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A unified nonlinear augmented Lagrangian approach for nonconvex vector optimization
A primal-dual algorithm for nonlinear programming exploiting negative curvature directions
1. | Sapienza University of Rome, Department of Computer and System Sciences ``A. Ruberti'', Via Ariosto 25, 00185 Roma, Italy, Italy |
2. | Italian National Research Council, Institute for Systems Analysis and Computer Science ``A. Ruberti'', Viale Manzoni 30, 00185 Roma, Italy |
References:
[1] |
A. Auslender, Penalty methods for computing points that satisfy second order necessary conditions, Mathematical Programming, 17 (1979), 229-238.
doi: 10.1007/BF01588245. |
[2] |
R. H. Byrd, R. B. Schnabel and G. A. Shultz, A trust region algorithm for nonlinearly constrained optimization, SIAM Journal on Numerical Analysis, 24 (1987), 1152-1170.
doi: 10.1137/0724076. |
[3] |
F. H. Clarke, "Optimization and Nonsmooth Analysis," John Wiley & Sons, New York, 1983. |
[4] |
T. F. Coleman, J. G. Liu and W. Yuan, A new trust-region algorithm for equality constrained optimization, Computational Optimization and Applications, 21 (2002), 177-199.
doi: 10.1023/A:1013764800871. |
[5] |
A. R. Conn, N. I. M. Gould, D. Orban and Ph. L. Toint, A primal-dual trust region algorithm for non-convex nonlinear programming, Math. Programming, Ser. B, 87 (2000), 215-249. |
[6] |
R. S. Dembo, S. C. Eisenstat and T. Steihaug, Inexact Newton methods, SIAM Journal on Numerical Anlysis, 19 (1982), 400-408.
doi: 10.1137/0719025. |
[7] |
J. E. Dennis and L. N. Vicente, On the convergence theory of trust-region-based algorithms for equality-constrained optimization, SIAM Journal on Optimization, 7 (1994), 527-550. |
[8] |
J. E. Dennis, M. Heinkenschloss and L. N. Vicente, Trust-region interior-point SQP algorithms for a class of nonlinear programming problems, SIAM Journal on Control and Optimization, 36 (1998), 1750-1794.
doi: 10.1137/S036012995279031. |
[9] |
G. Di Pillo and L. Grippo, Exact penalty functions in constrained optimization, SIAM J. on Control and Optimization, 27 (1989), 1333-1360.
doi: 10.1137/0327068. |
[10] |
G. Di Pillo, G. Liuzzi, S. Lucidi and L. Palagi, A truncated Newton method in an augmented Lagrangian framework for nonlinear programming, Computational Optimization and Applications, 45 (2010), 311-352.
doi: 10.1007/s10589-008-9216-3. |
[11] |
G. Di Pillo and S. Lucidi, On exact Augmented Lagrangian functions in nonlinear programming, In " Nonlinear Optimization and Applications"(eds. G. Di Pillo and F. Giannessi), Plenum Press, New York, (1996), 85-100. |
[12] |
G. Di Pillo and S. Lucidi, An augmented Lagrangian function with improved exactness properties, SIAM Journal on Optimization, 12 (2001), 376-406.
doi: 10.1137/S1052623497321894. |
[13] |
G. Di Pillo, S. Lucidi and L. Palagi, Convergence to second-order stationary points of a primal-dual algorithm model for nonlinear programming, Mathematics of Operations Research, 30 (2005), 897-915.
doi: 10.1287/moor.1050.0150. |
[14] |
M. M. El-Alem, Convergence to a second-order point of a trust-region algorithm with a nonmonotonic penalty parameter for constrained optimization, Journal of Optimization Theory and Applications, 91 (1996), 61-79.
doi: 10.1007/BF02192282. |
[15] |
F. Facchinei, Minimization of SC1 functions and the Maratos effect, Operations Research Letters, 17 (1995), 131-137.
doi: 10.1016/0167-6377(94)00059-F. |
[16] |
F. Facchinei, A. Fischer and C. Kanzow, On the accurate identification of active constraints, SIAM Journal on Optimization, 9 (1998), 14-32.
doi: 10.1137/S1052623496305882. |
[17] |
F. Facchinei and S. Lucidi, Quadratically and superlinearly convergent algorithms for the solution of inequality constrained minimization problems, Journal of Optimization Theory and Applications, 85 (1995), 265-289.
doi: 10.1007/BF02192227. |
[18] |
F. Facchinei and S. Lucidi, Convergence to second order stationary points in inequality constrained optimization, Mathematics of Operations Research, 23 (1998), 746-766.
doi: 10.1287/moor.23.3.746. |
[19] |
G. Fasano and S. Lucidi, A nonmonotone truncated Newton-Krylov method exploiting negative curvature directions, for large scale unconstrained optimization, Optimization Letters, 3 (2009), 521-535.
