메뉴 건너뛰기




Volumn 134, Issue 1, 2012, Pages 71-99

Smoothing methods for nonsmooth, nonconvex minimization

Author keywords

Eigenvalue optimization; Nonconvex minimization; Nonsmooth; Regularized minimization problems; Smoothing methods; Stochastic variational inequality problems

Indexed keywords

EIGENVALUE OPTIMIZATION; NON-CONVEX MINIMIZATION; NON-SMOOTH; REGULARIZED MINIMIZATION PROBLEMS; SMOOTHING METHODS; STOCHASTIC VARIATIONAL INEQUALITY PROBLEMS;

EID: 84865682357     PISSN: 00255610     EISSN: 14364646     Source Type: Journal    
DOI: 10.1007/s10107-012-0569-0     Document Type: Conference Paper
Times cited : (297)

References (107)
  • 1
    • 37549072235 scopus 로고    scopus 로고
    • A regularized projection method for complementarity problems with non-Lipschitzian functions
    • Alefeld, G., Chen, X.: A regularized projection method for complementarity problems with non-Lipschitzian functions. Math. Comput. 77, 379-395 (2008)
    • (2008) Math. Comput. , vol.77 , pp. 379-395
    • Alefeld, G.1    Chen, X.2
  • 2
    • 0002281861 scopus 로고    scopus 로고
    • How to deal with the unbounded in optimization: Theory and algorithm
    • Auslender, A.: How to deal with the unbounded in optimization: Theory and algorithm. Math. Program. 79, 3-18 (1997)
    • (1997) Math. Program. , vol.79 , pp. 3-18
    • Auslender, A.1
  • 4
    • 84865698786 scopus 로고    scopus 로고
    • Smoothing neural network for constrained non-Lipschitz optimizationwith applications
    • Bian,W., Chen, X.: Smoothing neural network for constrained non-Lipschitz optimizationwith applications. IEEE Trans. Neural Netw. Learn. Syst. 23, 399-411 (2012)
    • (2012) IEEE Trans. Neural Netw. Learn. Syst. , vol.23 , pp. 399-411
    • Bian, W.1    Chen, X.2
  • 7
    • 67649321486 scopus 로고    scopus 로고
    • Subgradient-based neural network for nonsmooth nonconvex optimization problem
    • Bian, W., Xue, X.P.: Subgradient-based neural network for nonsmooth nonconvex optimization problem. IEEE Trans. Neural Netw. 20, 1024-1038 (2009)
    • (2009) IEEE Trans. Neural Netw. , vol.20 , pp. 1024-1038
    • Bian, W.1    Xue, X.P.2
  • 8
    • 59749104367 scopus 로고    scopus 로고
    • From sparse solutions of systems of equations to sparse modeling of signals and images
    • Bruckstein, A.M., Donoho, D.L., Elad, M.: From sparse solutions of systems of equations to sparse modeling of signals and images. SIAM Rev. 51, 34-81 (2009)
    • (2009) SIAM Rev , vol.51 , pp. 34-81
    • Bruckstein, A.M.1    Donoho, D.L.2    Elad, M.3
  • 9
    • 0022361737 scopus 로고
    • Descent methods for composite nondifferentiable optimization problems
    • Burke, J.V.: Descent methods for composite nondifferentiable optimization problems. Math. Program. 33, 260-279 (1985)
    • (1985) Math. Program. , vol.33 , pp. 260-279
    • Burke, J.V.1
  • 10
    • 23844446458 scopus 로고    scopus 로고
    • A robust gradient sampling algorithm for nonsmooth, nonconvex optimization
    • Burke, J.V., Lewis, A.S., Overton, M.L.: A robust gradient sampling algorithm for nonsmooth, nonconvex optimization. SIAM J. Optim. 15, 751-779 (2005)
    • (2005) SIAM J. Optim. , vol.15 , pp. 751-779
    • Burke, J.V.1    Lewis, A.S.2    Overton, M.L.3
  • 12
    • 84856057771 scopus 로고    scopus 로고
    • On the evaluation complexity of composite function minimization with applications to nonconvex nonlinear programming
    • Catis, C., Gould, N.I.M., Toint, P.: On the evaluation complexity of composite function minimization with applications to nonconvex nonlinear programming. SIAM J. Optim. 21, 1721-1739 (2011)
    • (2011) SIAM J. Optim. , vol.21 , pp. 1721-1739
    • Catis, C.1    Gould, N.I.M.2    Toint, P.3
  • 13
    • 34548724437 scopus 로고    scopus 로고
    • Exact reconstruction of sparse signals via nonconvex minimization
    • Chartrand, R.: Exact reconstruction of sparse signals via nonconvex minimization. IEEE Signal Proc. Lett. 14, 707-710 (2007)
    • (2007) IEEE Signal Proc. Lett. , vol.14 , pp. 707-710
