求解非线性互补问题的光滑信赖域方法
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摘要
非线性互补问题在工程和经济中有许多重要应用,已经产生了大量的求解方法,同时也取得了全局收敛性和局部超线性收敛性结果.然而,学者们多采用NCP函数把非线性互补问题转化为非光滑方程组来求解.
本文采用Kanzow光滑逼近函数逼近Fischer-Burmeister函数,从而得到相应的光滑非线性方程组,并将其转化为优化问题,再通过引入自适应技术和非单调技术,与原来的较为可靠、稳定的信赖域方法相结合,提出了两种新的求解方法.
在第三章中,将自适应技术与信赖域算法结合提出了求解非线性互补问题的自适应光滑信赖域算法.该算法使得信赖域半径可以根据当前迭代点的参数进行自我调整;同时,若目标函数下降量足够大,则更新光滑逼近函数的光滑化系数.
第四章在第三章算法的基础上引入非单调技术,从而给出求解非线性互补问题的非单调自适应光滑信赖域算法.该算法在信赖域子问题的下降量估计中引入了“非单调比率”,当该比率可接受时接受该步迭代;同时,若目标函数下降量足够大,则更新光滑逼近函数的光滑化系数.
在假设F是P0函数的条件下,文中还证明了两种算法所产生的点列包含在一个水平集中;且在水平集为紧集的前提下,算法至少产生一个聚点,从而证明了算法的全局收敛性.进一步地,本文还给出了算法产生的点列在一定条件下有超线性收敛性及二次收敛性等性质.
在两种算法理论证明的基础上,第五章进行了数值实验,给出了比较结果,得到算法的有效性.
The nonlinear complementarity problem (NCP) has many important applications in engineering and economics, and there have developed many numerical methods to solve it at the same time, global and local convergence results have also been obtained. In the last years more attention has been devoted to reformulating the nonlinear complementarity problem as a system of nonsmooth equations by using some NCP function.
In this paper, we use the Kanzow function to approximate the Fischer-Burmeister function so that smooth and nonlinear equatons can be obtained and then changed into a optimization problem. With the combination of the adaptive technique、the nonmonotone technique and the trust-region method which is more stable and reliable than linear search method, two new algorithms for solving these equations are proposed in Chapter 3 and Chapter 4.
In Chapter 3, we combine the trust region subproblem with the adptive techique to propose a smoothing adptive trust region algorithm for the nonlinear complementarity problem. With the adaptive technique, the radius of trust regionΔk can be adjusted automatically to improve the efficiency of trust region methods. On the other hand, the smoothing coefficient of the Kanzow function is updated when the object function gets some satisfying reduction.
In Chapter 4, based on the algorithm in Chapter 3, the new algorithm adopts a 'nonmontone ratio' to approximate the reduction of the object functions. An iteration is accepted when the ratio is not very small. On the other hand, the smoothing coefficient of the Kanzow function is updated when the object function gets some satisfying reduction.
With the assumption that F is a P0 function, we prove that the sequence generated by the algorithm remains in some level set. Furthermore, the method will generate at least one accumulation point if the level set is compact, which leads to the global convergence. In addition, the sequence will converge to one point and the superlinear convergence or even quadratic convergence under some conditions.
In Chapter 5, a series of numerical examples are tested, which shows the algorithm is quite promising.
本文采用Kanzow光滑逼近函数逼近Fischer-Burmeister函数,从而得到相应的光滑非线性方程组,并将其转化为优化问题,再通过引入自适应技术和非单调技术,与原来的较为可靠、稳定的信赖域方法相结合,提出了两种新的求解方法.
在第三章中,将自适应技术与信赖域算法结合提出了求解非线性互补问题的自适应光滑信赖域算法.该算法使得信赖域半径可以根据当前迭代点的参数进行自我调整;同时,若目标函数下降量足够大,则更新光滑逼近函数的光滑化系数.
第四章在第三章算法的基础上引入非单调技术,从而给出求解非线性互补问题的非单调自适应光滑信赖域算法.该算法在信赖域子问题的下降量估计中引入了“非单调比率”,当该比率可接受时接受该步迭代;同时,若目标函数下降量足够大,则更新光滑逼近函数的光滑化系数.
在假设F是P0函数的条件下,文中还证明了两种算法所产生的点列包含在一个水平集中;且在水平集为紧集的前提下,算法至少产生一个聚点,从而证明了算法的全局收敛性.进一步地,本文还给出了算法产生的点列在一定条件下有超线性收敛性及二次收敛性等性质.
