Newton’s method for the nonlinear complementarity problem: a B- differentiable equation approach.

*(English)*Zbl 0724.90071The nonlinear complementarity problem \(F^ T(x)x=0\), \(F(x)\in R^ n_+\) can be equivalently formulated as a system of nonlinear equations \(H(x)=\min (x,F(x))=0\), where the ‘min’ operation is taken componentwise. The function H is not Fréchet differentiable in general, however, it is B-differentiable. S. Robinson was the first to study Newton’s method for equations with such functions. The present paper develops a previous work by the first author and J.-S. Pang [in: Computational solution of nonlinear systems of equations, Proc. SIAM-AMS Summer Semin., Ft. Collins/CO (USA) 1988, Lect. Appl. Math. 26, 265-284 (1990; Zbl 0699.65054)] converting the original problem into a system of equations through the use of a Minty map. At each step of the algorithm a linear system is solved and a line search for a given merit function is performed. Numerical results and a comparison with the traditional Josephy-Newton method are presented.

Reviewer: A.L.Dontchev (Ann Arbor)

##### MSC:

90C33 | Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming) |

49J50 | Fréchet and Gateaux differentiability in optimization |

65K05 | Numerical mathematical programming methods |

90C30 | Nonlinear programming |

##### Keywords:

nonlinear complementarity problem; B-differentiable; system of equations; Minty map; Josephy-Newton method
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\textit{P. T. Harker} and \textit{B. Xiao}, Math. Program. 48, No. 3 (B), 339--357 (1990; Zbl 0724.90071)

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##### References:

[1] | D.P. Bertsekas and E.M. Gafni, ”Projection methods for variational inequalities with application to the traffic assignment problem,”Mathematical Programming Study 17 (1982) 139–159. · Zbl 0478.90071 |

[2] | J.E. Dennis Jr. and R.B. Schnabel,Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Prentice-Hall, Englewood Cliffs, NJ, 1983). |

[3] | G.J. Habetler and M.M. Kostreva, ”On a direct algorithm for nonlinear complementarity problems,”SIAM Journal of Control and Optimization 16 (1978) 504–511. · Zbl 0392.90083 |

[4] | P.T. Harker, ”Alternative models of spatial competition,”Operations Research 34 (1986) 410–425. · Zbl 0602.90018 |

[5] | P.T. Harker, ”Accelerating the convergence of the diagonalization and projection algorithms for finite-dimensional variational inequalities,”Mathematical Programming 41 (1988) 29–59. · Zbl 0825.49019 |

[6] | P.T. Harker and J.S. Pang, ”Finite dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications,”Mathematical Programming B 48 (1990) 1–60. · Zbl 0738.90059 |

[7] | P.T. Harker and J.S. Pang, ”A damped-Newton method for the linear complementarity problem,” in: G. Allgower and K. Gears, eds.,Computational Solutions of Nonlinear Systems of Equations, Lectures in Applied Mathematics, Vol. 26 (American Mathematical Society, Providence, RI, 1990) 265–284. · Zbl 0699.65054 |

[8] | N.H. Josephy, ”Newton’s method for generalized equations,” Technical summary report No. 1965, Mathematics Research Center, University of Wisconsin (Madison, Wisconsin, 1979). |

[9] | N.H Josephy, ”Quasi-Newton method for generalized equations,” Technical Summary Report No. 1966, Mathematics Research Center, University of Wisconsin (Medison, WI, 1979). |

[10] | M. Kojima and S. Shindo, ”Extension of Newton and quasi-Newton methods to systems of PC equations,”Journal of the Operations Research Society of Japan 29 (1986) 352–374. · Zbl 0611.65032 |

[11] | M. Kojima, S. Mizuno and A. Yoshise, ”A polynomial-time algorithm for a class of linear complementarity problems,”Mathematical Programming 44 (1989) 1–26. · Zbl 0676.90087 |

[12] | M. Kojima, S. Mizuno and T. Name, ”A new continuation method for complementarity problems with uniform P-functions,”Mathematical Programming 43 (1989) 107–113. · Zbl 0673.90084 |

[13] | M. Kojima, S. Mizuno and T. Nome, ”Limiting behavior of trajectories generated by a continuation method for monotone complementarity problems,” Report No. B-199, Department of Information Sciences, Tokyo Institute of Technology (Tokyo, Japan, 1988). |

[14] | M.M. Kostreva, ”Direct algorithms for complementarity problems,” unpublished Ph.D. Dissertation, Department of Mathematics, Rensselaer Polytechnic Institute (Troy, NY, 1976). |