    • Chartrand, R.1
  • 14
    • 0039938688 scopus 로고    scopus 로고
    • A global and local superlinear continuation-smoothing method for P0 and R0 NCP or monotone NCP
    • Chen, B., Chen, X.: A global and local superlinear continuation-smoothing method for P0 and R0 NCP or monotone NCP. SIAM J. Optim. 9, 624-645 (1999)
    • (1999) SIAM J. Optim. , vol.9 , pp. 624-645
    • Chen, B.1    Chen, X.2
  • 15
    • 21344485287 scopus 로고
    • A non-interior-oint continuation method for linear complementarity problems
    • Chen, B., Harker, P.T.: A non-interior-oint continuation method for linear complementarity problems. SIAM J. Matrix Anal. Appl. 14, 1168-1190 (1993)
    • (1993) SIAM J. Matrix Anal. Appl. , vol.14 , pp. 1168-1190
    • Chen, B.1    Harker, P.T.2
  • 16
    • 0040531597 scopus 로고    scopus 로고
    • A global linear and local quadratic noninterior continuation method for nonlinear complementarity problems based on Chen-Mangasarian smoothing functions
    • Chen, B., Xiu, N.: A global linear and local quadratic noninterior continuation method for nonlinear complementarity problems based on Chen-Mangasarian smoothing functions. SIAM J. Optim. 9, 605-623 (1999)
    • (1999) SIAM J. Optim. , vol.9 , pp. 605-623
    • Chen, B.1    Xiu, N.2
  • 17
    • 0029206129 scopus 로고
    • A class of smoothing functions for nonlinear and mixed complementarity problems
    • Chen, C., Mangasarian, O.L.: A class of smoothing functions for nonlinear and mixed complementarity problems. Math. Program. 71, 51-70 (1995)
    • (1995) Math. Program. , vol.71 , pp. 51-70
    • Chen, C.1    Mangasarian, O.L.2
  • 18
    • 0034148176 scopus 로고    scopus 로고
    • Smoothing methods for complementarity problems and their applications: A survey
    • Chen, X.: Smoothing methods for complementarity problems and their applications: A survey. J. Oper. Res. Soc. Japan 43, 32-46 (2000)
    • (2000) J. Oper. Res. Soc. Japan , vol.43 , pp. 32-46
    • Chen, X.1
  • 19
    • 33747151978 scopus 로고    scopus 로고
    • A Superlinearly and globally convergent method for reaction and diffusion problems with a non-lipschitzian operator
    • Chen, X.: A superlinearly and globally convergent method for reaction and diffusion problems with a non-Lipschitzian operator. Computing[Suppl] 15, 79-90 (2001)
    • (2001) Computing[Suppl] , vol.15 , pp. 79-90
    • Chen, X.1
  • 20
    • 10244235403 scopus 로고    scopus 로고
    • First order conditions for nonsmooth discretized constrained optimal control problems
    • Chen, X.: First order conditions for nonsmooth discretized constrained optimal control problems. SIAM J. Control Optim. 42, 2004-2015 (2004)
    • (2004) SIAM J. Control Optim. , vol.42 , pp. 2004-2015
    • Chen, X.1
  • 21
    • 31144459305 scopus 로고    scopus 로고
    • Expected residual minimization method for stochastic linear complementarity problems
    • Chen, X., Fukushima, M.: Expected residual minimization method for stochastic linear complementarity problems. Math. Oper. Res. 30, 1022-1038 (2005)
    • (2005) Math. Oper. Res. , vol.30 , pp. 1022-1038
    • Chen, X.1    Fukushima, M.2
  • 22
    • 1342303387 scopus 로고    scopus 로고
    • A smoothing method for a mathematical program with P-matrix linear complementarity constraints
    • Chen, X., Fukushima, M.: A smoothing method for a mathematical program with P-matrix linear complementarity constraints. Comput. Optim. Appl. 27, 223-246 (2004)
    • (2004) Comput. Optim. Appl. , vol.27 , pp. 223-246
    • Chen, X.1    Fukushima, M.2
  • 24
    • 0034432264 scopus 로고    scopus 로고
    • Smoothing methods and semismooth methods for nondifferentiable operator equations
    • Chen, X., Nashed, Z., Qi, L.: Smoothing methods and semismooth methods for nondifferentiable operator equations. SIAM J. Numer. Anal. 38, 1200-1216 (2000)
    • (2000) SIAM J. Numer. Anal. , vol.38 , pp. 1200-1216
    • Chen, X.1    Nashed, Z.2    Qi, L.3
  • 26
    • 0028436825 scopus 로고
