  Motivated by the problem of finding dual representations for quasiconvex
systemic risk measures in financial mathematics, we study quasiconvex
compositions in an abstract infinite-dimensional setting. We calculate an
explicit formula for the penalty function of the composition in terms of the
penalty functions of the ingredient functions. The proof makes use of a
nonstandard minimax inequality (rather than equality as in the standard case)
that is available in the literature. In the second part of the paper, we apply
our results in concrete probabilistic settings for systemic risk measures, in
particular, in the context of Eisenberg-Noe clearing model. We also provide
novel economic interpretations of the dual representations in these settings.