[15] | M.M. Kostreva, ”Block pivot methods for solving the complementarity problem,”Linear Algebra and its Applications 21 (1978) 207–215. · Zbl 0395.65032 |

[16] | O.L. Mangasarian, ”Equivalence of the complementarity problem to a system of nonlinear equations,”SIAM Journal on Applied Mathematics 31 (1976) 89–92. · Zbl 0339.90051 |

[17] | L. Mathiesen, ”Computation of economic equilibria by a sequence of linear complementarity problems,”Mathematical Programming Study 23 (1985) 144–162. · Zbl 0579.90093 |

[18] | L. Mathiesen, ”Computational experience in solving equilibrium models by a sequence of linear complementarity problems,”Operations Research 33 (1985) 1225–1250. · Zbl 0583.90097 |

[19] | L. Mathiesen, ”An algorithm based on a sequence of linear complementarity problems applied to a Walrasian equilibrium model: an example,”Mathematical Programming 37 (1987) 1–18. · Zbl 0613.90098 |

[20] | K.G. Murty, ”Note on a Bard-type scheme for solving the complementarity problem,”Opsearch 11 (1974) 123–130. |

[21] | K.G. Murty,Linear Complementarity, Linear and Nonlinear Programming (Helderman, Berlin, 1988). · Zbl 0634.90037 |

[22] | J.M. Ortega and W.C. Rheinboldt,Iterative Solution of Nonlinear Equations in Several Variables (Academic Press, New York, 1970). · Zbl 0241.65046 |

[23] | J.S. Pang and D. Chan, ”Iterative methods for variational and complementarity problems,”Mathematical Programming 24 (1982) 284–313. · Zbl 0499.90074 |

[24] | J.S. Pang, ”Inexact Newton methods for the nonlinear complementarity problem,”Mathematical Programming 36 (1986) 54–71. · Zbl 0613.90097 |

[25] | J.S. Pang, ”Newton’s method for B-differentiable equations,”Mathematics of Operations Research, forthcoming. · Zbl 0716.90090 |

[26] | M.J.D. Powell, ”A method for nonlinear constraints in minimization problems,” in: R. Fletcher, ed.,Optimization (Academic Press, New York, 1969). · Zbl 0194.47701 |

[27] | P.V. Preckel, ”A modified Newton method for the nonlinear complementarity problem,” Paper presented at the ORSA/TIMS Joint National Meeting, Miami Beach, Florida, October 1986. |

[28] | S.M. Robinson, ”Implicit B-differentiability in generalized equations,” Technical Summary Report No. 2854, Mathematics Research Center, University of Wisconsin (Madison, WI, 1985). |

[29] | S.M. Robinson, ”Local structure of feasible sets in nonlinear programming, Part III: stability and sensitivity,”Mathematical Programming Study 30 (1987) 45–66. · Zbl 0629.90079 |

[30] | S.M. Robinson, ”An implicit function theorem for B-differentiable functions,” Working Paper, Department of Industrial Engineering, University of Wisconsin (Madison, WI, 1988). |

[31] | S.M. Robinson, ”Newton’s method for a class of nonsmooth functions,” Working Paper, Department of Industrial Engineering, University of Wisconsin (Madison, WI, 1988). |

[32] | R.T. Rockafellar, ”Monotone operators and the proximal point algorithm,”SIAM Journal on Control and Optimization 14 (1976) 877–898. · Zbl 0358.90053 |

[33] | H. Scarf,The Computation of Economic Equilibria, (Yale University Press, New Haven, CT, 1973). · Zbl 0311.90009 |

[34] | A. Shapiro, ”On concepts of directional differentiability,” Research Report 73/88(18), Department of Mathematics and Applied Mathematics, University of South Africa (Pretoria, South Africa, 1988). · Zbl 0682.49015 |

[35] | P.K. Subramanian, ”Gauss-Newton methods for the nonlinear complementarity problem,” Technical Summary Report No. 2845, Mathematics Research Center, University of Wisconsin (Madison, WI, 1985). |

[36] | P.K. Subramanian, ”A note on the least two norm solutions of monotone complementarity problems,”Applied Mathematics Letters 1 (1988) 395–397. · Zbl 0706.65061 |

[37] | R.L. Tobin, ”A variable dimension solution approach for the general spatial price equilibrium problem,”Mathematical Programming 40 (1988) 33–51. · Zbl 0645.90095 |

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