    • A parameterized Newton method and a quasi-Newton method for nonsmooth equations
    • Chen, X., Qi, L.: A parameterized Newton method and a quasi-Newton method for nonsmooth equations. Comput. Optim. Appl. 3, 157-179 (1994)
    • (1994) Comput. Optim. Appl. , vol.3 , pp. 157-179
    • Chen, X.1    Qi, L.2
  • 27
    • 0032380644 scopus 로고    scopus 로고
    • Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities
    • Chen, X., Qi, L., Sun, D.: Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities.Math. Comput. 67, 519-540 (1998)
    • (1998) Math. Comput. , vol.67 , pp. 519-540
    • Chen, X.1    Qi, L.2    Sun, D.3
  • 28
    • 84865693836 scopus 로고    scopus 로고
    • Stochastic variational inequalities: Residual minimization smoothing sample average approximations
    • to appear
    • Chen, X.,Wets, R.J-B., Zhang, Y.: Stochastic variational inequalities: Residual minimization smoothing/sample average approximations SIAM J. Optim. (to appear)
    • SIAM J. Optim
    • Chen, X.1    Wets, R.J.-B.2    Zhang, Y.3
  • 29
    • 79957506054 scopus 로고    scopus 로고
    • Minimizing the condition number of a Gram matrix
    • Chen, X., Womersley, R., Ye, J.: Minimizing the condition number of a Gram matrix. SIAM J. Optim. 21, 127-148 (2011)
    • (2011) SIAM J. Optim. , vol.21 , pp. 127-148
    • Chen, X.1    Womersley, R.2    Ye, J.3
  • 30
    • 78149326445 scopus 로고    scopus 로고
    • Lower bound theory of nonzero entries in solutions of l2 l p minimization
    • Chen, X., Xu, F., Ye, Y.: Lower bound theory of nonzero entries in solutions of l2 l p minimization. SIAM J. Sci. Comput. 32, 2832-2852 (2012)
    • (2012) SIAM J. Sci. Comput. , vol.32 , pp. 2832-2852
    • Chen, X.1    Xu, F.2    Ye, Y.3
  • 31
    • 0032201688 scopus 로고    scopus 로고
    • On homotopy-smoothing methods for variational inequalities
    • Chen, X., Ye, Y.: On homotopy-smoothing methods for variational inequalities. SIAM J. Control Optim. 37, 587-616 (1999)
    • (1999) SIAM J. Control Optim. , vol.37 , pp. 587-616
    • Chen, X.1    Ye, Y.2
  • 32
    • 46749143296 scopus 로고    scopus 로고
    • Robust solution of monotone stochastic linear complementarity problems
    • Chen, X., Zhang, C., Fukushima,M.: Robust solution of monotone stochastic linear complementarity problems. Math. Program. 117, 51-80 (2009)
    • (2009) Math. Program. , vol.117 , pp. 51-80
    • Chen, X.1    Zhang, C.2    Fukushima, M.3
  • 33
    • 78149291090 scopus 로고    scopus 로고
    • Smoothing nonlinear conjugate gradient method for image restoration using nonsmooth nonconvex minimization
    • Chen, X., Zhou, W.: Smoothing nonlinear conjugate gradient method for image restoration using nonsmooth nonconvex minimization. SIAM J. Imaging Sci. 3, 765-790 (2010)
    • (2010) SIAM J. Imaging Sci. , vol.3 , pp. 765-790
    • Chen, X.1    Zhou, W.2
  • 37
    • 70450219115 scopus 로고    scopus 로고
    • Identifying structure of nonsmooth convex functions by the bundle technique
    • Daniilidis, A., Sagastizbal, C., Solodov, M.: Identifying structure of nonsmooth convex functions by the bundle technique. SIAM J. Optim. 20, 820-840 (2009)
    • (2009) SIAM J. Optim. , vol.20 , pp. 820-840
    • Daniilidis, A.1    Sagastizbal, C.2    Solodov, M.3
  • 38
    • 77953562445 scopus 로고    scopus 로고
    • Distributionally robust optimization under monment uncertainty with application to data-driven problems
    • Delage, E., Ye, Y.: Distributionally robust optimization under monment uncertainty with application to data-driven problems. Oper. Res. 58, 595-612 (2010)
    • (2010) Oper. Res. , vol.58 , pp. 595-612
    • Delage, E.1    Ye, Y.2
  • 39
    • 0000425448 scopus 로고    scopus 로고
    • A smoothing method for mathematical programs with equilibrium constraints
    • Facchinei, F., Jiang, H., Qi, L.: A smoothing method for mathematical programs with equilibrium constraints. Math. Program. 85, 107-134 (1999)
    • (1999) Math. Program. , vol.85 , pp. 107-134