在两种算法理论证明的基础上,第五章进行了数值实验,给出了比较结果,得到算法的有效性.
The nonlinear complementarity problem (NCP) has many important applications in engineering and economics, and there have developed many numerical methods to solve it at the same time, global and local convergence results have also been obtained. In the last years more attention has been devoted to reformulating the nonlinear complementarity problem as a system of nonsmooth equations by using some NCP function.
In this paper, we use the Kanzow function to approximate the Fischer-Burmeister function so that smooth and nonlinear equatons can be obtained and then changed into a optimization problem. With the combination of the adaptive technique、the nonmonotone technique and the trust-region method which is more stable and reliable than linear search method, two new algorithms for solving these equations are proposed in Chapter 3 and Chapter 4.
In Chapter 3, we combine the trust region subproblem with the adptive techique to propose a smoothing adptive trust region algorithm for the nonlinear complementarity problem. With the adaptive technique, the radius of trust regionΔk can be adjusted automatically to improve the efficiency of trust region methods. On the other hand, the smoothing coefficient of the Kanzow function is updated when the object function gets some satisfying reduction.
In Chapter 4, based on the algorithm in Chapter 3, the new algorithm adopts a 'nonmontone ratio' to approximate the reduction of the object functions. An iteration is accepted when the ratio is not very small. On the other hand, the smoothing coefficient of the Kanzow function is updated when the object function gets some satisfying reduction.
With the assumption that F is a P0 function, we prove that the sequence generated by the algorithm remains in some level set. Furthermore, the method will generate at least one accumulation point if the level set is compact, which leads to the global convergence. In addition, the sequence will converge to one point and the superlinear convergence or even quadratic convergence under some conditions.
In Chapter 5, a series of numerical examples are tested, which shows the algorithm is quite promising.
引文
[1]韩继业,修乃华,戚厚铎.非线性互补理论与算法[M].上海:上海科学技术出版社.2006.
[2] Ferris M C, Pang J S .Engineering and economic applications of complementarity problems[J]. SIAM eview, 1997,39: 669-713.
[3] Harker P T, Pang J S. Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications. Math.Programming[J]. 1990,48: 161-220.
[4] Kummer B. Newton's method for nondifferentiable functions, In: Advances in Mathematical Optimization[M]. Berlin: Akademic-Verlag, 1998, 114-125.
[5] Qi L, Sun J. A nonsmooth version of Newton's method[J]. Mathematical Programming , 1993, 58: 353-367.
[6] Kanzow C, Zupke M. Inexact trust-region method for nonlinear complementarity problems. In: Reformulation: nosmooth, Piecewise Smooth, Semismooth and Smoothing Methods[M]. Norwell: Kluwer Academic Publisher, 1998, 221-233.
[7] Kanzow C, some noninterior continuation methods for linear complementarity problems[J]. SIAM Journal on Control and Optimization, 1995,38:402-418.
[8] Burke J, Xu S. The global linear convergence of a Non-interior path-following algorithm for linear complementarity problems[J]. Math Oper Res, 1998,23:719-734.
[9] Hotta K, Yoshise. A global convergence of a Class of Non-interior-Point algorithms Using Chen-Harker-Kanzow Functions for linear complementarity problems[J]. Mathematical Programming 1999,86:105-133.
[10] Xu S. The global linear convergence of aninfeasible non-interior path-following algorithm for complementarity problems with uniform P-functions[J]. Mathematical Programming 2000, 87, 501-517.
[11] Tseng P. Analysis of non-iterior continuation method based on Chen-mangasiarioan smoothing functions for complementarity problems. [C]. Boston: Kluwer Academic Publisher, 1998,381-404.
[12]高岩等.非光滑优化[M].北京:科学出版社.2008..
[13] Chen B, Harker P T. A non-interior-point continuation method for linear complementarity problems[J]. SIAM Journal of Matrix Analysis and Application, 1993, 14:1168-1190.
[14] Kanzow C. Some non-interior continuation methods for linear complementarity problems[J]. Siam Journal of Martrix Analysis and Application, 1996,17: 851-868.
[15]宋歹才,刘国新,刘庆怀等.线性互补补问题的一个高阶收敛性算法[J].吉林大学自然科学学报,1999,127:12-16.
[16] Fisher A, Jiang H. Merit functions for complementarity and related problems: asuvery[J]. Computational Optimization and Application. 2000,17:157-182.
[17] Chen C, Mangasarian O L. Smoothing methods for convex inequalities and linear complementarity problems[J]. Mathematical Programming, 1995,71:51-79.
[18] Qi H, Liao L. A smoothin newton method for extended vertical linear complementarity problems[J]. SIAM Journal of Martrix Analysis and Application, 1977, 21:45-66.
[19] Chen B, Xiu N. Superlinear noninterior one-step continuation method for monotone LCP in absence of strict complementarity[J]. Journal of Martrix Analysis and Application, 1997, 21:45-66.
[20]乌力吉,陈国庆.线性互补问题的一种新Lagrange乘子法[J].高等学校计算数学. 2004, 26:162-171.
[21] Harker P T, Pang J. Finite-dimensional variational inequality and nonlinear complementarity problem: a survey of theory, algorithms and applications[J]. Mathematical Programming, 1990,48: 161-220.
[22]修乃华,高自友.互补问题算法的新进展[J].数学进展,1999,28:193-210.
[23] Qi L, Chen X. A globally convergent successive approximation method for nonsmooth equations[J]. SIAM Journal on Control and Optimization, 1995, 38: 402-418.
[24] Kanzow C. Some noninterior continuation methods for linear complementarity problems[J]. SIAM J Matrix Anal Appl, 1996,17: 851-868.
[25] Chen X, Qi L, Sun D. Global and superlinear of the smoothing. Newton methods and its application to general box constrained Variational inequalities[J]. Mathematics of Computation, 1998, 222:519-540.
[26] Kanzow C, Pieper H. Jacobian smoothing methods for nonlinear complementarity problems[J]. SIAM Journal on Optimization, 1999,9: 342-373.
[27] Deng N Y, Xiao Y, Zhou F T. Nonmonotonic trust region algorithm[J]. JOTA, 1993. 26: 259-285.
[28] Toint Ph L. A non-monotone trust trgion algorithm for nonlinear optimization subject to cnves constraints[J]. Mathematical Programming, 1997, 77: 69-94.
[29] Coleman T F, Li Y. An interior trust region approach for nonlinear minimization subject to bounds[J]. SIAM J Opt, 1996,6: 418-445.
[30] Nocedai J, Yuan Y. Combing trust region and line search techniques in Advances in Nonlinear Programming[J]. Kluwer, 1998, 153-175.
[31] Ou Y.G. Trust Region Algorithm for a Class of Nonlinear Complementarity Problem[J]. Chin. Quart. J. of Math. 2007,22(4): 558-566.
[32] Yang Y.F. and Qi L. Smoothing trust region methods for nonlinear complementarity problems with P0-functions[J]. Annals of Operations Research, 2005, 133: 99-177
[33]马昌凤.求解非线性互补问题的一个光滑信赖域算法[J].工程数学学报.2006(23).
[34]陈为民,杨余飞.求解一般非线性互补问题的光滑化方法[J].运筹学学报.2008(12).
[35] Clarke F.H, Optimization and Nonsmooth Analysis[M]. New York: John Wiley & Son, 1983 224-256.
[36] Qi L, C-differentiablity, C-differential Operators and generalized Newton methods[D]. Tech. Report AMR96/5, School of mathematics, the university of New South Wales, Sydney, Australia, 1996.
[37] Mifflin R. Semismooth and semiconvex functions in constrained optimization[J]. SIAM Journal on Control and Optimization, 1997,15:957-972.
[38] Qi L, Sun J. A nonsmooth version of Newton's method[J]. Mathematical programming, 1993, 58:353-367.
[39] Jiang H.Y, M.Fukushima, Qi. L, and Sun D. A trust Region Method for Solving Generalized Complementarity Problems[J]. SIAM Journal on Optimization, 1998,8: 140-157.
[40] Fisher, A. Solution of monotone complementarity problems with locally Lipschitzian functions[J]. Mathematical Programming, 1997,76: 513-532.
[41] Facchinei F, Soares J. A new merit function for nonlinear complementarity problems and arelated algorithm[J]. SIAM Journal on Optimization, 1997,7:225-247.
[42] Kanzow C. Some noninterior continuation methods for linear complementarity problems[J]. SIAM J Matrix Anal Appl, 1996, 17:851-868.
[43] Kanzow C. A New approach to continuation methods for complementarity problems with uniform P-functions[J]. Oper Res Lett, 1997,20:85-92.
[44] Chen, X. Smoothing methods for Complementarity Problems and Their Applications: A Survey[J]. Journall of the Operations Research Society of Japan 2000,43,32-47.
