  We introduce a new technique for proving membership of problems in FIXP - the
class capturing the complexity of computing a fixed-point of an algebraic
circuit. Our technique constructs a "pseudogate" which can be used as a black
box when building FIXP circuits. This pseudogate, which we term the "OPT-gate",
can solve most convex optimization problems. Using the OPT-gate, we prove new
FIXP-membership results, and we generalize and simplify several known results
from the literature on fair division, game theory and competitive markets.
  In particular, we prove complexity results for two classic problems:
computing a market equilibrium in the Arrow-Debreu model with general concave
utilities is in FIXP, and computing an envy-free division of a cake with very
general valuations is FIXP-complete. We further showcase the wide applicability
of our technique, by using it to obtain simplified proofs and extensions of
known FIXP-membership results for equilibrium computation for various types of
strategic games, as well as the pseudomarket mechanism of Hylland and
Zeckhauser.