    • Facchinei, F.1    Jiang, H.2    Qi, L.3
  • 41
    • 0009935552 scopus 로고    scopus 로고
    • Comments on "Wavelets in Statistic: A review"i by A. Antoniadis
    • Fan, J.: Comments on "Wavelets in Statistic: A review" by A. Antoniadis. Stat. Method. Appl. 6, 131-138 (1997)
    • (1997) Stat. Method. Appl. , vol.6 , pp. 131-138
    • Fan, J.1
  • 42
    • 1542784498 scopus 로고    scopus 로고
    • Variable selection via nonconcave penalized likelihood and its oracle properties
    • Fan, J., Li, R.: Variable selection via nonconcave penalized likelihood and its oracle properties. J. Am. Stat. Assoc. 96, 1348-1360 (2001)
    • (2001) J. Am. Stat. Assoc. , vol.96 , pp. 1348-1360
    • Fan, J.1    Li, R.2
  • 43
    • 42449155183 scopus 로고    scopus 로고
    • Stochastic R0 matrix linear complementarity problems
    • Fang, H., Chen, X., Fukushima, M.: Stochastic R0 matrix linear complementarity problems. SIAM J. Optim. 18, 482-506 (2007)
    • (2007) SIAM J. Optim. , vol.18 , pp. 482-506
    • Fang, H.1    Chen, X.2    Fukushima, M.3
  • 44
    • 84865692730 scopus 로고    scopus 로고
    • Extended mathematical programming: Competition and stochasticity
    • Ferris, M.: Extended mathematical programming: Competition and stochasticity. SIAM News 45, 1-2 (2012)
    • (2012) SIAM News , vol.45 , pp. 1-2
    • Ferris, M.1
  • 45
    • 84948263182 scopus 로고
    • A special Newton-type optimization method
    • Fischer, A.: A special Newton-type optimization method. Optimization 24, 269-284 (1992)
    • (1992) Optimization , vol.24 , pp. 269-284
    • Fischer, A.1
  • 46
    • 0032047752 scopus 로고    scopus 로고
    • A globally convergent sequential quadratic programming algorithm for mathematical programs with linear complementarity constraints
    • Fukushima, M., Luo, Z.-Q., Pang, J.-S.: A globally convergent sequential quadratic programming algorithm for mathematical programs with linear complementarity constraints. Comput. Optim. Appl. 10, 5-34 (1998)
    • (1998) Comput. Optim. Appl. , vol.10 , pp. 5-34
    • Fukushima, M.1    Luo, Z.-Q.2    Pang, J.-S.3
  • 47
    • 0036013023 scopus 로고    scopus 로고
    • Smoothing functions for second-order-cone complementarity problems
    • Fukushima, M., Luo, Z.-Q., Tseng, P.: Smoothing functions for second-order-cone complementarity problems. SIAM J. Optim. 12, 436-460 (2002)
    • (2002) SIAM J. Optim. , vol.12 , pp. 436-460
    • Fukushima, M.1    Luo, Z.-Q.2    Tseng, P.3
  • 48
    • 0001787420 scopus 로고    scopus 로고
    • Convergence of a smoothing continuation method for mathematical programs with complementarity constraints
    • In: Thera, M., Tichatschke, R. (eds.) Springer, Berlin
    • Fukushima, M., Pang, J.-S.: Convergence of a smoothing continuation method for mathematical programs with complementarity constraints. In: Thera, M., Tichatschke, R. (eds.) Lecture Notes in Economics and Mathematical Systems, vol. 477, pp. 99-110. Springer, Berlin (1999)
    • (1999) Lecture Notes in Economics and Mathematical Systems , vol.477
    • Fukushima, M.1    Pang, J.-S.2
  • 49
    • 0002579086 scopus 로고    scopus 로고
    • Smoothing of mixed complementarity problems
    • In: Ferris, M.C., Pang, J.S. (eds.) SIAM, Philadelphia
    • Gabriel S.A., More J.J.: Smoothing of mixed complementarity problems. In: Ferris, M.C., Pang, J.S. (eds.) Complementarity and Variational Problems: State of the Art, pp. 105-116. SIAM, Philadelphia (1997)
    • (1997) Complementarity and Variational Problems: State of the Art , pp. 105-116
    • Gabriel, S.A.1    More, J.J.2
  • 51
    • 84865688967 scopus 로고    scopus 로고
    • A note on the complexity of L p minimization
    • Ge, D., Jiang, X., Ye, Y.: A note on the complexity of L p minimization. Math. Program. 21, 1721-1739 (2011)
    • (2011) Math. Program. , vol.21 , pp. 1721-1739
    • Ge, D.1    Jiang, X.2    Ye, Y.3
  • 52
    • 0026835074 scopus 로고
    • Constrained restoration and the recovery of discontinuities
