A quasisecant method for solving a system of nonsmooth equations

In this paper, the solution of nonsmooth equations is studied. We first transform the problem into an equivalent nonsmooth optimization problem and then the quasisecant method is introduced to solve it. Some nonsmooth equations that have arisen from bilevel programming problems are solved by our proposed method. The numerical results show the effectiveness and efficiency of our proposed method.

[1]  Jong-Shi Pang,et al.  Iterative methods for variational and complementarity problems , 1982, Math. Program..

[2]  L. Qi,et al.  A Survey of Some Nonsmooth Equations and Smoothing Newton Methods , 1999 .

[3]  Adil M. Bagirov,et al.  Subgradient Method for Nonconvex Nonsmooth Optimization , 2013, J. Optim. Theory Appl..

[4]  Huifu Xu,et al.  New Version of the Newton Method for Nonsmooth Equations , 1997 .

[5]  A. Shapiro On concepts of directional differentiability , 1990 .

[6]  S. M. Robinson Generalized equations and their solutions, Part I: Basic theory , 1979 .

[7]  Krzysztof C. Kiwiel,et al.  Exact penalty functions in proximal bundle methods for constrained convex nondifferentiable minimization , 1991, Math. Program..

[8]  Christian Kanzow,et al.  A continuation method for (strongly) monotone variational inequalities , 1998, Math. Program..

[9]  Adil M. Bagirov,et al.  A quasisecant method for minimizing nonsmooth functions , 2010, Optim. Methods Softw..

[10]  Olvi L. Mangasarian,et al.  A class of smoothing functions for nonlinear and mixed complementarity problems , 1996, Comput. Optim. Appl..

[11]  Krzysztof C. Kiwiel,et al.  Proximity control in bundle methods for convex nondifferentiable minimization , 1990, Math. Program..

[12]  Christian Kanzow,et al.  Some Noninterior Continuation Methods for Linear Complementarity Problems , 1996, SIAM J. Matrix Anal. Appl..

[13]  Defeng Sun,et al.  Secant methods for semismooth equations , 1998, Numerische Mathematik.

[14]  Kaisa Miettinen,et al.  New limited memory bundle method for large-scale nonsmooth optimization , 2004, Optim. Methods Softw..

[15]  Adil M. Bagirov,et al.  A derivative-free method for linearly constrained nonsmooth optimization , 2006 .

[16]  B. Curtis Eaves,et al.  On the basic theorem of complementarity , 1971, Math. Program..

[17]  A. Bagirov,et al.  Discrete Gradient Method: Derivative-Free Method for Nonsmooth Optimization , 2008 .

[18]  J. J. Moré,et al.  Smoothing of mixed complementarity problems , 1995 .

[19]  Christian Kanzow,et al.  A QP-free constrained Newton-type method for variational inequality problems , 1999, Math. Program..

[20]  C. T. Kelley,et al.  Pseudo-Transient Continuation for Nonsmooth Nonlinear Equations , 2005, SIAM J. Numer. Anal..

[21]  P. Neittaanmäki,et al.  Nonsmooth Optimization: Analysis and Algorithms with Applications to Optimal Control , 1992 .

[22]  Yu. S. Ledyaev,et al.  Nonsmooth analysis and control theory , 1998 .

[23]  L. Qi,et al.  A Globally Convergent Successive Approximation Method for Severely Nonsmooth Equations , 1995 .

[24]  Patrick T. Harker,et al.  Smooth Approximations to Nonlinear Complementarity Problems , 1997, SIAM J. Optim..

[25]  Kaisa Miettinen,et al.  Globally convergent limited memory bundle method for large-scale nonsmooth optimization , 2007, Math. Program..

[26]  Michal Kočvara,et al.  Nonsmooth approach to optimization problems with equilibrium constraints : theory, applications, and numerical results , 1998 .

[27]  Stephen M. Robinson,et al.  Strongly Regular Generalized Equations , 1980, Math. Oper. Res..

[28]  Jonathan F. Bard,et al.  Bundle Trust-Region Algorithm for Bilinear Bilevel Programming , 2001 .

[29]  M. Fukushima,et al.  Nonsmooth Equation Based BFGS Method for Solving KKT Systems in Mathematical Programming , 2001 .

[30]  Liqun Qi,et al.  Trust Region Algorithms for Solving Nonsmooth Equations , 1995, SIAM J. Optim..

[31]  N. Josephy Newton's Method for Generalized Equations. , 1979 .

[32]  Defeng Sun,et al.  Newton and Quasi-Newton Methods for a Class of Nonsmooth Equations and Related Problems , 1997, SIAM J. Optim..

[33]  N. Josephy Quasi-Newton methods for generalized equations , 1979 .

[34]  Jong-Shi Pang,et al.  Newton's Method for B-Differentiable Equations , 1990, Math. Oper. Res..