Mixed Complementarity Problem (MCP) is a problem formulation in mathematical programming

Mathematical Programming

Mathematical Programming, established in 1971, and published by Springer Science+Business Media, is the official scientific journal of the Mathematical Optimization Society. It currently consists of two series: A and B. The "A" series contains general publications. The "B" series focuses on topical...

. Many well-known problem types are special cases of, or may be reduced to MCP. It is a generalization of Nonlinear complementarity problem

Nonlinear complementarity problem

In applied mathematics, a nonlinear complementarity problem with respect to a mapping ƒ : Rn → Rn, denoted by NCPƒ, is to find a vector x ∈ Rn such that...

 (NCP).

Definition

The mixed complementarity problem is defined by a mapping , lower values and upper values .

The solution of the MCP is a vector such that for each index one of the following alternatives holds:

• ;
• ;
• .

Another definition for MCP is: it is a variational inequality

Variational inequality

In mathematics, a variational inequality is an inequality involving a functional, which has to be solved for all the value of a given variable, belonging usually to a convex set...

 on the parallelepiped

Parallelepiped

In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms. By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclidean geometry, its definition encompasses all four concepts...

.