    • Geman, D., Reynolds, G.: Constrained restoration and the recovery of discontinuities. IEEE Trans. Pattern Anal. Mach. Intell. 14, 357-383 (1992)
    • (1992) IEEE Trans. Pattern Anal. Mach. Intell. , vol.14 , pp. 357-383
    • Geman, D.1    Reynolds, G.2
  • 53
    • 0007954921 scopus 로고    scopus 로고
    • Sample-ath solution of stochastic variational inequalities
    • Gürkan, G., Özge, A.Y., Robinson, S.M.: Sample-ath solution of stochastic variational inequalities. Math. Program. 84, 313-333 (1999)
    • (1999) Math. Program. , vol.84 , pp. 313-333
    • Gürkan, G.1    Özge, A.Y.2    Robinson, S.M.3
  • 54
    • 80054958066 scopus 로고    scopus 로고
    • Pricing American options with uncertain volatility through stochastic linear complementarity models
    • Hamatani, K., Fukushima, M.: Pricing American options with uncertain volatility through stochastic linear complementarity models. Comput. Optim. Appl. 50, 263-286 (2011)
    • (2011) Comput. Optim. Appl. , vol.50 , pp. 263-286
    • Hamatani, K.1    Fukushima, M.2
  • 56
    • 18744383641 scopus 로고    scopus 로고
    • A combined smoothing and regularization method for monotone second-order cone complementarity problems
    • Hayashi, S., Yamashita, N., Fukushima, M.: A combined smoothing and regularization method for monotone second-order cone complementarity problems. SIAM J. Optim. 15, 593-615 (2005)
    • (2005) SIAM J. Optim. , vol.15 , pp. 593-615
    • Hayashi, S.1    Yamashita, N.2    Fukushima, M.3
  • 57
    • 0026993855 scopus 로고
    • Fast algorithms for nonsmooth compact fixed-oint problems
    • Heinkenschloss, M., KelleyC.T. Tran, H.T.: Fast algorithms for nonsmooth compact fixed-oint problems. SIAM J. Numer. Anal. 29, 1769-1792 (1992)
    • (1992) SIAM J. Numer. Anal. , vol.29 , pp. 1769-1792
    • Heinkenschloss, M.1    Kelley, C.T.2    Tran, H.T.3
  • 60
    • 84861910612 scopus 로고    scopus 로고
    • Smoothing approach to Nash equilibrium formulations for a class of equilibrium problems with shared complementarity constraints
    • (to appear
    • Hu, M., Fukushima, M.: Smoothing approach to Nash equilibrium formulations for a class of equilibrium problems with shared complementarity constraints. Comput. Optim. Appl. (to appear)
    • Comput. Optim. Appl
    • Hu, M.1    Fukushima, M.2
  • 61
    • 49949115667 scopus 로고    scopus 로고
    • Asymptotic properties of bridge estimators in sparse high-dimensional regression models
    • Huang, J., Horowitz, J.L., Ma, S.: Asymptotic properties of bridge estimators in sparse high-dimensional regression models. Ann. Stat. 36, 587-613 (2008)
    • (2008) Ann. Stat. , vol.36 , pp. 587-613
    • Huang, J.1    Horowitz, J.L.2    Ma, S.3
  • 62
    • 66249102619 scopus 로고    scopus 로고
    • A group bridge approach for variable selection
    • Huang, J., Ma, S., Xie, H., Zhang, C.-H.: A group bridge approach for variable selection. Biometrika 96, 339-355 (2009)
    • (2009) Biometrika , vol.96 , pp. 339-355
    • Huang, J.1    Ma, S.2    Xie, H.3    Zhang, C.-H.4
  • 63
    • 12244279249 scopus 로고    scopus 로고
    • Sub-quadratic convergence of a smoothing Newton algorithm for the P0-and monotone LCP
    • Huang, Z., Qi, L., Sun, D.: Sub-quadratic convergence of a smoothing Newton algorithm for the P0-and monotone LCP. Math. Program. 99, 423-441 (2004)
    • (2004) Math. Program. , vol.99 , pp. 423-441
    • Huang, Z.1    Qi, L.2    Sun, D.3
  • 64
    • 0037394962 scopus 로고    scopus 로고
    • Smooth SQP methods for mathematical programs with nonlinear complementarity constraints
    • Jiang, H., Ralph, D.: Smooth SQP methods for mathematical programs with nonlinear complementarity constraints. Comput. Optim. Appl. 25, 123-150 (2002)
    • (2002) Comput. Optim. Appl. , vol.25 , pp. 123-150
    • Jiang, H.1    Ralph, D.2
  • 65
    • 51449092038 scopus 로고    scopus 로고
    • Stochastic approximation approaches to the stochastic variational inequality problem
    • Jiang, H., Xu, H.: Stochastic approximation approaches to the stochastic variational inequality problem. IEEE. Trans. Autom. Control 53, 1462-1475 (2008)
