基于几类束方法的VU-分解理论
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摘要
在过去的十年里,许多从事非光滑优化研究的学者们构造了一类函数和集合,尽管它们本身是非光滑的,然而存在某种光滑的子结构.这种结构可以被用于设计快速收敛的算法,给出计算准则,展开灵敏性分析.2000年,Lemarechal, Mifflin, Sagastizabal和Oustry对这类特殊的函数提出了vu-分解理论.其基本思想是将Rn空间分解为两个正交的子空间u和v的直和,使得原函数在u空间上的一阶近似是线性的,而其不光滑特征集中于v空间中,借助一个中间函数,u-拉格朗日函数,得到原函数在切于u的某个光滑轨道上的二阶展式.2004年,Mifflin和Sagastizabal给出了非凸函数的vu-分解理论.但是在约束问题的vu-分解方法以及vu-分解方法应用方面的研究还很初步.
本文围绕上述问题展开研究,主要工作如下：
1.第二章主要研究一类约束非光滑凸规划问题的超线性空间分解方法.我们假设该规划问题的目标函数是分片二阶连续可微的凸函数,约束是由光滑凸函数组成的不等式约束.运用精确罚函数,此规划问题被转化为一个无约束问题,利用无约束问题目标函数具有与vu-空间分解相关的原始对偶结构这一性质,计算出一条光滑轨道,并得到函数在其上的二阶展式.提出解决约束规划问题的vu-空间分解算法.在一定条件下证明了算法的收敛性.最后通过数值实验验证算法有效性.
2.第三章主要研究非光滑凸规划问题的近似vu-分解方法.对于凸的非光滑优化问题,文献[1]给出了一个vu-空间分解算法.算法的不足之处在于每次迭代都需要计算目标函数的精确次梯度.这在实际计算中是很困难的.针对这一问题,本章引入近似u-拉格朗日函数概念并给出相关性质.提出只需要计算函数近似次梯度的近似分解算法框架.根据迫近点落在原始轨道上的理论,将近似分解算法可执行化.最后给出数值实验说明算法的有效性.
3.第四章将vu-分解理论应用到二阶锥规划问题上.给出相应的vu-空间分解和原始对偶函数,并得到相应结论.提出解决二阶锥规划问题的vu-分解算法,证明了算法收敛性.最后给出数值算例说明算法的有效性.
Over the past decade, several independent researches studying nonsmooth optimization have developed classes of functions and sets that, though nonsmooth themselves, have underly-ing smooth substructure. These substructures are exploited for algorithmic purposes, to create calculus rules, and develop sensitivity analysis. Basing on this substructure, Lemarechal, Mif-flin, Sagastizabal and Oustry introduced Vu-theory in 2000. The basic idea is to decompose Rn into two orthogonal subspaces u and V so that f's nonsmoothness is concentrated essentially in V. It is possible to find smooth trajectories, via the intermediate function, called u-Lagrangian, yielding a second-order expansion for f. In 2004, Mifflin and Sagastizabal presented the VU-smoothness and proximal point results for some nonconvex functions. However, the study on constraint problem and the application of vu-decomposition method is not enough. The main results of this dissertation can be summarized as follows:
1. Chapter 2 studies a superlinear space-decomposition algorithm for constrained nonsmooth convex programs. We consider the constrained nonsmooth convex optimization problems, that is, piecewise C2 convex objectives with smooth convex inequality constraints. With the help of exact penalty function, these programs are transformed into unconstrained nonsmooth convex programs. The objective functions of these unconstrained programs are particular cases of functions with primal-dual gradient structure which has connection with Vu-space decomposition. Based on this special structure, a smooth trajectory can be computed and the second-order expansion of f can be obtained. A Vu-space decomposi-tion method for solving these constrained programs is given. This method is proved to be convergent with local superlinear rate under certain assumptions. An illustrative example is given to show how this method works.
2. Chapter 3 discusses an approximate decomposition algorithm for convex minimization. For nonsmooth convex optimization, Mifflin and Sagastizabal introduced a Vu-space decom-position algorithm. However, a drawback is that it needs the exact subgradient of the objective function, which is expensive to compute. To solve this problem, this chapter in-troduces a conception of approximate u-Lagrangian and gives its corresponding results. An approximate decomposition algorithm frame that is capable to deal with approximate subgradients is given. This method can implement with the theory of proximal point sequence follows the primal track. Numerical tests emphasize the theoretical findings.
