.: On the formulation and theory of Newton interior-point method for nonlinear programming. J. Optim. Theory Appl. 89, 507-541 (1996)
    • (1996) J. Optim. Theory Appl , vol.89 , pp. 507-541
    • El-Bakry, A.S.1    Tapia, R.A.2    Tsuchiya, T.3    Zhang, Y.4
  • 23
    • 0036776589 scopus 로고    scopus 로고
    • Inertia-controlling factorizations for optimization algorithms
    • Forsgren, A.: Inertia-controlling factorizations for optimization algorithms. Appl. Numer. Math. 43, 91-107 (2002)
    • (2002) Appl. Numer. Math , vol.43 , pp. 91-107
    • Forsgren, A.1
  • 24
    • 0037427551 scopus 로고    scopus 로고
    • The Newton-arithmetic mean method for the solution of systems of nonlinear equations
    • Galligani, E.: The Newton-arithmetic mean method for the solution of systems of nonlinear equations. Appl. Math. Comput. 134, 9-34 (2003)
    • (2003) Appl. Math. Comput , vol.134 , pp. 9-34
    • Galligani, E.1
  • 25
    • 0027901882 scopus 로고
    • Numerical solution of equality-constrained quadratic programming problems on vector-parallel computers
    • Galligani, I., Ruggiero, V.: Numerical solution of equality-constrained quadratic programming problems on vector-parallel computers. Optim. Methods Softw. 2, 233-247 (1993)
    • (1993) Optim. Methods Softw , vol.2 , pp. 233-247
    • Galligani, I.1    Ruggiero, V.2
  • 26
    • 11644274626 scopus 로고    scopus 로고
    • A minimization method for the solution of large symmetric eigenproblems
    • Galligani, E., Ruggiero, V., Zanni, L.: A minimization method for the solution of large symmetric eigenproblems. Int. J. Comput. Math. 70, 99-115 (1998)
    • (1998) Int. J. Comput. Math , vol.70 , pp. 99-115
    • Galligani, E.1    Ruggiero, V.2    Zanni, L.3
  • 28
    • 0030532067 scopus 로고    scopus 로고
    • On the stability of Choleski factorization for symmetric quasidefinite systems
    • Gill, P.E., Saunders, M.A., Shinned, J.R.: On the stability of Choleski factorization for symmetric quasidefinite systems. SIAM J. Matrix Anal. Appl. 17, 35-46 (1996)
    • (1996) SIAM J. Matrix Anal. Appl , vol.17 , pp. 35-46
    • Gill, P.E.1    Saunders, M.A.2    Shinned, J.R.3
  • 29
    • 0347577906 scopus 로고    scopus 로고
    • On solving block-structured indefinite linear systems
    • Golub, G., Greif, C.: On solving block-structured indefinite linear systems. SIAM J. Sci. Comput. 24, 2076-2092 (2003)
    • (2003) SIAM J. Sci. Comput , vol.24 , pp. 2076-2092
    • Golub, G.1    Greif, C.2
  • 30
    • 0041437231 scopus 로고    scopus 로고
    • SOR-like methods for augmented system
    • Golub, G., Wu, X., Yuan, J.Y.: SOR-like methods for augmented system. BIT 41, 71-85 (2001)
    • (2001) BIT , vol.41 , pp. 71-85
    • Golub, G.1    Wu, X.2    Yuan, J.Y.3
  • 31
    • 34047158958 scopus 로고    scopus 로고
    • Harwell Subroutine Library: A Catalogue of Subroutines (HSL 2000). AEA Technology, Harwell, Oxfordshire (2002)
    • Harwell Subroutine Library: A Catalogue of Subroutines (HSL 2000). AEA Technology, Harwell, Oxfordshire (2002)
  • 32
    • 0014604308 scopus 로고
    • Multiplier and gradient methods
    • Hestenes, M.R.: Multiplier and gradient methods. J. Optim. Theory Appl. 4, 303-320 (1969)
    • (1969) J. Optim. Theory Appl , vol.4 , pp. 303-320
    • Hestenes, M.R.1
  • 34
    • 0034145598 scopus 로고    scopus 로고
    • Constraint preconditioners for indefinite linear systems
    • Keller, C., Gould, N.I.M., Wathen, A.J.: Constraint preconditioners for indefinite linear systems. SIAM J. Matrix Anal. Appl. 21, 1300-1317 (2000)
    • (2000) SIAM J. Matrix Anal. Appl , vol.21 , pp. 1300-1317
    • Keller, C.1    Gould, N.I.M.2    Wathen, A.J.3
  • 35
    • 21144443182 scopus 로고    scopus 로고
    • Optimum parameter for the SOR-like method for augmented system