    • (2008) IEEE. Trans. Autom. Control , vol.53 , pp. 1462-1475
    • Jiang, H.1    Xu, H.2
  • 66
    • 0030517292 scopus 로고    scopus 로고
    • Some noninterior continuation methods for linear cornplementarity problems
    • Kanzow, C.: Some noninterior continuation methods for linear cornplementarity problems. SIAM J. Matrix Anal. Appl. 17, 851-868 (1996)
    • (1996) SIAM J. Matrix Anal. Appl. , vol.17 , pp. 851-868
    • Kanzow, C.1
  • 67
    • 0034287156 scopus 로고    scopus 로고
    • Asymptotics for lasso-type estimators
    • Knight, K., Fu, W.J.: Asymptotics for lasso-type estimators. Ann. Stat. 28, 1356-1378 (2000)
    • (2000) Ann. Stat. , vol.28 , pp. 1356-1378
    • Knight, K.1    Fu, W.J.2
  • 69
    • 44649165114 scopus 로고    scopus 로고
    • Convergence of the gradient sampling algorithm for nonsmooth nonconvex optimization
    • Kiwiel, K.C.: Convergence of the gradient sampling algorithm for nonsmooth nonconvex optimization. SIAM J. Optim. 18, 379-388 (2007)
    • (2007) SIAM J. Optim. , vol.18 , pp. 379-388
    • Kiwiel, K.C.1
  • 70
    • 84865678322 scopus 로고    scopus 로고
    • Lipschitz behavior of the robust regularization
    • Lewis, A.S., Pang, C.H.J.: Lipschitz behavior of the robust regularization. SIAM, J. Control Optim. 48, 3080-3104 (2010)
    • (2010) SIAM, J. Control Optim. , vol.48 , pp. 3080-3104
    • Lewis, A.S.1    Pang, C.H.J.2
  • 71
  • 72
    • 0034481503 scopus 로고    scopus 로고
    • Smoothing Newton and quasi-Newton methods for mixed complementarity problems
    • Li, D., Fukushima, M.: Smoothing Newton and quasi-Newton methods for mixed complementarity problems. Comput. Optim. Appl. 17, 203-230 (2000)
    • (2000) Comput. Optim. Appl. , vol.17 , pp. 203-230
    • Li, D.1    Fukushima, M.2
  • 73
    • 45749085726 scopus 로고    scopus 로고
    • Solving stochastic mathematical programs with equilibrium constraints via approximation and smoothing implicit programming with penalization
    • Lin, G.H., Chen, X., Fukushima, M.: Solving stochastic mathematical programs with equilibrium constraints via approximation and smoothing implicit programming with penalization. Math. Program. 116, 343-368 (2009)
    • (2009) Math. Program. , vol.116 , pp. 343-368
    • Lin, G.H.1    Chen, X.2    Fukushima, M.3
  • 74
    • 78049451897 scopus 로고    scopus 로고
    • Stochastic equilibrium problems and stochastic mathematical programs with equilibrium constraints: A survey
    • Lin, G.H., Fukushima, M.: Stochastic equilibrium problems and stochastic mathematical programs with equilibrium constraints: A survey. Pac. J. Optim. 6, 455-482 (2010)
    • (2010) Pac. J. Optim. , vol.6 , pp. 455-482
    • Lin, G.H.1    Fukushima, M.2
  • 75
    • 0041540601 scopus 로고    scopus 로고
    • Exact penalization and stationarity conditions of mathematical programs with equilibrium constraints
    • Luo, Z-Q., Pang, J-S., Ralph, D., Wu, S.: Exact penalization and stationarity conditions of mathematical programs with equilibrium constraints. Math. Program. 75, 19-76 (1996)
    • (1996) Math. Program. , vol.75 , pp. 19-76
    • Luo, Z.-Q.1    Pang, J.-S.2    Ralph, D.3    Wu, S.4
  • 76
    • 70450205215 scopus 로고    scopus 로고
    • Optimizing condition numbers
    • Maréchal, P., Ye, J.J.: Optimizing condition numbers. SIAM J. Optim. 20, 935-947 (2009)
    • (2009) SIAM J. Optim. , vol.20 , pp. 935-947
    • Maréchal, P.1    Ye, J.J.2
  • 77
    • 0001083255 scopus 로고    scopus 로고
    • A trust region method for minimization of nonsmooth functions with linear constraints
    • Martínez, J.M., Moretti, A.C.: A trust region method for minimization of nonsmooth functions with linear constraints. Math. Program. 76, 431-449 (1997)
    • (1997) Math. Program. , vol.76 , pp. 431-449
    • Martínez, J.M.1    Moretti, A.C.2
  • 78
    • 79251502006 scopus 로고    scopus 로고
    • Optimality conditions via exact penalty functions
