메뉴 건너뛰기




Volumn 23, Issue 3, 2013, Pages 1718-1741

Worst-case complexity of smoothing quadratic regularization methods for non-lipschitzian optimization

Author keywords

Convergence; Nonsmooth nonconvex optimization; Quadratic regularization; Smoothing approximation; Stationary point; Worst case complexity

Indexed keywords

CONVERGENCE; NONSMOOTH NONCONVEX OPTIMIZATION; QUADRATIC REGULARIZATION; SMOOTHING APPROXIMATION; STATIONARY POINTS; WORST-CASE COMPLEXITY;

EID: 84886302151     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/120864908     Document Type: Article
Times cited : (48)

References (33)
  • 1
    • 0442312210 scopus 로고    scopus 로고
    • Regularization of wavelets approximations
    • A. Antoniadias and J. Fan, Regularization of wavelets approximations, J. Amer. Statist. Assoc., 96 (2001), pp. 929-967.
    • (2001) J. Amer. Statist. Assoc , vol.96 , pp. 929-967
    • Antoniadias, A.1    Fan, J.2
  • 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 Netw. Learning Syst., 23 (2012), pp. 399-411.
    • (2012) IEEE Trans. Neural Netw. Learning Syst , vol.23 , pp. 399-411
    • Bian, W.1    Chen, X.2
  • 3
    • 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
  • 4
    • 29144439194 scopus 로고    scopus 로고
    • Decoding by linear programming
    • E. Candes and T. Tao, Decoding by linear programming, IEEE Trans. Inform. Theory, 51 (2005), pp. 4203-4215.
    • (2005) IEEE Trans. Inform. Theory , vol.51 , pp. 4203-4215
    • Candes, E.1    Tao, T.2
  • 5
    • 57349174008 scopus 로고    scopus 로고
    • Enhancing sparsity by reweighted eell;1 minimization
    • E. Candes, M. Wakin, and S. Boyd, Enhancing sparsity by reweighted eell;1 minimization, J. Fourier Anal. Appl., 14 (2008), pp. 877-905.
    • (2008) J. Fourier Anal. Appl , Issue.14 , pp. 877-905
    • Candes, E.1    Wakin, M.2    Boyd, S.3
  • 6
    • 84856057771 scopus 로고    scopus 로고
    • On the evaluation complexity of composite function minimization with applications to nonconvex nonlinear programming
    • C. Cartis, 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
    • Cartis, C.1    Gould, N.I.M.2    Toint, L.3
  • 7
    • 79952763936 scopus 로고    scopus 로고
    • Adaptive cubic regularization methods for unconstrained optimization, part I: Motivation, convergence and numerical results
    • C. Cartis, N. I. M. Gould, and Ph. L. Toint, Adaptive cubic regularization methods for unconstrained optimization, Part I: Motivation, convergence and numerical results, Math. Program., 127 (2011), pp. 245-295.
    • (2011) Math. Program , vol.127 , pp. 245-295
    • Cartis, C.1    Gould, N.I.M.2    Toint, P.L.3
  • 8
    • 81255179401 scopus 로고    scopus 로고
    • Adaptive cubic regularization methods for unconstrained optimization, part II: Worst-case function- and derivative-evaluation complexity
    • C. Cartis, N. I. M. Gould, and Ph. L. Toint, Adaptive cubic regularization methods for unconstrained optimization, Part II: Worst-case function- and derivative-evaluation complexity, Math. Program., 130 (2011), pp. 295-319.
    • (2011) Math. Program , vol.130 , pp. 295-319
    • Cartis, C.1    Gould, N.I.M.2    Toint, P.L.3
  • 9
    • 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 Probl., 24 (2008), pp. 1-14.
    • (2008) Inverse Probl , vol.24 , pp. 1-14
    • Chartrand, R.1    Staneva, V.2
  • 10
    • 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
  • 11
    • 84903554150 scopus 로고    scopus 로고
    • Complexity of unconstrained L2-Lp minimization
    • X. Chen, D. Ge, Z. Wang, and Y. Ye, Complexity of unconstrained L2-Lp minimization, Math. Program., to appear.
    • Math. Program
    • Chen, X.1    Ge, D.2    Wang, Z.3    Ye, Y.4
  • 12
    • 84869480932 scopus 로고    scopus 로고
    • Nonconvex lp regularization and box constrained model for image restoration
    • X. Chen, M. Ng, and C. Zhang, Nonconvex lp regularization and box constrained model for image restoration, IEEE Trans. Image Process., 21 (2012), pp. 4709-4721.
    • (2012) IEEE Trans. Image Process , vol.21 , pp. 4709-4721
    • Chen, X.1    Ng, M.2    Zhang, C.3
  • 13
    • 78149326445 scopus 로고    scopus 로고
    • Lower bound theory of nonzero entries in solutions of l2-lp 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
  • 14
    • 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. Imaging Sci., 3 (2010), pp. 765-790.
    • (2010) SIAM J. Imaging Sci , vol.3 , pp. 765-790
    • Chen, X.1    Zhou, W.2
  • 16
    • 30844437386 scopus 로고    scopus 로고
    • Neighborly polytopes and sparse solutions of underdetermined linear equations
    • Stanford University, Stanford, CA