    • Li, C., Li, Z., Shao, X., Nie, Y., Evans, D.J.: Optimum parameter for the SOR-like method for augmented system. Int. J. Comput. Math. 81, 749-763 (2004)
    • (2004) Int. J. Comput. Math , vol.81 , pp. 749-763
    • Li, C.1    Li, Z.2    Shao, X.3    Nie, Y.4    Evans, D.J.5
  • 36
    • 21144482411 scopus 로고
    • On finding supernodes for sparse matrix computations
    • Liu, J.W., Ng, E.G., Peyton, B.W.: On finding supernodes for sparse matrix computations. SIAM J. Matrix Anal. Appl. 14, 242-252 (1993)
    • (1993) SIAM J. Matrix Anal. Appl , vol.14 , pp. 242-252
    • Liu, J.W.1    Ng, E.G.2    Peyton, B.W.3
  • 38
    • 10044222210 scopus 로고    scopus 로고
    • Interior-point method for non-linear non-convex optimization
    • Lukšan, L., Matonoha, C., Vlček, J.: Interior-point method for non-linear non-convex optimization. Numer. Linear Algebra Appl. 11, 431-453 (2004)
    • (2004) Numer. Linear Algebra Appl , vol.11 , pp. 431-453
    • Lukšan, L.1    Matonoha, C.2    Vlček, J.3
  • 39
    • 0345293940 scopus 로고    scopus 로고
    • Indefinitely preconditioned inexact Newton method for large sparse equality constrained non-linear programming problems
    • Lukšan, L., Vlček, J.: Indefinitely preconditioned inexact Newton method for large sparse equality constrained non-linear programming problems. Numer. Linear Algebra Appl. 5, 219-247 (1998)
    • (1998) Numer. Linear Algebra Appl , vol.5 , pp. 219-247
    • Lukšan, L.1    Vlček, J.2
  • 40
    • 0033738978 scopus 로고    scopus 로고
    • Optimization techniques for solving elliptic control problems with control and state constraints: Part 1. Boundary control
    • Maurer, H., Mittelmann, H.D.: Optimization techniques for solving elliptic control problems with control and state constraints: Part 1. Boundary control. Comput. Optim. Appl. 16, 29-55 (2000)
    • (2000) Comput. Optim. Appl , vol.16 , pp. 29-55
    • Maurer, H.1    Mittelmann, H.D.2
  • 41
    • 0035248626 scopus 로고    scopus 로고
    • Optimization techniques for solving elliptic control problems with control and state constraints: Part 2. Distributed control
    • Maurer, H., Mittelmann, H.D.: Optimization techniques for solving elliptic control problems with control and state constraints: Part 2. Distributed control. Comput. Optim. Appl. 18, 141-160 (2001)
    • (2001) Comput. Optim. Appl , vol.18 , pp. 141-160
    • Maurer, H.1    Mittelmann, H.D.2
  • 42
    • 0035480023 scopus 로고    scopus 로고
    • Verification of second-order sufficient optimality conditions for semilinear elliptic and parabolic control problems
    • Mittelmann, H.D.: Verification of second-order sufficient optimality conditions for semilinear elliptic and parabolic control problems. Comput. Optim. Appl. 20, 93-110 (2001)
    • (2001) Comput. Optim. Appl , vol.20 , pp. 93-110
    • Mittelmann, H.D.1
  • 43
    • 34047130437 scopus 로고    scopus 로고
    • Private communication
    • Mittelmann, H.D.: Private communication (2004)
    • (2004)
    • Mittelmann, H.D.1
  • 44
    • 0034255203 scopus 로고    scopus 로고
    • Solving elliptic control problems with interior point and SQP methods: Control and state constraints
    • Mittelmann, H.D., Maurer, H.: Solving elliptic control problems with interior point and SQP methods: control and state constraints. J. Comput. Appl. Math. 120, 175-195 (2000)
    • (2000) J. Comput. Appl. Math , vol.120 , pp. 175-195
    • Mittelmann, H.D.1    Maurer, H.2
  • 45
    • 0013105967 scopus 로고    scopus 로고
    • Assessing the potential of interior methods for nonlinear optimization
    • Technical report OTC 2001/04, Optimization Technology Center, North-western University, Evanston