3. In Chapter 4, we apply the Vu-theory to the second-order cone programming problem. The Vu-space decomposition method and primal-dual function are given. A Vu-algorithm is presented to solve the second-order cone programming problem. The algorithm is proved to converge with superlinear rate of convergence under certain assumptions. An illustrative example is given to show how the method works.
本文围绕上述问题展开研究,主要工作如下：
1.第二章主要研究一类约束非光滑凸规划问题的超线性空间分解方法.我们假设该规划问题的目标函数是分片二阶连续可微的凸函数,约束是由光滑凸函数组成的不等式约束.运用精确罚函数,此规划问题被转化为一个无约束问题,利用无约束问题目标函数具有与vu-空间分解相关的原始对偶结构这一性质,计算出一条光滑轨道,并得到函数在其上的二阶展式.提出解决约束规划问题的vu-空间分解算法.在一定条件下证明了算法的收敛性.最后通过数值实验验证算法有效性.
2.第三章主要研究非光滑凸规划问题的近似vu-分解方法.对于凸的非光滑优化问题,文献[1]给出了一个vu-空间分解算法.算法的不足之处在于每次迭代都需要计算目标函数的精确次梯度.这在实际计算中是很困难的.针对这一问题,本章引入近似u-拉格朗日函数概念并给出相关性质.提出只需要计算函数近似次梯度的近似分解算法框架.根据迫近点落在原始轨道上的理论,将近似分解算法可执行化.最后给出数值实验说明算法的有效性.
3.第四章将vu-分解理论应用到二阶锥规划问题上.给出相应的vu-空间分解和原始对偶函数,并得到相应结论.提出解决二阶锥规划问题的vu-分解算法,证明了算法收敛性.最后给出数值算例说明算法的有效性.
Over the past decade, several independent researches studying nonsmooth optimization have developed classes of functions and sets that, though nonsmooth themselves, have underly-ing smooth substructure. These substructures are exploited for algorithmic purposes, to create calculus rules, and develop sensitivity analysis. Basing on this substructure, Lemarechal, Mif-flin, Sagastizabal and Oustry introduced Vu-theory in 2000. The basic idea is to decompose Rn into two orthogonal subspaces u and V so that f's nonsmoothness is concentrated essentially in V. It is possible to find smooth trajectories, via the intermediate function, called u-Lagrangian, yielding a second-order expansion for f. In 2004, Mifflin and Sagastizabal presented the VU-smoothness and proximal point results for some nonconvex functions. However, the study on constraint problem and the application of vu-decomposition method is not enough. The main results of this dissertation can be summarized as follows:
1. Chapter 2 studies a superlinear space-decomposition algorithm for constrained nonsmooth convex programs. We consider the constrained nonsmooth convex optimization problems, that is, piecewise C2 convex objectives with smooth convex inequality constraints. With the help of exact penalty function, these programs are transformed into unconstrained nonsmooth convex programs. The objective functions of these unconstrained programs are particular cases of functions with primal-dual gradient structure which has connection with Vu-space decomposition. Based on this special structure, a smooth trajectory can be computed and the second-order expansion of f can be obtained. A Vu-space decomposi-tion method for solving these constrained programs is given. This method is proved to be convergent with local superlinear rate under certain assumptions. An illustrative example is given to show how this method works.
2. Chapter 3 discusses an approximate decomposition algorithm for convex minimization. For nonsmooth convex optimization, Mifflin and Sagastizabal introduced a Vu-space decom-position algorithm. However, a drawback is that it needs the exact subgradient of the objective function, which is expensive to compute. To solve this problem, this chapter in-troduces a conception of approximate u-Lagrangian and gives its corresponding results. An approximate decomposition algorithm frame that is capable to deal with approximate subgradients is given. This method can implement with the theory of proximal point sequence follows the primal track. Numerical tests emphasize the theoretical findings.
3. In Chapter 4, we apply the Vu-theory to the second-order cone programming problem. The Vu-space decomposition method and primal-dual function are given. A Vu-algorithm is presented to solve the second-order cone programming problem. The algorithm is proved to converge with superlinear rate of convergence under certain assumptions. An illustrative example is given to show how the method works.
