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Volumn 23, Issue 3, 2013, Pages 1528-1552

Optimality conditions and a smoothing trust region newton method for nonlipschitz optimization

Author keywords

Convergence; Non Lipschitz; Nonsmooth nonconvex optimization; Penalty function; Regularized optimization; Smoothing methods; Trust region Newton method

Indexed keywords

CONVERGENCE; NON-LIPSCHITZ; NONSMOOTH NONCONVEX OPTIMIZATION; PENALTY FUNCTION; REGULARIZED OPTIMIZATIONS; SMOOTHING METHODS; TRUST REGION;

EID: 84886296616     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/120871390     Document Type: Article
Times cited : (91)

References (47)
  • 1
    • 43949152558 scopus 로고
    • A trust region algorithm for nonsmooth optimization
    • T. Bannert, A trust region algorithm for nonsmooth optimization, Math. Program., 67 (1994), pp. 247-264.
    • (1994) Math. Program , vol.67 , pp. 247-264
    • Bannert, T.1
  • 2
    • 84865698786 scopus 로고    scopus 로고
    • Smoothing neural network for constrained non-Lipschitz optimization with applications
    • W. Bian and X. Chen, Smoothing neural network for constrained non-Lipschitz optimization with applications, IEEE Trans. Neural Network, 23 (2012), pp. 399-411.
    • (2012) IEEE Trans. Neural Network , vol.23 , pp. 399-411
    • Bian, W.1    Chen, X.2
  • 3
    • 59749104367 scopus 로고    scopus 로고
    • From sparse solutions of systems of equations to sparse modeling of signals and images
    • A.M. Bruckstein, D.L. Donoho, and M. Elad, From sparse solutions of systems of equations to sparse modeling of signals and images, SIAM Rev., 51 (2009), pp. 34-81.
    • (2009) SIAM Rev. , vol.51 , pp. 34-81
    • Bruckstein, A.M.1    Donoho, D.L.2    Elad, M.3
  • 4
    • 23844446458 scopus 로고    scopus 로고
    • A robust gradient sampling algorithm for nonsmooth, nonconvex optimization
    • J.V. Burke, A.S. Lewis, and M.L. Overton, A robust gradient sampling algorithm for nonsmooth, nonconvex optimization, SIAM J. Optim., 15 (2005), pp. 751-779.
    • (2005) SIAM J. Optim , vol.15 , pp. 751-779
    • Burke, J.V.1    Lewis, A.S.2    Overton, M.L.3
  • 5
    • 84856057771 scopus 로고    scopus 로고
    • On the evaluation complexity of composite function minimization with applications to nonconvex nonlinear programming
    • C. Catis, N.I.M. Gould, and Ph.L. Toint, On the evaluation complexity of composite function minimization with applications to nonconvex nonlinear programming, SIAM J. Optim., 21 (2011), pp. 1721-1739.
    • (2011) SIAM J. Optim. , vol.21 , pp. 1721-1739
    • Catis, C.1    Gould, N.I.M.2    Toint, Ph.L.3
  • 6
    • 44449127493 scopus 로고    scopus 로고
    • Restricted isometry properties and nonconvex compressive sensing
    • R. Chartrand and V. Staneva, Restricted isometry properties and nonconvex compressive sensing, Inverse Problems, 24 (2008), pp. 1-14.
    • (2008) Inverse Problems , vol.24 , pp. 1-14
    • Chartrand, R.1    Staneva, V.2
  • 7
    • 84865682357 scopus 로고    scopus 로고
    • Smoothing methods for nonsmooth, nonconvex minimization
    • X. Chen, Smoothing methods for nonsmooth, nonconvex minimization, Math. Program., 134 (2012), pp. 71-99.
    • (2012) Math. Program , vol.134 , pp. 71-99
    • Chen, X.1
  • 9
    • 78149326445 scopus 로고    scopus 로고
    • p minimization
    • X. Chen, F. Xu, and Y. Ye, Lower bound theory of nonzero entries in solutions of l2-lp minimization, SIAM J. Sci. Comput., 32 (2010), pp. 2832-2852.
    • (2010) SIAM J. Sci. Comput. , vol.32 , pp. 2832-2852
    • Chen, X.1    Xu, F.2    Ye, Y.3
  • 10
    • 78149291090 scopus 로고    scopus 로고
    • Smoothing nonlinear conjugate gradient method for image restoration using nonsmooth nonconvex minimization
    • X. Chen and W. Zhou, Smoothing nonlinear conjugate gradient method for image restoration using nonsmooth nonconvex minimization, SIAM J. Imag. Sci., 3 (2010), pp. 765-790.
    • (2010) SIAM J. Imag. Sci. , vol.3 , pp. 765-790
    • Chen, X.1    Zhou, W.2
  • 13
    • 58149365128 scopus 로고
    • A unified approach to global convergence of trust region methods for nonsmooth optimization
    • J.E. Dennis Jr., S.-B. Li, and R.A. Tapia, A unified approach to global convergence of trust region methods for nonsmooth optimization, Math. Program., 68 (1995), pp. 319-346.