doi: 10.1007/s11590-009-0132-y. |
[20] |
A. Forsgren and W. Murray, Newton methods for large-scale linear inequality constrained minimization, SIAM Journal on Optimization, 7 (1997), 162-176.
doi: 10.1137/S1052623494279122. |
[21] |
L. Grippo, F. Lampariello and S. Lucidi, A truncated Newton method with nomonotone linesearch for unconstrained optimization, Journal of Optimization Theory and Applications, 60 (1989), 401-419.
doi: 10.1007/BF00940345. |
[22] |
J. B. Hiriart-Urruty, J. J. Strodiot and V. H. Nguyen, Generalized Hessian matrix and second-order optimality conditions for problems with C1,1 data, Applied Mathematics and Optimization, 11 (1984), 43-56.
doi: 10.1007/BF01442169. |
[23] |
S. Lucidi, F. Rochetich and M. Roma, Curvilinear stabilization techniques for truncated Newton methods in large scale unconstrained optimization: The complete results, SIAM Journal on Optimization, 8 (1998), 916-939.
doi: 10.1137/S1052623495295250. |
[24] |
G. P. McCormick, "Nonlinear Programming: Theory, Algorithms and Applications," John Wiley & Sons, New York, 1983. |
[25] |
J. M. Moguerza and F. J. Prieto, An augmented Lagrangian interior point method using directions of negative curvature, Mathematical Programming, 95 (2003), 573-616.
doi: 10.1007/s10107-002-0360-8. |
[26] |
J. J. Moré and D. C. Sorensen, On the use of directions of negative curvature in a modified Newton method, Mathematical Programming, 16 (1979), 1-20.
doi: 10.1007/BF01582091. |
[27] |
J. J. Moré and D. C. Sorensen, Computing a trust region step, SIAM Journal on Scientific and Statistical Computing, 4 (1983), 553-572. |
[28] |
H. Mukai and E. Polak, A second-order method for the general nonlinear programming problem, Journal of Optimization Theory and Applications, 26 (1978), 515-532.
doi: 10.1007/BF00933150. |
[29] |
L. Qi and J. Sun, A nonsmooth version of Newton's method, Mathematical Programming, 58 (1993), 353-367.
doi: 10.1007/BF01581275. |
[30] |
D. C. Sorensen, Newton's method with a model trust region modification, SIAM Journal on Numerical Analysis, 19 (1982), 409-426.
doi: 10.1137/0719026. |
[31] |
P. Tseng, A convergent infeasible interior-point trust-region method for constrained optimization, SIAM Journal on Optimization, 13 (2002), 432-469.
doi: 10.1137/S1052623499357945. |
show all references
References:
[1] |
A. Auslender, Penalty methods for computing points that satisfy second order necessary conditions, Mathematical Programming, 17 (1979), 229-238.
doi: 10.1007/BF01588245. |
[2] |
R. H. Byrd, R. B. Schnabel and G. A. Shultz, A trust region algorithm for nonlinearly constrained optimization, SIAM Journal on Numerical Analysis, 24 (1987), 1152-1170.
doi: 10.1137/0724076. |
[3] |
F. H. Clarke, "Optimization and Nonsmooth Analysis," John Wiley & Sons, New York, 1983. |
[4] |
T. F. Coleman, J. G. Liu and W. Yuan, A new trust-region algorithm for equality constrained optimization, Computational Optimization and Applications, 21 (2002), 177-199.
doi: 10.1023/A:1013764800871. |
[5] |
A. R. Conn, N. I. M. Gould, D. Orban and Ph. L. Toint, A primal-dual trust region algorithm for non-convex nonlinear programming, Math. Programming, Ser. B, 87 (2000), 215-249. |
[6] |
R. S. Dembo, S. C. Eisenstat and T. Steihaug, Inexact Newton methods, SIAM Journal on Numerical Anlysis, 19 (1982), 400-408.
doi: 10.1137/0719025. |
[7] |
J. E. Dennis and L. N. Vicente, On the convergence theory of trust-region-based algorithms for equality-constrained optimization, SIAM Journal on Optimization, 7 (1994), 527-550. |
[8] |
J. E. Dennis, M. Heinkenschloss and L. N. Vicente, Trust-region interior-point SQP algorithms for a class of nonlinear programming problems, SIAM Journal on Control and Optimization, 36 (1998), 1750-1794.
doi: 10.1137/S036012995279031. |
[9] |
G. Di Pillo and L. Grippo, Exact penalty functions in constrained optimization, SIAM J. on Control and Optimization, 27 (1989), 1333-1360.
doi: 10.1137/0327068. |
[10] |
G. Di Pillo, G. Liuzzi, S. Lucidi and L. Palagi, A truncated Newton method in an augmented Lagrangian framework for nonlinear programming, Computational Optimization and Applications, 45 (2010), 311-352.
