GAMS/PATH User Guide Version 4.3

Chapter 3
Advanced Topics
This chapter discusses some of the difficulties encountered when dealing with
complementarity problems. We start off with a very formal definition of a
complementarity problem which is used in later sections on merit functions
and ill-defined, poorly-scaled, and singular models.
3.1 Formal Definition of MCP
The mixed complementarity problem is defined by a function, F : D → R n
where D ⊆ R n is the domain of F , and possibly infinite lower and upper
bounds, ℓ and u. Let C := {x ∈ R n | ℓ ≤ x ≤ u}, a Cartesian product of
closed (possibly infinite) intervals. The problem is given as
MCP : find x ∈ C ∩ D s.t. 〈F (x), y− x〉 ≥0, ∀y ∈ C.
This formulation is a special case of the variational inequality problem defined
by F and a (nonempty, closed, convex) set C. Special choices of ℓ and u lead
to the familiar cases of a system of nonlinear equations
F (x) =0
(generated by ℓ ≡−∞, u ≡ +∞) and the nonlinear complementarity problem
0 ≤ x ⊥ F (x) ≥ 0
(generated using ℓ ≡ 0, u ≡ +∞).
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