引文
[1]R. Mifflin and C. Sagastizabal. A Vu-algorithm for convex minimization. Math. Program.,2005, 104(Ser.B):583-608.
[2]J. Outrata, M. Kocvara and J. Zowe. Nonsmooth approach to optimization problems with equilibrium constraints. Theory, Applications and Numerical Results. Kluwer Academic Publishers, Dordrecht, 1998.
[3]J.J. Morear, P.D. Panagiotopoulos and G. Strang. Topics in nonsmooth mechanics. Birkhauser Verlag, Basel,1998.
[4]E.S. Mistakidis and G.E. Stavroulakis. Nonconvex optimization in mechanics:smooth and nonsmooth algorithm. Heuristics and Engineering Applications by the F. E. M. Kluwer Academic Publisher, Dor-drecht,1998.
[5]F.H. Clarke, Y.S. Ledyaev, R.J. Stern and P.R. Wolenski. Nonsmooth Analysis and Control Theory, Springer, New York,1998.
[6]K. Miettnin, M.M. Makela and T. Mannikko. Optimal control of continuous casting by nondifferentiable multiobjective optimization. Computational Optimization and Applications,1998,11:177-197.
[7]C. Lemarechal. Nondifferentiable optimization. In:G.L. Nemhauser, A.H.G. Rinnooy Kan and M.J. Todd(Eds), Optimization, North Holland, Amsterdam,1989,529-572.
[8]J. Haslinger and P. Neittaanmaki. Finite element approximation for optimal shape. Material and Topol-ogy Design. J. Wiley and Sons, Chichester,1996.
[9]A. Auslender. Numerical methods for nondifferential convex optimization. Mathematical Programming Study,1987,30:102-126.
[10]R.W. Chaney. On second derivatives for nonsmooth functions. Nonlinear Analysis:Theory, Methods and Applications,1985,9(11):1189-1209.
[11]J.-P. Crouzeix. A relationship between the second derivative of a convex function and its conjugate. Mathematical Programming,1977,13:364-365.
[12]J.-B. Jiriart-Urruty. Lipschitz r-continuity of the approximate subdifferential of a convex function. Math-ematica Scandinavica,1980,47:123-134.
[13]J.-B. Jiriart-Urruty. The approximate first-order and second-order directional derivatives for a convex function. In J.-P. Cecconi and T. Zolezzi, editors. Mathematical Theories of Optimization, number 979 in Lecture Notes in Mathematics, Springer-Verlag,1983,154-166.
[14]M.L. Overton and R.S. Womersley. Second derivatives for optimizing eigenvalues of symmetric matri-ces. Journal of Mathematical Analysis and Applications,1995,3:667-718.
[15]M.L. Overton and X.J. Ye. Towards second-order methods for structured nonsmooth optimization. Ad-vances in Optimization and Numerical Analysis (S. Gomez and J.P. Hennart, eds.), Kluwer,1994,97-109.
[16]R.T. Rockafellar. Maximal monotone relations and the second derivatives of nonsmooth functions. An-nales de l'Institut Henri Poincare, Analyse non lineaire,1985,3:167-186.
[17]R.T. Rockafellar. First and second-order epi-differentiability in nonlinear programming. Transactions of the American Mathematical Society,1988,307:75-107.
[18]M.S. Bazaraa and C.M. Shetty. Nonlinear Programming, Theory and Algorithms. John Wiley and Sons, New York,1979.
[19]K.L. Teo and C.J. Goh. On constrained optimization problems with nonsmooth cost functionals. Applied mathematics and Optimization,1988,18:181-190.
[20]M.M. Makela and P. Neittaanmaki. Nonsmooth Optimization:Analysis and Algorithms with Applica-tions to Optimal Control. World Scientific Publishing Co., Singapore,1992.
[21]N.Z. Shor. Minimization Methods for Nondifferentiable Functions. Springer Verlag, Berlin,1985.
[22]J. Moreau. Proximite et dualite dans un espace Hilbertien. Bulletin de la Societe Mathematique de France,1965,93:273-299.
[23]R.T. Rockaffellar. Monotone operators and the proximal point algorithm. SIAM Journal on Control and Optimization,1976,14:877-898.
[24]M.Fukushima. A descent algorithm for nonsmooth convex optimization. Math. Program,1984,30: 163-175
[25]J.-B. Hiriart-Urruty and C. Lemarechal. Convex Analysis and Minimization Algorithms. Springer-Verlag, Berlin,1993.