    • (1995) Math. Program. , vol.68 , pp. 319-346
    • Dennis Jr., J.E.1    Li, S.-B.2    Tapia, R.A.3
  • 14
    • 0001151466 scopus 로고    scopus 로고
    • A nonsmooth inexact Newton method for the solution of largescale nonlinear complementarity problems
    • F. Facchinei and C. Kanzow, A nonsmooth inexact Newton method for the solution of largescale nonlinear complementarity problems, Math. Program., 76 (1997), pp. 493-512.
    • (1997) Math. Program , vol.76 , pp. 493-512
    • Facchinei, F.1    Kanzow, C.2
  • 15
    • 1542784498 scopus 로고    scopus 로고
    • Variable selection via nonconcave penalized likelihood and its oracle properties
    • J. Fan and R. Li, Variable selection via nonconcave penalized likelihood and its oracle properties, J. Amer. Statist. Assoc., 96 (2011), pp. 1348-1360.
    • (2011) J. Amer. Statist. Assoc. , vol.96 , pp. 1348-1360
    • Fan, J.1    Li, R.2
  • 16
    • 0001963239 scopus 로고
    • A model algorithm for composite non-differentiable optimization problem
    • F. Fletcher, A model algorithm for composite non-differentiable optimization problem, Math. Program. Studies, 17 (1982), pp. 67-76.
    • (1982) Math. Program. Studies , vol.17 , pp. 67-76
    • Fletcher, F.1
  • 18
    • 62549128663 scopus 로고    scopus 로고
    • q-minimization for 0 < q ≤ 1
    • S. Foucart and M.J. Lai, Sparsest solutions of underdetermined linear systems via ℓq-minimization for 0 < q ≤ 1, Appl. Comput. Harmonic Anal., 26 (2009), pp. 395-407.
    • (2009) Appl. Comput. Harmonic Anal. , vol.26 , pp. 395-407
    • Foucart, S.1    Lai, M.J.2
  • 19
    • 84886277792 scopus 로고    scopus 로고
    • Smoothing and worst case complexity for direct-search methods in non-smooth optimization
    • to appear
    • R. Garmanjani and L.N. Vicente, Smoothing and worst case complexity for direct-search methods in non-smooth optimization, IMA Numer. Anal., to appear.
    • IMA Numer. Anal.
    • Garmanjani, R.1    Vicente, L.N.2
  • 24
    • 49949115667 scopus 로고    scopus 로고
    • Asymptotic properties of bridge estimators in sparse high-dimensional regression models
    • J. Huang, J.L. Horowitz, and S. Ma, Asymptotic properties of bridge estimators in sparse high-dimensional regression models, Ann. Stat., 36 (2008), pp. 587-613.
    • (2008) Ann. Stat. , vol.36 , pp. 587-613
    • Huang, J.1    Horowitz, J.L.2    Ma, S.3
  • 26
    • 0032331517 scopus 로고    scopus 로고
    • A trust region method for solving generalized complementarity problems
    • H. Jiang, M. Fukushima, L. Qi, and D. Sun, A trust region method for solving generalized complementarity problems, SIAM J. Optim., 8 (1998), pp. 140-157.
    • (1998) SIAM J. Optim. , vol.8 , pp. 140-157
    • Jiang, H.1    Fukushima, M.2    Qi, L.3    Sun, D.4
  • 27
    • 0034287156 scopus 로고    scopus 로고
    • Asymptotics for lasso-type estimators
    • K. Knight and W.J. Fu, Asymptotics for lasso-type estimators, Ann. Statist., 28 (2000), pp. 1356-1378.
    • (2000) Ann. Statist. , vol.28 , pp. 1356-1378
    • Knight, K.1    Fu, W.J.2
  • 28
    • 44649165114 scopus 로고    scopus 로고
    • Convergence of the gradient sampling algorithm for nonsmooth nonconvex optimization
    • K.C. Kiwiel, Convergence of the gradient sampling algorithm for nonsmooth nonconvex optimization, SIAM J. Optim., 18 (2007), pp. 379-388.
    • (2007) SIAM J. Optim. , vol.18 , pp. 379-388
    • Kiwiel, K.C.1
  • 29
    • 0001083255 scopus 로고    scopus 로고
    • A trust region method for minimization of nonsmooth functions with linear constraints
    • J.M. Martínez and A.C. Moretti, A trust region method for minimization of nonsmooth functions with linear constraints, Math. Program., 76 (1997), pp. 431-449.
    • (1997) Math. Program , vol.76 , pp. 431-449
    • Martínez, J.M.1    Moretti, A.C.2
  • 30
    • 58149364313 scopus 로고
    • A trust-region strategy for minimization on arbitrary domains
    • J.M. Martínez and S.A. Santos, A trust-region strategy for minimization on arbitrary domains, Math. Program., 68 (1995), pp. 267-301.
    • (1995) Math. Program , vol.68 , pp. 267-301
    • Martínez, J.M.1    Santos, S.A.2
  • 32
    • 70350096085 scopus 로고
    • Large sample estimation and hypothesis testing
    • North-Holland, Amsterdam
    • W.K. Newey and D. McFadden, Large sample estimation and hypothesis testing, Handbook of Econometrics, Vol. IV, Handbooks in Econom. 2, North-Holland, Amsterdam, 1994, pp. 2111-2245.
    • (1994) Handbook of Econometrics, Handbooks in Econom. , vol.4 , Issue.2 , pp. 2111-2245
    • Newey, W.K.1    McFadden, D.2
  • 33
    • 17444406259 scopus 로고    scopus 로고
    • Smooth minimization of nonsmooth functions
    • Y. Nesterov, Smooth minimization of nonsmooth functions, Math. Program., 103 (2005), pp. 127-152.
    • (2005) Math. Program , vol.103 , pp. 127-152
    • Nesterov, Y.1
  • 34
    • 33947732413 scopus 로고    scopus 로고
    • Smoothing technique and its applications in semi definite optimization
    • Y. Nesterov, Smoothing technique and its applications in semi definite optimization, Math. Program., 110 (2007), pp. 245-259.
    • (2007) Math. Program , vol.110 , pp. 245-259
    • Nesterov, Y.1
  • 35
    • 84907772842 scopus 로고    scopus 로고
    • Efficient reconstruction of piecewise constant images using nonsmooth nonconvex minimization
    • M. Nikolova, M.K. Ng, S. Zhang, and W. Ching, Efficient reconstruction of piecewise constant images using nonsmooth nonconvex minimization, SIAM J. Imag. Sci., 1 (2008), pp. 2-25.
    • (2008) SIAM J. Imag. Sci. , vol.1 , pp. 2-25
    • Nikolova, M.1    Ng, M.K.2    Zhang, S.3    Ching, W.4
  • 38
    • 23644434856 scopus 로고
    • 1 optimization problems
    • L. Qi, Superlinearly convergent approximate Newton methods for LC1 optimization problems, Math. Program., 64 (1994), pp. 277-294.
    • (1994) Math. Program , vol.64 , pp. 277-294
    • Qi, L.1
  • 39
    • 0028480086 scopus 로고
    • A trust region algorithm for minimization of locally Lipschitzian functions
    • L. Qi and J. Sun, A trust region algorithm for minimization of locally Lipschitzian functions, Math. Program., 66 (1994), pp. 25-43.
    • (1994) Math. Program , vol.66 , pp. 25-43
    • Qi, L.1    Sun, J.2
  • 40
    • 0004267646 scopus 로고
    • Princeton University Press, Princeton, NJ
    • R.T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ, 1970.
    • (1970) Convex Analysis
    • Rockafellar, R.T.1
  • 42
    • 0024599612 scopus 로고
    • Prostate specific antigen in the diagnosis and treatment of adenocarcinoma of the prostate. II. Radical prostatectomy treated patients
    • T. Stamey, J. Kabalin, J. McNeal, I. Johnstone, F. Freiha, E. Redwine, and N. Yang, Prostate specific antigen in the diagnosis and treatment of adenocarcinoma of the prostate. II. Radical prostatectomy treated patients, J. Urology, 141 (1989), pp. 1076-1083.
    • (1989) J. Urology , vol.141 , pp. 1076-1083
    • Stamey, T.1    Kabalin, J.2    McNeal, J.3    Johnstone, I.4    Freiha, F.5    Redwine, E.6    Yang, N.7
  • 43
    • 85194972808 scopus 로고    scopus 로고
    • Regression shrinkage and selection via the lasso
    • R. Tibshirani, Regression shrinkage and selection via the lasso, J. Roy. Statist. Soc. B, 58 (1996), pp. 267-288.
    • (1996) J. Roy. Statist. Soc. B , vol.58 , pp. 267-288
    • Tibshirani, R.1
  • 44
    • 0022010767 scopus 로고
    • Conditions for convergence of a trust-region method for nonsmooth optimization
    • Y. Yuan, Conditions for convergence of a trust-region method for nonsmooth optimization, Math. Program., 31 (1985), pp. 220-228.
    • (1985) Math. Program , vol.31 , pp. 220-228
    • Yuan, Y.1
  • 45
    • 0022028162 scopus 로고
    • On the superliner convergence of a trust region algorithm for nonsmooth optimization
    • Y. Yuan, On the superliner convergence of a trust region algorithm for nonsmooth optimization, Math. Program., 31 (1985), pp. 269-285.
    • (1985) Math. Program , vol.31 , pp. 269-285
    • Yuan, Y.1
  • 46
    • 0347433923 scopus 로고    scopus 로고
    • On the truncated conjugate gradient method
    • Y. Yuan, On the truncated conjugate gradient method, Math. Program., 87 (2000), pp. 561-571.
    • (2000) Math. Program , vol.87 , pp. 561-571
    • Yuan, Y.1
  • 47
    • 77649284492 scopus 로고    scopus 로고
    • Nearly unbiased variable selection under minimax concave penalty
    • C.-H. Zhang, Nearly unbiased variable selection under minimax concave penalty, Ann. Statist., 38 (2010), pp. 894-942.
    • (2010) Ann. Statist. , vol.38 , pp. 894-942
    • Zhang, C.-H.1


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