doi: 10.1007/s10589-008-9216-3. |
[11] |
G. Di Pillo and S. Lucidi, On exact Augmented Lagrangian functions in nonlinear programming, In " Nonlinear Optimization and Applications"(eds. G. Di Pillo and F. Giannessi), Plenum Press, New York, (1996), 85-100. |
[12] |
G. Di Pillo and S. Lucidi, An augmented Lagrangian function with improved exactness properties, SIAM Journal on Optimization, 12 (2001), 376-406.
doi: 10.1137/S1052623497321894. |
[13] |
G. Di Pillo, S. Lucidi and L. Palagi, Convergence to second-order stationary points of a primal-dual algorithm model for nonlinear programming, Mathematics of Operations Research, 30 (2005), 897-915.
doi: 10.1287/moor.1050.0150. |
[14] |
M. M. El-Alem, Convergence to a second-order point of a trust-region algorithm with a nonmonotonic penalty parameter for constrained optimization, Journal of Optimization Theory and Applications, 91 (1996), 61-79.
doi: 10.1007/BF02192282. |
[15] |
F. Facchinei, Minimization of SC1 functions and the Maratos effect, Operations Research Letters, 17 (1995), 131-137.
doi: 10.1016/0167-6377(94)00059-F. |
[16] |
F. Facchinei, A. Fischer and C. Kanzow, On the accurate identification of active constraints, SIAM Journal on Optimization, 9 (1998), 14-32.
doi: 10.1137/S1052623496305882. |
[17] |
F. Facchinei and S. Lucidi, Quadratically and superlinearly convergent algorithms for the solution of inequality constrained minimization problems, Journal of Optimization Theory and Applications, 85 (1995), 265-289.
doi: 10.1007/BF02192227. |
[18] |
F. Facchinei and S. Lucidi, Convergence to second order stationary points in inequality constrained optimization, Mathematics of Operations Research, 23 (1998), 746-766.
doi: 10.1287/moor.23.3.746. |
[19] |
G. Fasano and S. Lucidi, A nonmonotone truncated Newton-Krylov method exploiting negative curvature directions, for large scale unconstrained optimization, Optimization Letters, 3 (2009), 521-535.
doi: 10.1007/s11590-009-0132-y. |
[20] |
A. Forsgren and W. Murray, Newton methods for large-scale linear inequality constrained minimization, SIAM Journal on Optimization, 7 (1997), 162-176.
doi: 10.1137/S1052623494279122. |
[21] |
L. Grippo, F. Lampariello and S. Lucidi, A truncated Newton method with nomonotone linesearch for unconstrained optimization, Journal of Optimization Theory and Applications, 60 (1989), 401-419.
doi: 10.1007/BF00940345. |
[22] |
J. B. Hiriart-Urruty, J. J. Strodiot and V. H. Nguyen, Generalized Hessian matrix and second-order optimality conditions for problems with C1,1 data, Applied Mathematics and Optimization, 11 (1984), 43-56.
doi: 10.1007/BF01442169. |
[23] |
S. Lucidi, F. Rochetich and M. Roma, Curvilinear stabilization techniques for truncated Newton methods in large scale unconstrained optimization: The complete results, SIAM Journal on Optimization, 8 (1998), 916-939.
doi: 10.1137/S1052623495295250. |
[24] |
G. P. McCormick, "Nonlinear Programming: Theory, Algorithms and Applications," John Wiley & Sons, New York, 1983. |
[25] |
J. M. Moguerza and F. J. Prieto, An augmented Lagrangian interior point method using directions of negative curvature, Mathematical Programming, 95 (2003), 573-616.
doi: 10.1007/s10107-002-0360-8. |
[26] |
J. J. Moré and D. C. Sorensen, On the use of directions of negative curvature in a modified Newton method, Mathematical Programming, 16 (1979), 1-20.
doi: 10.1007/BF01582091. |
[27] |
J. J. Moré and D. C. Sorensen, Computing a trust region step, SIAM Journal on Scientific and Statistical Computing, 4 (1983), 553-572. |
[28] |
H. Mukai and E. Polak, A second-order method for the general nonlinear programming problem, Journal of Optimization Theory and Applications, 26 (1978), 515-532.
doi: 10.1007/BF00933150. |
[29] |
L. Qi and J. Sun, A nonsmooth version of Newton's method, Mathematical Programming, 58 (1993), 353-367.
doi: 10.1007/BF01581275. |
[30] |
D. C. Sorensen, Newton's method with a model trust region modification, SIAM Journal on Numerical Analysis, 19 (1982), 409-426.
doi: 10.1137/0719026. |
[31] |
P. Tseng, A convergent infeasible interior-point trust-region method for constrained optimization, SIAM Journal on Optimization, 13 (2002), 432-469.
doi: 10.1137/S1052623499357945. |
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