[26]X. Chen and M. Fukushima. Proximal quasi-Newton methods for nondifferentiable convex optimiza-tion. Math. Program.,1999,85(2, Ser. A):313-334.
[27]C. Lemarechal and C. Sagastizabal. An approach to variable metric bundle methods. In:J. Henry, J.P. Yvon, (eds.), Systems Modeling and Optimization, no.197 in Lecture Notes in Control and Information Sciences, Springer-Verlag,1994,144-162.
[28]C. Lemarechal and C. Sagastizabal. More than first-order developments of convex function:primal-dual relations. Journal of Convex Analysis,1996,3(2):1-14.
[29]C. Lemarechal and C. Sagastizabal. Variable metric bundle methods:from conceptual to implementable forms. Math. Program.,1997,76(Ser. A):393-410.
[30]R. Mifflin. A quasi-second-order proximal bundle algorithm. Math. Program,1996,73(1):51-72.
[31]R. Mifflin, D. Sun and L. Qi. Quasi-Newton bundle-type methods for nondifferentiable convex opti-mization. SIAM Journal on Optimization,1998,8(2):583-603.
[32]L. Qi and X. Chen. A preconditioning proximal Newton for nondifferentiable convex optimization. Math. Program,1997,76(3, Ser. B):411-.429.
[33]A. Rauf and M. Fukushima. Globally convergent BFGS method for nonsmooth convex optimization. J. Optim. Theory Appl,2000,104(3):539-558.
[34]C. Lemarechal. Nondifferentiable optimization, subgradient and e-subgradient methods. Optimization and Operations Research, Lecture Notes in Economics and Mathematical Systems. Springer-Verlag, Berlin,1976,117:191-199.
[35]C. Lemarechal. An extension of davidon methods to nondifferentiable problems. Mathematical Pro-gramming Study,1975,3:95-109.
[36]P. Wolfe. A method for conjugate subgradients for minimizing nondifferentiable functions. Nondiffer-entiable Optimization. Mathematical Programming Study,1975,3:145-173.
[37]R.T. Rockafellar. Generalized second derivatives of convex function and saddle functions. Transactions of the American Mathematical Society,1990,322(1):51-78.
[38]C. Lemarechal and F. Oustry. Growth conditions and u-Lagrangians. Set-Valued Analysis,2001,9: 123-129.
[39]J.-B. Hiriart-Urruty. A new set-valued second order derivative for convex functions. Fermat Days 85: Mathematics for Optimization (J.-B. Hiriar-Urruty, ed.), Mathematics Studies, North-Holland,1986, 129:157-182.
[40]B.S. Mordukhovich. Generalized differential calculus for nonsmooth and set-valued mappings Journal of Mathematical Analysis and Applications,1993,183(1):250-288.
[41]R.A. Poliquin. Proto-differentiation of subgradient set-valued mappings. Canadian Journal of Mathe-matics,1990,42(3):520-532.
[42]J.-B. Hiriart-Urruty and C. Lemarechal. Convex Analysis and Minimization Algorithm. Springer Verlag, Berlin,1993.
[43]T.F. Coleman and A. R. Conn. Nonlinear programming via an exact penalty function:asymptotic anal-ysis. Mathematical Programming,1982,24:123-136.
[44]M.J.D. Power. The convergence of variable metric methods for nonlinearly constrained optimization calculations. Nonlinear Programming(O.L. Mangasarian, R.R. Meyer and S.M. Robinson, eds.),1978, 3:27-63.
[45]C. Lemarechal, F. Oustry and C. Sagastizabal. The u-Lagrangian of a convex function. Transactions of the American Mathematical Society,2000,352(2):711-729.
[46]R. Mifflin and C. Sagastizabal. Vu-decomposition derivatives for convex max-functions. In R. Tichatschke and M. Thera, editors, Ill-posed variational Problems and Regularization Techniques,477 in Lecture Notes in Economics and Mathematical Systems, Springer-Verlag Berlin Heidelberg,1999, 167-186.
[47]R. Mifflin and C. Sagastizabal. Functions with primal-dual gradient structure and u-Hessians. In G.Di Pillo and F. Giannessi, editors, Nonlinear Optimization and Related Topics,2000,36:219-233 in Ap-plied Optimization, Kluwer Academic Publishers B.V..