[45] A.Sartenear. Automatic determination of an initial trust region in nonlinear programming[J]. SIAM J. Sci. Compuit, 1997,18: 1788-1803.
[46] X.S.Zhang, JL.Zhang, L.Z.Liao. An adaptive trust region algorithm for unconstrained optimization[J]. Science in China, 2002,45(5):620-631.
[47] X.S.Zhang, Z.W.Chen, J.L.Zhang. A self-adaptive trust region method for unconstrained optimization[J]. OR Transactions, 2001,5(1):53-62.
[48] J.L.Zhang. Y.Wang. Anew trust region method for nonlinear equations[J]. Math Meth Oper Res, 2003,58:283-298.
[49] J.L.Zhang. Trust region algorithm for nonlinear optimization[D]. Ph. D Thesis, Institute of Applied Mathematics, Chinese Academic of Science, 2001.
[50] Powell, M.J.D Convergence proterties of a Class of Minimization algorithms[M]. In L.L.Mangasarian, R.R.Meyer and S.M.Robinson(eds.), Nonlinear Programming, 1975, Vol.2. New York: Academic Press,pp, 1-27.
[51] Jiang H.Y, M.Fukushima, Qi L, and Sun D. A Trust Region Method for Solving Generalized Complementarity problems[J]. Mathematical Programming 1996,75, 407-439.
[52] Robinson, S.M.. Strongly Regular Generalized Equation[J]. Math. Oper. Res.(1980)5,43-62.
[53] Facchinei,F. and J. Soares. A New Merit Function for Nonlinear Complementarity Problems and A Related Algorithm[J]. SIAM Journal on Optimization 7, 225-247.
[54] Kanzow,C. and H.Pieper. Jacobian Smoothing Methods for Newton-Type Method for Variational Inequality Problems[J]. Mathematical Programming Ser. 1999 A 85, 81-106.
[55] Y.H.Dai, ON the Nonmonotone Line Search[J]. Journal of Optimization and Application, 2002,112:315-330.
[56] Chamberlain.R.M.,Powell, M.J.D., Lemarechal, C., and pedersen, H.C., The Watchdog Technique for Forcing Convergence in Algorithms for Constrained Optimization[J]. Mathematical Programming Study, 1986,26,1-17.
[57] Grippo, L, Lampariello, F, and Lucidi, S., A Nonmonotone Line Search Technique for Newton's Method[J]. SIAM Journal on Numerical Analysis, 1986,23:707-716.
[58] Fischer, A. Solution of Monotone Complementarity Problems with Locally Lipschitzian Functions[J]. Mathematical Programming 1997,76, 513–532.
[59] Chen, X. and Y. Ye. On Homotopy-Smoothing Methods for Box-Constrained Variational Inequalities[J]. SIAM Journal on Control and Optimization 1999,37,589-616.
[60] Facchinei, F. STRuctural and Stability Properties of P0 Nonlinear Complementarity Problems[J]. Math. Oper. Res. 1998, 23, 735-745.
[61] Liu, J. Strong Stability in Variational Inequalities[J]. SIAM Journal on Control and Optimization 1995,33, 725–749.
[62] Pang J S, Gabriel S A. NE/SQP: A Robust Algorithm for Nonlinear complementarity problems[J]. Mathematical Programming, 1993,60:295-337.
[2] Ferris M C, Pang J S .Engineering and economic applications of complementarity problems[J]. SIAM eview, 1997,39: 669-713.
[3] Harker P T, Pang J S. Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications. Math.Programming[J]. 1990,48: 161-220.
[4] Kummer B. Newton's method for nondifferentiable functions, In: Advances in Mathematical Optimization[M]. Berlin: Akademic-Verlag, 1998, 114-125.
[5] Qi L, Sun J. A nonsmooth version of Newton's method[J]. Mathematical Programming , 1993, 58: 353-367.
[6] Kanzow C, Zupke M. Inexact trust-region method for nonlinear complementarity problems. In: Reformulation: nosmooth, Piecewise Smooth, Semismooth and Smoothing Methods[M]. Norwell: Kluwer Academic Publisher, 1998, 221-233.
[7] Kanzow C, some noninterior continuation methods for linear complementarity problems[J]. SIAM Journal on Control and Optimization, 1995,38:402-418.
[8] Burke J, Xu S. The global linear convergence of a Non-interior path-following algorithm for linear complementarity problems[J]. Math Oper Res, 1998,23:719-734.
[9] Hotta K, Yoshise. A global convergence of a Class of Non-interior-Point algorithms Using Chen-Harker-Kanzow Functions for linear complementarity problems[J]. Mathematical Programming 1999,86:105-133.
