Mixed complementarity problem explained

Mixed Complementarity Problem (MCP) is a problem formulation in mathematical programming. Many well-known problem types are special cases of, or may be reduced to MCP. It is a generalization of nonlinear complementarity problem (NCP).

Definition

The mixed complementarity problem is defined by a mapping

F(x): R ^{n} \to R ^{n} , lower values \ell _{i} \in R \cup \{ -inf ty\} and upper values u _{i} \in R \cup\{ inf ty\} .

The solution of the MCP is a vector

x \in R ^{n} such that for each index i \in \{ 1 , \ldots, n\} one of the following alternatives holds:x _{i} = \ell _{i,} F _{i(x)} \ge 0 ;\ell _{i} < x _{i} < u _{i,} F _{i(x)} = 0 ;x _{i} = u _{i,} F _{i(x)} \le 0 .[\ell, u] .

See also

• Complementarity theory

References

• Web site: Stephen C. Billups. [https:/ftp.cs.wisc.edu/math-prog/tech-reports/95-14.ps Algorithms for complementarity problems and generalized equations]. 1995. PS. 2006-08-14.
• Book: Francisco Facchinei, Jong-Shi Pang. Finite-Dimensional Variational Inequalities and Complementarity Problems, Volume I. 2003.