[48]F. Shan, L.P. Pang, L.M. Zhu and Z.Q. Xia. A uV-decomposed method for solving an MPEC problem. Applied Mathematics and Mechanics,2008,29(4):535-540.
[49]R. Mifflin and C. Sagastizabal. Vu-smoothness and proximal point results for some nonconvex func-tions. Optimization Methods and Software,2004,19 (5):463-478.
[50]R.T. Rockafellar and R.J.-B. Wets. Variational analysis, volume 317 of Grundlehren der Mathema-tischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1998.
[51]R. Mifflin and C. Sagastizabal. Proximal points are on the fast track. Journal of Convex Analysis,2002, 9(2):563-579.
[52]F.W. Meng and G. Y. Zhao. On second-order properties of the Moreau-Yosida regulation for constrained nonsmooth convex programs. Numerical Functional Analysis and Optimization,2007,25:515-530.
[53]R. Mifflin and C. Sagastizabal. Vu-decomposition derivatives for convex max-functions. In:Ill-posed Variational Problems and Regularization Techniques by Tichatschke and Thera (eds), Lecture Notes in Economics and Mathematical Systems 1999,477:167-186.
[54]R. Mifflin and C. Sagastizabal. On Vu-theory for functions with primal-dual gradient strcture. SIAM Journal on Optimization,2000,352:547-571.
[55]R. Mifflin and C. Sagastizabal. Primal-dual gradient structured functions:second-order results; links to epi-derivatives and partly smooth functions. SIAM Journal on Optimization,2003,13:1174-1197.
[56]R. T. Rockafellar. Monotone operators and the proximal point algorithm. SIAM Journal on Control and Optimization,1976,14:877-898.
[57]S.J. Wright. Identifiable surfaces in constrained optimization. SIAM Journal on Control and Optimiza-tion,1993,31:1063-1079.
[58]A. Lewis. Active sets, nonsmoothness and sensitivity. SIAM Journal on Optimization,2002,13:702-725.
[59]R.T. Rockafellar. Convex Analysis. Princeton University Press,1970.
[60]R. Correa and C. Lemarechal. Convergence of some algorithm for convex minimization. Mathematical Programming,1993,62(2):261-275.
[61]M. Hintermuller. A proximal bundle method based on approximate subgradients. Computational Opti-mization and Applications,2001,20:245-266.
[62]F. Alizadeh and S. Schmieta. Symmetric cones, potential reduction methods. In:H. Wolkowicz, R. Saigal, L. Vandenberghe(eds.), Handbook of Semidefinite Programming, Kluwer, Boston,2000,195-233.
[63]H.Y. Benson and R.J. Vanderbei. Solving problems with semidefinite and related constraints using in-terior point methods for nonlinear programming. Mathematical Programming,2003,95:279-302.
[64]M. Fukushima, Z.Q. Luo and P. Tseng. Smoothing functions for second-order-cone complementarity problems. SIAM Journal on Control and Optimization,2001,12:436-460.
[65]C. Kanzow, I. Ferenczi and M. Fukushima. Semismooth methods for linear and nonlinear second-order cone programs. Technical Report 2006-005, Department of Applied Mathematics and Physics, Kyoto University,2006.
[66]刘勇进,张立卫,王银河.线性二阶锥规划的一个光滑化方法及其收敛性.数学进展,2007,36(4)：491-502.
[67]Y.J. Liu and L.W. Zhang. Convergence of augmented lagrangian method for nonlinear optimization problems over second-order cones. Journal of Optimization Theory and Applications,2008,139:557-575.
[68]Y.Q. Bai and G.Q. Wang. Primal-dual interior-point algorithms for second-order cone optimization based on a new parametric kernel function. Acta Mathematica Sinica, English Series,2007,23(11): 2027-2042.
[69]J.F. Bonnans and C.H. Ramirez. Perturbation analysis of second order cone programming problems. Mathematical Programming,2005,104(Ser. B):205-227.
[70]W. Hare and C. Sagastizabal. A redistributed proximal bundle method for nonconvex optimization, http://www. optimization-online.org/DB-HTML/2009/04/2273.html.
[71]W. Hare and C. Sagastizabal. Computing proximal points of nonconvex functions. Math. Program., 2009,116(1-2, Ser.B):221-258.
[2]J. Outrata, M. Kocvara and J. Zowe. Nonsmooth approach to optimization problems with equilibrium constraints. Theory, Applications and Numerical Results. Kluwer Academic Publishers, Dordrecht, 1998.
[3]J.J. Morear, P.D. Panagiotopoulos and G. Strang. Topics in nonsmooth mechanics. Birkhauser Verlag, Basel,1998.
[4]E.S. Mistakidis and G.E. Stavroulakis. Nonconvex optimization in mechanics:smooth and nonsmooth algorithm. Heuristics and Engineering Applications by the F. E. M. Kluwer Academic Publisher, Dor-drecht,1998.
[5]F.H. Clarke, Y.S. Ledyaev, R.J. Stern and P.R. Wolenski. Nonsmooth Analysis and Control Theory, Springer, New York,1998.
[6]K. Miettnin, M.M. Makela and T. Mannikko. Optimal control of continuous casting by nondifferentiable multiobjective optimization. Computational Optimization and Applications,1998,11:177-197.
[7]C. Lemarechal. Nondifferentiable optimization. In:G.L. Nemhauser, A.H.G. Rinnooy Kan and M.J. Todd(Eds), Optimization, North Holland, Amsterdam,1989,529-572.
[8]J. Haslinger and P. Neittaanmaki. Finite element approximation for optimal shape. Material and Topol-ogy Design. J. Wiley and Sons, Chichester,1996.
[9]A. Auslender. Numerical methods for nondifferential convex optimization. Mathematical Programming Study,1987,30:102-126.
[10]R.W. Chaney. On second derivatives for nonsmooth functions. Nonlinear Analysis:Theory, Methods and Applications,1985,9(11):1189-1209.
[11]J.-P. Crouzeix. A relationship between the second derivative of a convex function and its conjugate. Mathematical Programming,1977,13:364-365.
[12]J.-B. Jiriart-Urruty. Lipschitz r-continuity of the approximate subdifferential of a convex function. Math-ematica Scandinavica,1980,47:123-134.
[13]J.-B. Jiriart-Urruty. The approximate first-order and second-order directional derivatives for a convex function. In J.-P. Cecconi and T. Zolezzi, editors. Mathematical Theories of Optimization, number 979 in Lecture Notes in Mathematics, Springer-Verlag,1983,154-166.
[14]M.L. Overton and R.S. Womersley. Second derivatives for optimizing eigenvalues of symmetric matri-ces. Journal of Mathematical Analysis and Applications,1995,3:667-718.
[15]M.L. Overton and X.J. Ye. Towards second-order methods for structured nonsmooth optimization. Ad-vances in Optimization and Numerical Analysis (S. Gomez and J.P. Hennart, eds.), Kluwer,1994,97-109.
[16]R.T. Rockafellar. Maximal monotone relations and the second derivatives of nonsmooth functions. An-nales de l'Institut Henri Poincare, Analyse non lineaire,1985,3:167-186.
[17]R.T. Rockafellar. First and second-order epi-differentiability in nonlinear programming. Transactions of the American Mathematical Society,1988,307:75-107.
[18]M.S. Bazaraa and C.M. Shetty. Nonlinear Programming, Theory and Algorithms. John Wiley and Sons, New York,1979.
[19]K.L. Teo and C.J. Goh. On constrained optimization problems with nonsmooth cost functionals. Applied mathematics and Optimization,1988,18:181-190.
[20]M.M. Makela and P. Neittaanmaki. Nonsmooth Optimization:Analysis and Algorithms with Applica-tions to Optimal Control. World Scientific Publishing Co., Singapore,1992.
[21]N.Z. Shor. Minimization Methods for Nondifferentiable Functions. Springer Verlag, Berlin,1985.
[22]J. Moreau. Proximite et dualite dans un espace Hilbertien. Bulletin de la Societe Mathematique de France,1965,93:273-299.
[23]R.T. Rockaffellar. Monotone operators and the proximal point algorithm. SIAM Journal on Control and Optimization,1976,14:877-898.
[24]M.Fukushima. A descent algorithm for nonsmooth convex optimization. Math. Program,1984,30: 163-175
[25]J.-B. Hiriart-Urruty and C. Lemarechal. Convex Analysis and Minimization Algorithms. Springer-Verlag, Berlin,1993.
[26]X. Chen and M. Fukushima. Proximal quasi-Newton methods for nondifferentiable convex optimiza-tion. Math. Program.,1999,85(2, Ser. A):313-334.
[27]C. Lemarechal and C. Sagastizabal. An approach to variable metric bundle methods. In:J. Henry, J.P. Yvon, (eds.), Systems Modeling and Optimization, no.197 in Lecture Notes in Control and Information Sciences, Springer-Verlag,1994,144-162.
[28]C. Lemarechal and C. Sagastizabal. More than first-order developments of convex function:primal-dual relations. Journal of Convex Analysis,1996,3(2):1-14.
[29]C. Lemarechal and C. Sagastizabal. Variable metric bundle methods:from conceptual to implementable forms. Math. Program.,1997,76(Ser. A):393-410.
[30]R. Mifflin. A quasi-second-order proximal bundle algorithm. Math. Program,1996,73(1):51-72.
[31]R. Mifflin, D. Sun and L. Qi. Quasi-Newton bundle-type methods for nondifferentiable convex opti-mization. SIAM Journal on Optimization,1998,8(2):583-603.
[32]L. Qi and X. Chen. A preconditioning proximal Newton for nondifferentiable convex optimization. Math. Program,1997,76(3, Ser. B):411-.429.
[33]A. Rauf and M. Fukushima. Globally convergent BFGS method for nonsmooth convex optimization. J. Optim. Theory Appl,2000,104(3):539-558.
[34]C. Lemarechal. Nondifferentiable optimization, subgradient and e-subgradient methods. Optimization and Operations Research, Lecture Notes in Economics and Mathematical Systems. Springer-Verlag, Berlin,1976,117:191-199.
[35]C. Lemarechal. An extension of davidon methods to nondifferentiable problems. Mathematical Pro-gramming Study,1975,3:95-109.
[36]P. Wolfe. A method for conjugate subgradients for minimizing nondifferentiable functions. Nondiffer-entiable Optimization. Mathematical Programming Study,1975,3:145-173.
[37]R.T. Rockafellar. Generalized second derivatives of convex function and saddle functions. Transactions of the American Mathematical Society,1990,322(1):51-78.
[38]C. Lemarechal and F. Oustry. Growth conditions and u-Lagrangians. Set-Valued Analysis,2001,9: 123-129.
[39]J.-B. Hiriart-Urruty. A new set-valued second order derivative for convex functions. Fermat Days 85: Mathematics for Optimization (J.-B. Hiriar-Urruty, ed.), Mathematics Studies, North-Holland,1986, 129:157-182.
[40]B.S. Mordukhovich. Generalized differential calculus for nonsmooth and set-valued mappings Journal of Mathematical Analysis and Applications,1993,183(1):250-288.
[41]R.A. Poliquin. Proto-differentiation of subgradient set-valued mappings. Canadian Journal of Mathe-matics,1990,42(3):520-532.
[42]J.-B. Hiriart-Urruty and C. Lemarechal. Convex Analysis and Minimization Algorithm. Springer Verlag, Berlin,1993.
[43]T.F. Coleman and A. R. Conn. Nonlinear programming via an exact penalty function:asymptotic anal-ysis. Mathematical Programming,1982,24:123-136.
[44]M.J.D. Power. The convergence of variable metric methods for nonlinearly constrained optimization calculations. Nonlinear Programming(O.L. Mangasarian, R.R. Meyer and S.M. Robinson, eds.),1978, 3:27-63.
[45]C. Lemarechal, F. Oustry and C. Sagastizabal. The u-Lagrangian of a convex function. Transactions of the American Mathematical Society,2000,352(2):711-729.
[46]R. Mifflin and C. Sagastizabal. Vu-decomposition derivatives for convex max-functions. In R. Tichatschke and M. Thera, editors, Ill-posed variational Problems and Regularization Techniques,477 in Lecture Notes in Economics and Mathematical Systems, Springer-Verlag Berlin Heidelberg,1999, 167-186.
[47]R. Mifflin and C. Sagastizabal. Functions with primal-dual gradient structure and u-Hessians. In G.Di Pillo and F. Giannessi, editors, Nonlinear Optimization and Related Topics,2000,36:219-233 in Ap-plied Optimization, Kluwer Academic Publishers B.V..
