# Nonlinear complementarity problem

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Mathematics problem

In [applied mathematics](/source/Applied_mathematics), a **nonlinear complementarity problem** (**NCP**) with respect to a mapping *ƒ* : **R***n* → **R***n*, denoted by NCP*ƒ*, is to find a vector *x* ∈ **R***n* such that

- x ≥ 0 , f ( x ) ≥ 0 and x T f ( x ) = 0 {\displaystyle x\geq 0,\ f(x)\geq 0{\text{ and }}x^{T}f(x)=0}

where *ƒ*(*x*) is a smooth mapping. The case of a discontinuous mapping was discussed by Habetler and Kostreva (1978).

## References

- Ahuja, Kapil; Watson, Layne T.; Billups, Stephen C. (December 2008). "Probability-one homotopy maps for mixed complementarity problems". *Computational Optimization and Applications*. **41** (3): 363–375. [doi](/source/Doi_(identifier)):[10.1007/s10589-007-9107-z](https://doi.org/10.1007%2Fs10589-007-9107-z). [hdl](/source/Hdl_(identifier)):[10919/31539](https://hdl.handle.net/10919%2F31539).

- Cottle, Richard W.; Pang, Jong-Shi; Stone, Richard E. (1992). *The linear complementarity problem*. Computer Science and Scientific Computing. Boston, MA: Academic Press, Inc. pp. xxiv+762 pp. [ISBN](/source/ISBN_(identifier)) [0-12-192350-9](https://en.wikipedia.org/wiki/Special:BookSources/0-12-192350-9). [MR](/source/MR_(identifier)) [1150683](https://mathscinet.ams.org/mathscinet-getitem?mr=1150683).

v t e Complementarity problems and algorithms Complementarity Problems Linear programming (LP) Quadratic programming (QP) Linear complementarity problem (LCP) Mixed linear (MLCP) Mixed (MCP) Nonlinear (NCP) Basis-exchange algorithms Simplex (Dantzig) Revised simplex Criss-cross Lemke

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