[10] Xu S. The global linear convergence of aninfeasible non-interior path-following algorithm for complementarity problems with uniform P-functions[J]. Mathematical Programming 2000, 87, 501-517.
[11] Tseng P. Analysis of non-iterior continuation method based on Chen-mangasiarioan smoothing functions for complementarity problems. [C]. Boston: Kluwer Academic Publisher, 1998,381-404.
[12]高岩等.非光滑优化[M].北京:科学出版社.2008..
[13] Chen B, Harker P T. A non-interior-point continuation method for linear complementarity problems[J]. SIAM Journal of Matrix Analysis and Application, 1993, 14:1168-1190.
[14] Kanzow C. Some non-interior continuation methods for linear complementarity problems[J]. Siam Journal of Martrix Analysis and Application, 1996,17: 851-868.
[15]宋歹才,刘国新,刘庆怀等.线性互补补问题的一个高阶收敛性算法[J].吉林大学自然科学学报,1999,127:12-16.
[16] Fisher A, Jiang H. Merit functions for complementarity and related problems: asuvery[J]. Computational Optimization and Application. 2000,17:157-182.
[17] Chen C, Mangasarian O L. Smoothing methods for convex inequalities and linear complementarity problems[J]. Mathematical Programming, 1995,71:51-79.
[18] Qi H, Liao L. A smoothin newton method for extended vertical linear complementarity problems[J]. SIAM Journal of Martrix Analysis and Application, 1977, 21:45-66.
[19] Chen B, Xiu N. Superlinear noninterior one-step continuation method for monotone LCP in absence of strict complementarity[J]. Journal of Martrix Analysis and Application, 1997, 21:45-66.
[20]乌力吉,陈国庆.线性互补问题的一种新Lagrange乘子法[J].高等学校计算数学. 2004, 26:162-171.
[21] Harker P T, Pang J. Finite-dimensional variational inequality and nonlinear complementarity problem: a survey of theory, algorithms and applications[J]. Mathematical Programming, 1990,48: 161-220.
[22]修乃华,高自友.互补问题算法的新进展[J].数学进展,1999,28:193-210.
[23] Qi L, Chen X. A globally convergent successive approximation method for nonsmooth equations[J]. SIAM Journal on Control and Optimization, 1995, 38: 402-418.
[24] Kanzow C. Some noninterior continuation methods for linear complementarity problems[J]. SIAM J Matrix Anal Appl, 1996,17: 851-868.
[25] Chen X, Qi L, Sun D. Global and superlinear of the smoothing. Newton methods and its application to general box constrained Variational inequalities[J]. Mathematics of Computation, 1998, 222:519-540.
[26] Kanzow C, Pieper H. Jacobian smoothing methods for nonlinear complementarity problems[J]. SIAM Journal on Optimization, 1999,9: 342-373.
[27] Deng N Y, Xiao Y, Zhou F T. Nonmonotonic trust region algorithm[J]. JOTA, 1993. 26: 259-285.
[28] Toint Ph L. A non-monotone trust trgion algorithm for nonlinear optimization subject to cnves constraints[J]. Mathematical Programming, 1997, 77: 69-94.
[29] Coleman T F, Li Y. An interior trust region approach for nonlinear minimization subject to bounds[J]. SIAM J Opt, 1996,6: 418-445.
[30] Nocedai J, Yuan Y. Combing trust region and line search techniques in Advances in Nonlinear Programming[J]. Kluwer, 1998, 153-175.
[31] Ou Y.G. Trust Region Algorithm for a Class of Nonlinear Complementarity Problem[J]. Chin. Quart. J. of Math. 2007,22(4): 558-566.
[32] Yang Y.F. and Qi L. Smoothing trust region methods for nonlinear complementarity problems with P0-functions[J]. Annals of Operations Research, 2005, 133: 99-177
[33]马昌凤.求解非线性互补问题的一个光滑信赖域算法[J].工程数学学报.2006(23).
[34]陈为民,杨余飞.求解一般非线性互补问题的光滑化方法[J].运筹学学报.2008(12).
[35] Clarke F.H, Optimization and Nonsmooth Analysis[M]. New York: John Wiley & Son, 1983 224-256.
[36] Qi L, C-differentiablity, C-differential Operators and generalized Newton methods[D]. Tech. Report AMR96/5, School of mathematics, the university of New South Wales, Sydney, Australia, 1996.