[48]F. Shan, L.P. Pang, L.M. Zhu and Z.Q. Xia. A uV-decomposed method for solving an MPEC problem. Applied Mathematics and Mechanics,2008,29(4):535-540.
[49]R. Mifflin and C. Sagastizabal. Vu-smoothness and proximal point results for some nonconvex func-tions. Optimization Methods and Software,2004,19 (5):463-478.
[50]R.T. Rockafellar and R.J.-B. Wets. Variational analysis, volume 317 of Grundlehren der Mathema-tischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1998.
[51]R. Mifflin and C. Sagastizabal. Proximal points are on the fast track. Journal of Convex Analysis,2002, 9(2):563-579.
[52]F.W. Meng and G. Y. Zhao. On second-order properties of the Moreau-Yosida regulation for constrained nonsmooth convex programs. Numerical Functional Analysis and Optimization,2007,25:515-530.
[53]R. Mifflin and C. Sagastizabal. Vu-decomposition derivatives for convex max-functions. In:Ill-posed Variational Problems and Regularization Techniques by Tichatschke and Thera (eds), Lecture Notes in Economics and Mathematical Systems 1999,477:167-186.
[54]R. Mifflin and C. Sagastizabal. On Vu-theory for functions with primal-dual gradient strcture. SIAM Journal on Optimization,2000,352:547-571.
[55]R. Mifflin and C. Sagastizabal. Primal-dual gradient structured functions:second-order results; links to epi-derivatives and partly smooth functions. SIAM Journal on Optimization,2003,13:1174-1197.
[56]R. T. Rockafellar. Monotone operators and the proximal point algorithm. SIAM Journal on Control and Optimization,1976,14:877-898.
[57]S.J. Wright. Identifiable surfaces in constrained optimization. SIAM Journal on Control and Optimiza-tion,1993,31:1063-1079.
[58]A. Lewis. Active sets, nonsmoothness and sensitivity. SIAM Journal on Optimization,2002,13:702-725.
[59]R.T. Rockafellar. Convex Analysis. Princeton University Press,1970.
[60]R. Correa and C. Lemarechal. Convergence of some algorithm for convex minimization. Mathematical Programming,1993,62(2):261-275.
[61]M. Hintermuller. A proximal bundle method based on approximate subgradients. Computational Opti-mization and Applications,2001,20:245-266.
[62]F. Alizadeh and S. Schmieta. Symmetric cones, potential reduction methods. In:H. Wolkowicz, R. Saigal, L. Vandenberghe(eds.), Handbook of Semidefinite Programming, Kluwer, Boston,2000,195-233.
[63]H.Y. Benson and R.J. Vanderbei. Solving problems with semidefinite and related constraints using in-terior point methods for nonlinear programming. Mathematical Programming,2003,95:279-302.
[64]M. Fukushima, Z.Q. Luo and P. Tseng. Smoothing functions for second-order-cone complementarity problems. SIAM Journal on Control and Optimization,2001,12:436-460.
[65]C. Kanzow, I. Ferenczi and M. Fukushima. Semismooth methods for linear and nonlinear second-order cone programs. Technical Report 2006-005, Department of Applied Mathematics and Physics, Kyoto University,2006.
[66]刘勇进,张立卫,王银河.线性二阶锥规划的一个光滑化方法及其收敛性.数学进展,2007,36(4)：491-502.
[67]Y.J. Liu and L.W. Zhang. Convergence of augmented lagrangian method for nonlinear optimization problems over second-order cones. Journal of Optimization Theory and Applications,2008,139:557-575.
[68]Y.Q. Bai and G.Q. Wang. Primal-dual interior-point algorithms for second-order cone optimization based on a new parametric kernel function. Acta Mathematica Sinica, English Series,2007,23(11): 2027-2042.
[69]J.F. Bonnans and C.H. Ramirez. Perturbation analysis of second order cone programming problems. Mathematical Programming,2005,104(Ser. B):205-227.
[70]W. Hare and C. Sagastizabal. A redistributed proximal bundle method for nonconvex optimization, http://www. optimization-online.org/DB-HTML/2009/04/2273.html.
[71]W. Hare and C. Sagastizabal. Computing proximal points of nonconvex functions. Math. Program., 2009,116(1-2, Ser.B):221-258.