    • D. L. Donoho, Neighborly Polytopes and Sparse Solutions of Underdetermined Linear Equations, Technical report, Stanford University, Stanford, CA, 2005.
    • (2005) Technical Report
    • Donoho, D.L.1
  • 17
    • 0009935552 scopus 로고    scopus 로고
    • Comments on 'wavelets in stastics: A review' by A. Antoniadis
    • J. Fan, Comments on 'Wavelets in stastics: a review' by A. Antoniadis, Stat. Method. Appl., 6 (1997), pp. 131-138.
    • (1997) Stat. Method. Appl , vol.6 , pp. 131-138
    • Fan, J.1
  • 18
    • 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 (2001), pp. 1348-1360.
    • (2001) J. Amer. Statist. Assoc , vol.96 , pp. 1348-1360
    • Fan, J.1    Li, R.2
  • 19
    • 84886277792 scopus 로고    scopus 로고
    • Smoothing and worst case complexity for direct-search methods in non-smooth optimization
    • R. Garmanjani and L. N. Vicente, Smoothing and worst case complexity for direct-search methods in non-smooth optimization, IMA J. Numer. Anal., to appear.
    • IMA J. Numer. Anal
    • Garmanjani, R.1    Vicente, L.N.2
  • 20
    • 84865688967 scopus 로고    scopus 로고
    • A note on the complexity of Lp minimization
    • D. Ge, X. Jiang, and Y. Ye, A note on the complexity of Lp minimization, Math. Program., 21 (2011), pp. 1721-1739.
    • (2011) Math. Program , vol.21 , pp. 1721-1739
    • Ge, D.1    Jiang, X.2    Ye, Y.3
  • 21
    • 0002396293 scopus 로고
    • On the convergence rate of subgradient optimization methods
    • J.-L. Goffin, On the convergence rate of subgradient optimization methods, Math. Program., 13 (1977), pp. 329-347.
    • (1977) Math. Program , vol.13 , pp. 329-347
    • Goffin, J.-L.1
  • 22
    • 77951283936 scopus 로고    scopus 로고
    • A second derivative SQP method: Global convergence
    • N. I. M. Gould and D. P. Robinson, A second derivative SQP method: Global convergence, SIAM J. Optim., 20 (2010), pp. 2023-2048.
    • (2010) SIAM J. Optim , vol.20 , pp. 2023-2048
    • Gould, N.I.M.1    Robinson, D.P.2
  • 23
    • 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. Statist., 36 (2008), pp. 587-613.
    • (2008) Ann. Statist , vol.36 , pp. 587-613
    • Huang, J.1    Horowitz, J.L.2    Ma, S.3
  • 24
    • 66249102619 scopus 로고    scopus 로고
    • A group bridge approach for variable selection
    • J. Huang, S. Ma, H. Xue, and C. Zhang, A group bridge approach for variable selection, Biometrika, 96 (2009), pp. 339-355.
    • (2009) Biometrika , vol.96 , pp. 339-355
    • Huang, J.1    Ma, S.2    Xue, H.3    Zhang, C.4
  • 25
    • 85194972808 scopus 로고    scopus 로고
    • Shrinkage and selection via the Lasso
    • R. Tibshirani, Shrinkage and selection via the Lasso, J. Roy. Statist. Soc. Ser. B, 58 (1996), pp. 267-288.
    • (1996) J. Roy. Statist. Soc. Ser. B , vol.58 , pp. 267-288
    • Tibshirani, R.1
  • 26
    • 17444406259 scopus 로고    scopus 로고
    • Smooth minimization of non-smooth functions
    • Yu. Nesterov, Smooth minimization of non-smooth functions, Math. Program., 103 (2005),pp. 127-152.
    • (2005) Math. Program , vol.103 , pp. 127-152
    • Nesterov, Yu.1
  • 27
    • 33947611624 scopus 로고    scopus 로고
    • Modified Gauss-Newton scheme with worst-case guarantees for global performance
    • Yu. Nesterov, Modified Gauss-Newton scheme with worst-case guarantees for global performance, Optim. Method Softw., 22 (2007), pp. 413-431.
    • (2007) Optim. Method Softw , vol.22 , pp. 413-431
    • Nesterov, Yu.1
  • 28
    • 33646730150 scopus 로고    scopus 로고
    • Cubic regularization of Newton's method and its global performance
    • Yu. Nesterov and B. T. Polyak, Cubic regularization of Newton's method and its global performance, Math. Program., 108 (2006), pp. 177-205.
    • (2006) Math. Program , vol.108 , pp. 177-205
    • Nesterov, Yu.1    Polyak, B.T.2
  • 29
    • 84907772842 scopus 로고    scopus 로고
    • Efficient reconstruction of piecewise constant images using nonsmooth nonconvex minimization
    • M. Nikolova, M. K. Ng, S. Zhang, and W.-K. Ching, Efficient reconstruction of piecewise constant images using nonsmooth nonconvex minimization, SIAM J. Imaging Sci., 1 (2008), pp. 2-25.
    • (2008) SIAM J. Imaging Sci , vol.1 , pp. 2-25
    • Nikolova, M.1    Ng, M.K.2    Zhang, S.3    Ching, W.-K.4
  • 31
    • 85194972808 scopus 로고    scopus 로고
    • Regression shrinkage and selection via the Lasso
    • R. J. Tibshirani, Regression shrinkage and selection via the Lasso, J. Roy. Statist. Soc. Ser. B, 58 (1996), pp. 267-288.
    • (1996) J. Roy. Statist. Soc. Ser. B , vol.58 , pp. 267-288
    • Tibshirani, R.J.1
  • 33
    • 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


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.