## Abstract

In this paper, we present a new iterative method for solving the nonlinear complementarity problem. This method, which we call NE/SQP (for Nonsmooth Equations/Successive Quadratic Programming), is a damped Gauss—Newton algorithm applied to solve a certain nonsmooth-equation formulation of the complementarity problem; it is intended to overcome a major deficiency of several previous methods of this type. Unlike these earlier algorithms whose convergence critically depends on a solvability assumption on the subproblems, the NE/SQP method involves solving a sequence of nonnegatively constrained convex quadratic programs of the least-squares type; the latter programs are always solvable and their solution can be obtained by a host of efficient quadratic programming subroutines. Hence, the new method is a robust procedure which, not only is very easy to describe and simple to implement, but also has the potential advantage of being capable of solving problems of very large size. Besides the desirable feature of robustness and ease of implementation, the NE/SQP method retains two fundamental attractions of a typical member in the Gauss—Newton family of algorithms; namely, it is globally and locally quadratically convergent. Besides presenting the detailed description of the NE/SQP method and the associated convergence theory, we also report the numerical results of an extensive computational study which is aimed at demonstrating the practical efficiency of the method for solving a wide variety of realistic nonlinear complementarity problems.

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This work was based on research supported by the National Science Foundation under grants ECS-8717968 and DDM-9104078.

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Pang, JS., Gabriel, S.A. NE/SQP: A robust algorithm for the nonlinear complementarity problem.
*Mathematical Programming* **60**, 295–337 (1993). https://doi.org/10.1007/BF01580617

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DOI: https://doi.org/10.1007/BF01580617

### Key words

- Nonlinear complementarity problem
- nonsmooth equations
- sequential quadratic programming
- equilibrium programming