[37] Mifflin R. Semismooth and semiconvex functions in constrained optimization[J]. SIAM Journal on Control and Optimization, 1997,15:957-972.
[38] Qi L, Sun J. A nonsmooth version of Newton's method[J]. Mathematical programming, 1993, 58:353-367.
[39] Jiang H.Y, M.Fukushima, Qi. L, and Sun D. A trust Region Method for Solving Generalized Complementarity Problems[J]. SIAM Journal on Optimization, 1998,8: 140-157.
[40] Fisher, A. Solution of monotone complementarity problems with locally Lipschitzian functions[J]. Mathematical Programming, 1997,76: 513-532.
[41] Facchinei F, Soares J. A new merit function for nonlinear complementarity problems and arelated algorithm[J]. SIAM Journal on Optimization, 1997,7:225-247.
[42] Kanzow C. Some noninterior continuation methods for linear complementarity problems[J]. SIAM J Matrix Anal Appl, 1996, 17:851-868.
[43] Kanzow C. A New approach to continuation methods for complementarity problems with uniform P-functions[J]. Oper Res Lett, 1997,20:85-92.
[44] Chen, X. Smoothing methods for Complementarity Problems and Their Applications: A Survey[J]. Journall of the Operations Research Society of Japan 2000,43,32-47.
[45] A.Sartenear. Automatic determination of an initial trust region in nonlinear programming[J]. SIAM J. Sci. Compuit, 1997,18: 1788-1803.
[46] X.S.Zhang, JL.Zhang, L.Z.Liao. An adaptive trust region algorithm for unconstrained optimization[J]. Science in China, 2002,45(5):620-631.
[47] X.S.Zhang, Z.W.Chen, J.L.Zhang. A self-adaptive trust region method for unconstrained optimization[J]. OR Transactions, 2001,5(1):53-62.
[48] J.L.Zhang. Y.Wang. Anew trust region method for nonlinear equations[J]. Math Meth Oper Res, 2003,58:283-298.
[49] J.L.Zhang. Trust region algorithm for nonlinear optimization[D]. Ph. D Thesis, Institute of Applied Mathematics, Chinese Academic of Science, 2001.
[50] Powell, M.J.D Convergence proterties of a Class of Minimization algorithms[M]. In L.L.Mangasarian, R.R.Meyer and S.M.Robinson(eds.), Nonlinear Programming, 1975, Vol.2. New York: Academic Press,pp, 1-27.
[51] Jiang H.Y, M.Fukushima, Qi L, and Sun D. A Trust Region Method for Solving Generalized Complementarity problems[J]. Mathematical Programming 1996,75, 407-439.
[52] Robinson, S.M.. Strongly Regular Generalized Equation[J]. Math. Oper. Res.(1980)5,43-62.
[53] Facchinei,F. and J. Soares. A New Merit Function for Nonlinear Complementarity Problems and A Related Algorithm[J]. SIAM Journal on Optimization 7, 225-247.
[54] Kanzow,C. and H.Pieper. Jacobian Smoothing Methods for Newton-Type Method for Variational Inequality Problems[J]. Mathematical Programming Ser. 1999 A 85, 81-106.
[55] Y.H.Dai, ON the Nonmonotone Line Search[J]. Journal of Optimization and Application, 2002,112:315-330.
[56] Chamberlain.R.M.,Powell, M.J.D., Lemarechal, C., and pedersen, H.C., The Watchdog Technique for Forcing Convergence in Algorithms for Constrained Optimization[J]. Mathematical Programming Study, 1986,26,1-17.
[57] Grippo, L, Lampariello, F, and Lucidi, S., A Nonmonotone Line Search Technique for Newton's Method[J]. SIAM Journal on Numerical Analysis, 1986,23:707-716.
[58] Fischer, A. Solution of Monotone Complementarity Problems with Locally Lipschitzian Functions[J]. Mathematical Programming 1997,76, 513–532.
[59] Chen, X. and Y. Ye. On Homotopy-Smoothing Methods for Box-Constrained Variational Inequalities[J]. SIAM Journal on Control and Optimization 1999,37,589-616.
[60] Facchinei, F. STRuctural and Stability Properties of P0 Nonlinear Complementarity Problems[J]. Math. Oper. Res. 1998, 23, 735-745.
[61] Liu, J. Strong Stability in Variational Inequalities[J]. SIAM Journal on Control and Optimization 1995,33, 725–749.
[62] Pang J S, Gabriel S A. NE/SQP: A Robust Algorithm for Nonlinear complementarity problems[J]. Mathematical Programming, 1993,60:295-337.