混合互补问题-mixed complementarity problem-mcp，比线性互补和非线性互补更广泛

最新推荐文章于 2025-08-13 15:02:16 发布

原创最新推荐文章于 2025-08-13 15:02:16 发布 · 2k 阅读

3 ·

CC 4.0 BY-SA版权

文章标签：

#运筹学 #mpec

笔记专栏收录该内容

10 篇文章

订阅专栏

本文详细解析了数学优化中MCP（混合互补问题）、LCP（线性互补问题）及NCP（非线性互补问题）的概念与形式，并通过一个GAMS实例展示了如何实现MCP。

MCP is more general than Linear Complementarity Problems (LCP) and Nonlinear Complementarity Problems (NCP).

The form of MCP is as follows:

F(x) ⟂ lb ≤ x ≤ ub

which means

x = lb, then F(x) ≥ 0 #lb means lower bound
lb < x < ub, then F(x) = 0
x = ub, then F(x) ≤ 0 #ub means upper bound

When there is no upper bound ub, and the lower bound lb=0, then it is a regular Nonlinear Complementarity Problem (NCP) of the form:

0 ≤ F(x) ⟂ x ≥ 0

which means, for each $x_i$ , we can get the below expressions:

F(x)' x = 0, F(x) ≥ 0, x ≥ 0

or, we can write: $F_i(x)$ ⟂ $x_i$ $\in$ $[lb,ub]$ .

When F(x) is a linear mapping such as F(x) = Ax + q with matrix A and vector q, then it is a Linear Complementarity Problem (LCP).

Note: ⟂ denotes perpendicular symbol.

An example from GAMS tutorial doc:

Among which,

where,

Then implement it in GAMS with the codes as:

$TITLE simple mpec example

variable f, x1, x2, y1, y2;
positive variable y1;
y2.lo = -1;
y2.up =  1;

equations cost, g, h1, h2;

* define the function f
cost..  f =E= x1 + x2; 
g..     sqr(x1) + sqr(x2) =L= 1;
h1..    x1 =G= y1 - y2 + 1;
h2..    x2 + y2 =N= 0;

model example / cost, g, h1.y1, h2.y2 /;
solve example using mpec min f;

the results are shown below pic with red mark: