  In a Mathematical Program with Generalized Complementarity Constraints
(MPGCC), complementarity relationships are imposed between each pair of
variable blocks. MPGCC includes the traditional Mathematical Program with
Complementarity Constraints (MPCC) as a special case. On account of the
disjunctive feasible region, MPCC and MPGCC are generally difficult to handle.
The $\ell_1$ penalty method, often adopted in computation, opens a way of
circumventing the difficulty. Yet it remains unclear about the exactness of the
$\ell_1$ penalty function, namely, whether there exists a sufficiently large
penalty parameter so that the penalty problem shares the optimal solution set
with the original one. In this paper, we consider a class of MPGCCs that are of
multi-affine objective functions. This problem class finds applications in
various fields, e.g., the multi-marginal optimal transport problems in
many-body quantum physics and the pricing problem in network transportation. We
first provide an instance from this class, the exactness of whose $\ell_1$
penalty function cannot be derived by existing tools. We then establish the
exactness results under rather mild conditions. Our results cover those
existing ones for MPCC and apply to multi-block contexts.