    • Morales, J.L., Nocedal, J., Waltz, R.A., Liu, G., Goux, J.P.: Assessing the potential of interior methods for nonlinear optimization. Technical report OTC 2001/04, Optimization Technology Center, North-western University, Evanston (2001)
    • (2001)
    • Morales, J.L.1    Nocedal, J.2    Waltz, R.A.3    Liu, G.4    Goux, J.P.5
  • 46
    • 0036344992 scopus 로고    scopus 로고
    • On the solution of equality constrained quadratic programming problems arising in optimization
    • Nocedal, J., Hribar, M.E., Gould, N.I.M.: On the solution of equality constrained quadratic programming problems arising in optimization. SIAM J. Sci. Comput. 23, 1375-1394 (2001)
    • (2001) SIAM J. Sci. Comput , vol.23 , pp. 1375-1394
    • Nocedal, J.1    Hribar, M.E.2    Gould, N.I.M.3
  • 48
    • 0002380692 scopus 로고
    • A method for nonlinear constraints in minimization problems
    • Fletcher, R, ed, Academic, London
    • Powell, M.J.D.: A method for nonlinear constraints in minimization problems. In: Fletcher, R. (ed.) Optimization, pp. 283-298. Academic, London (1969)
    • (1969) Optimization , pp. 283-298
    • Powell, M.J.D.1
  • 51
    • 34047161254 scopus 로고    scopus 로고
    • Saunders, M., Tomlin, J.A.: Solving regularized linear programs using barrier methods and KKT systems. Technical report SOL 96-4, Systems Optimization Lab., Dept. Oper. Res., Stanford University, Stanford (1996)
    • Saunders, M., Tomlin, J.A.: Solving regularized linear programs using barrier methods and KKT systems. Technical report SOL 96-4, Systems Optimization Lab., Dept. Oper. Res., Stanford University, Stanford (1996)
  • 52
    • 0022146092 scopus 로고
    • On the method of weighting for equality-constrained least-squares problems
    • Van Loan, C.: On the method of weighting for equality-constrained least-squares problems. SIAM J. Numer. Anal. 22, 851-864 (1985)
    • (1985) SIAM J. Numer. Anal , vol.22 , pp. 851-864
    • Van Loan, C.1
  • 53
    • 0002090173 scopus 로고
    • Symmetric quasidefinite matrices
    • Vanderbei, R.J.: Symmetric quasidefinite matrices. SIAM J. Optim. 5, 100-113 (1995)
    • (1995) SIAM J. Optim , vol.5 , pp. 100-113
    • Vanderbei, R.J.1
  • 54
    • 0000287204 scopus 로고    scopus 로고
    • An interior-point algorithm for nonconvex nonlinear programming
    • Vanderbei, R.J., Shanno, D.F.: An interior-point algorithm for nonconvex nonlinear programming. Comput. Optim. Appl. 13, 231-252 (1999)
    • (1999) Comput. Optim. Appl , vol.13 , pp. 231-252
    • Vanderbei, R.J.1    Shanno, D.F.2
  • 56
    • 0000892372 scopus 로고    scopus 로고
    • Failure of global convergence for a class of interior point methods for nonlinear programming
    • Wächter, A., Biegler, L.T.: Failure of global convergence for a class of interior point methods for nonlinear programming. Math. Program. 88, 565-574 (2000)
    • (2000) Math. Program , vol.88 , pp. 565-574
    • Wächter, A.1    Biegler, L.T.2
  • 57
    • 0029712553 scopus 로고    scopus 로고
    • On the monotone convergence of multisplitting method for a class of systems of weakly nonlinear equations
    • Wang, D., Bai, Z.Z., Evans, D.J.: On the monotone convergence of multisplitting method for a class of systems of weakly nonlinear equations. Int. J. Comput. Math. 60, 229-242 (1996)
    • (1996) Int. J. Comput. Math , vol.60 , pp. 229-242
    • Wang, D.1    Bai, Z.Z.2    Evans, D.J.3
  • 58
    • 13544264699 scopus 로고    scopus 로고
    • The interior-point revolution in optimization: History, recent developments, and lasting consequences
    • Wright, M.: The interior-point revolution in optimization: history, recent developments, and lasting consequences. Bull. Amer. Math. Soc. 42, 39-56 (2005)
    • (2005) Bull. Amer. Math. Soc , vol.42 , pp. 39-56
    • Wright, M.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.