By Pang J.

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**Extra resources for A B-differentiable equation-based, globally and locally quadratically convergent algorithm for nonlinear programs, complementarity and variational inequality problems**

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T. S. Pang, "Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications," Mathematical Programming (Series 13) 48 (1990) 161-220. T. S. L. Allgower and K. , Computational Solution of Nonlinear ,Systems of Equations. Lectures in Applied Mathematics No. 26 (American Mathematical Society, Providence, RI, 1990) pp. 265-284. T. Harker and B. Xiao, "Newton's method for the nonlinear complementarity problem: a B-ditterentiable equation approach," Mathematical Programming (Series B) 48 (1990) 339-358.

Murty, Linear Complementarity, Linear and Nonlinear Programming (Helderman-Verlag, Berlin, 1988). M. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables (Academic Press, New York, 1970). S. Pang, "Newton's method for B-differentiable equations," Mathematics of Operations Research 15 (1990) 331-341. S. Pang and D. Chan, "Iterative methods for variational and complementarity problems," Mathematical programming 24 (1982) 284-313. M. Robinson, "Strongly regular generalized equations," Mathematics of Operations Research 5 (1980) 43-62.

By the differentiability assumptions and the mean-value theorem, it follows that the right-hand vector is bounded above in norm by where c' is a certain constant dependent only on ~. )~)~ exists, and is bounded in norm by a constant dependent only on ~. Consequently, there exists a constant c > 0 such that for all k sufficiently large, 1| / llz -Sll Now, the desired inequality (42) follows by observing that the left-hand quantity in the above inequality is equal to IIz~*'-~ll. This completes the p r o o f of the theorem.