    • Meng, K., Yang, X.: Optimality conditions via exact penalty functions. SIAM J. Optim. 20, 3205-3231 (2010)
    • (2010) SIAM J. Optim. , vol.20 , pp. 3205-3231
    • Meng, K.1    Yang, X.2
  • 79
    • 0000012763 scopus 로고
    • Semismooth and semiconvex functions in constrained optimization
    • Mifflin, R.: Semismooth and semiconvex functions in constrained optimization. SIAM J. Control Optim. 15, 957-972 (1977)
    • (1977) SIAM J. Control Optim. , vol.15 , pp. 957-972
    • Mifflin, R.1
  • 81
    • 17444406259 scopus 로고    scopus 로고
    • Smooth minimization of nonsmooth functions
    • Nesterov, Y.: Smooth minimization of nonsmooth functions. Math. Program. 103, 127-152 (2005)
    • (2005) Math. Program. , vol.103 , pp. 127-152
    • Nesterov, Y.1
  • 82
    • 33947732413 scopus 로고    scopus 로고
    • Smoothing technique and its applications in semidefinite optimization
    • Nesterov, Y.: Smoothing technique and its applications in semidefinite optimization. Math. Program. 110, 245-259 (2007)
    • (2007) Math. Program. , vol.110 , pp. 245-259
    • Nesterov, Y.1
  • 83
    • 70350096085 scopus 로고
    • Large sample estimation and hypothesis testing
    • North-Holland, Amsterdam
    • Newey, W.K., McFadden, D.: Large sample estimation and hypothesis testing. In: Handbook of econometrics, vol. IV, pp. 2111-2245. North-Holland, Amsterdam (1994)
    • (1994) Handbook of Econometrics , vol.4 , pp. 2111-2245
    • Newey, W.K.1    McFadden, D.2
  • 84
    • 33748416181 scopus 로고    scopus 로고
    • Analysis of the recovery of edges in images and signals by minimizing nonconvex regularized least-squares
    • Nikolova, M.: Analysis of the recovery of edges in images and signals by minimizing nonconvex regularized least-squares. SIAM J. Multiscale Model. Simul. 4, 960-991 (2005)
    • (2005) SIAM J. Multiscale Model. Simul. , vol.4 , pp. 960-991
    • Nikolova, M.1
  • 85
    • 84907772842 scopus 로고    scopus 로고
    • Efficient reconstruction of piecewise constant images using nonsmooth nonconvex minimization
    • Nikolova,M., Ng,M.K., Zhang, S., Ching,W.: Efficient reconstruction of piecewise constant images using nonsmooth nonconvex minimization. SIAM J. Imaging Sci. 1, 2-25 (2008)
    • (2008) SIAM J. Imaging Sci. , vol.1 , pp. 2-25
    • Nikolova, M.1    Ng, M.K.2    Zhang, S.3    Ching, W.4
  • 88
    • 4043111445 scopus 로고    scopus 로고
    • A nonlinear rescaling vs. smoothing technique in convex optimization
    • Polyak, R.A.: A nonlinear rescaling vs. smoothing technique in convex optimization. Math. Program. 92, 197-235 (2002)
    • (2002) Math. Program. , vol.92 , pp. 197-235
    • Polyak, R.A.1
  • 89
    • 0033630888 scopus 로고    scopus 로고
    • A smoothing Newton method for extended vertical linear complementarity problems
    • Qi, H.D., Liao, L.Z.: A smoothing Newton method for extended vertical linear complementarity problems. SIAM J. Matrix Anal. Appl. 21, 45-66 (1999)
    • (1999) SIAM J. Matrix Anal. Appl. , vol.21 , pp. 45-66
    • Qi, H.D.1    Liao, L.Z.2
  • 90
    • 0029276590 scopus 로고
    • A globally convergent successive approximation method for severely nonsmooth equations
    • Qi, L., Chen, X.: A globally convergent successive approximation method for severely nonsmooth equations. SIAM J. Control Optim. 33, 402-418 (1995)
    • (1995) SIAM J. Control Optim. , vol.33 , pp. 402-418
    • Qi, L.1    Chen, X.2
  • 91
    • 58149507007 scopus 로고    scopus 로고
    • A smoothing projected Newton-type algorithm for semi-infinite programming
    • Qi, L., Ling, C., Tong, X., Zhou, G.: A smoothing projected Newton-type algorithm for semi-infinite programming. Comput. Optim. Appl. 42, 1-30 (2009)
    • (2009) Comput. Optim. Appl. , vol.42 , pp. 1-30
    • Qi, L.1    Ling, C.2    Tong, X.3    Zhou, G.4
  • 92
    • 0002287357 scopus 로고    scopus 로고
    • A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities
    • Qi, L., Sun, D., Zhou, G.: A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities. Math. Program. 87, 1-35 (2000)
