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.

翻译：我们提出了一种新方法，用于证明问题属于FIXP——这一复杂性类描述了代数电路不动点计算的复杂度。该方法构造了一种“伪门”，可在构建FIXP电路时作为黑盒使用。我们称这种伪门为“OPT门”，它能够解决大多数凸优化问题。借助OPT门，我们证明了新的FIXP成员性结果，并归纳简化了公平分配、博弈论和竞争市场文献中若干已知结论。具体而言，我们为两个经典问题给出了复杂度结果：带一般凹效用的Arrow-Debreu模型中的市场均衡计算属于FIXP，以及极宽泛估值下的无嫉妒蛋糕分配问题是FIXP完全的。我们还通过利用该技术简化证明并扩展了已知的FIXP成员性结果（涵盖各类策略博弈的均衡计算及Hylland-Zeckhauser伪市场机制），进一步展示了其广泛适用性。