    • (2000) Math. Program. , vol.87 , pp. 1-35
    • Qi, L.1    Sun, D.2    Zhou, G.3
  • 93
    • 0037393283 scopus 로고    scopus 로고
    • A property of piecewise smooth functions
    • Rockafellar, R.T.: A property of piecewise smooth functions. Comput. Optim. Appl. 25, 247-250 (2003)
    • (2003) Comput. Optim. Appl. , vol.25 , pp. 247-250
    • Rockafellar, R.T.1
  • 97
    • 4043052308 scopus 로고    scopus 로고
    • A squared smoothing Newton method for nonsmooth matrix equations and its applications in semidefinite optimization problems
    • Sun, J., Sun, D., Qi, L.: A squared smoothing Newton method for nonsmooth matrix equations and its applications in semidefinite optimization problems. SIAM J. Optim. 14, 783-806 (2004)
    • (2004) SIAM J. Optim. , vol.14 , pp. 783-806
    • Sun, J.1    Sun, D.2    Qi, L.3
  • 99
    • 9944241576 scopus 로고
    • Convergence of an inexact algorithm for composite nonsmooth optimization
    • Wright, S.J.: Convergence of an inexact algorithm for composite nonsmooth optimization. IMA J. Numer. Anal. 9, 299-321 (1990)
    • (1990) IMA J. Numer. Anal. , vol.9 , pp. 299-321
    • Wright, S.J.1
  • 100
    • 77952188559 scopus 로고    scopus 로고
    • Sample average approximation methods for a class of stochastic variational inequality problems
    • Xu, H.: Sample average approximation methods for a class of stochastic variational inequality problems. Asia Pac. J. Oper. Res. 27, 103-119 (2010)
    • (2010) Asia Pac. J. Oper. Res. , vol.27 , pp. 103-119
    • Xu, H.1
  • 101
    • 84860280270 scopus 로고    scopus 로고
    • L1/2 regularizer. Sci. China Ser
    • Xu, Z., Zhang, H., Wang, Y., Chang, X.: L1/2 regularizer. Sci. China Ser. F-Inf Sci. 53, 1159-1169 (2010)
    • (2010) F-Inf Sci , vol.53 , pp. 1159-1169
    • Xu, Z.1    Zhang, H.2    Wang, Y.3    Chang, X.4
  • 102
    • 0022010767 scopus 로고
    • Conditions for convergence of a trust-region method for nonsmooth optimization
    • Yuan, Y.: Conditions for convergence of a trust-region method for nonsmooth optimization. Math. Program. 31, 220-228 (1985)
    • (1985) Math. Program. , vol.31 , pp. 220-228
    • Yuan, Y.1
  • 103
    • 0002359182 scopus 로고
    • A smoothing-out technique for min-max optimization
    • Zang, I.: A smoothing-out technique for min-max optimization. Math. Program. 19, 61-71 (1980)
    • (1980) Math. Program. , vol.19 , pp. 61-71
    • Zang, I.1
  • 104
    • 70450260767 scopus 로고    scopus 로고
    • Smoothing projected gradient method and its application to stochastic linear complementarity problems
    • Zhang, C., Chen, X.: Smoothing projected gradient method and its application to stochastic linear complementarity problems. SIAM J. Optim. 20, 627-649 (2009)
    • (2009) SIAM J. Optim. , vol.20 , pp. 627-649
    • Zhang, C.1    Chen, X.2
  • 105
    • 79951509705 scopus 로고    scopus 로고
    • Wardrop's user equilibrium assignment under stochastic environment
    • Zhang, C., Chen, X., Sumalee, A.: Wardrop's user equilibrium assignment under stochastic environment. Transp. Res. B 45, 534-552 (2011)
    • (2011) Transp. Res. B , vol.45 , pp. 534-552
    • Zhang, C.1    Chen, X.2    Sumalee, A.3
  • 106
    • 77649284492 scopus 로고    scopus 로고
    • Nearly unbiased variable selection under minimax concave penalty
    • Zhang, C.-H.: Nearly unbiased variable selection under minimax concave penalty. Ann. Stat. 38, 894-942 (2010)
    • (2010) Ann. Stat. , vol.38 , pp. 894-942
    • Zhang, C.-H.1
  • 107
    • 77951274573 scopus 로고    scopus 로고
    • L: Asuperlinearly convergent method for a class of complementarity problems with non-Lipschitzian functions
    • Zhou, G.L., Caccetta, L., Teo, K.L: Asuperlinearly convergent method for a class of complementarity problems with non-Lipschitzian functions. SIAM J. Optim. 20, 1811-1827 (2010)
    • (2010) SIAM J. Optim. , vol.20 , pp. 1811-1827
    • Zhou, G.L.1    Caccetta, L.2    Teo, K.3


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.