We introduce a general technique for proving membership of search problems with exact rational solutions in PPAD, one of the most well-known classes containing total search problems with polynomial-time verifiable solutions. In particular, we construct a "pseudogate", coined the linear-OPT-gate, which can be used as a "plug-and-play" component in a piecewise-linear (PL) arithmetic circuit, as an integral component of the "Linear-FIXP" equivalent definition of the class. The linear-OPT-gate can solve several convex optimization programs, including quadratic programs, which often appear organically in the simplest existence proofs for these problems. This effectively transforms existence proofs to PPAD-membership proofs, and consequently establishes the existence of solutions described by rational numbers. Using the linear-OPT-gate, we are able to significantly simplify and generalize almost all known PPAD-membership proofs for finding exact solutions in the application domains of game theory, competitive markets, auto-bidding auctions, and fair division, as well as to obtain new PPAD-membership results for problems in these domains.

翻译：我们提出了一种通用技术，用于证明具有精确有理数解搜索问题属于PPAD类——这是最著名的类之一，包含所有具有多项式时间可验证解的总搜索问题。具体而言，我们构造了一种名为"线性-OPT-门"的"伪门"，可作为分段线性算术电路中的"即插即用"组件，成为该类"Linear-FIXP"等价定义中不可或缺的组成部分。该线性-OPT-门可求解包括二次规划在内的多种凸优化问题，而这类问题往往自然出现在这些问题的最简存在性证明中。这有效地将存在性证明转化为PPAD隶属度证明，从而确立了有理数解的存在性。通过使用线性-OPT-门，我们能够显著简化并泛化博弈论、竞争市场、自动竞价拍卖和公平分配等领域中几乎所有已知的精确解PPAD隶属度证明，同时在这些领域中获得新的PPAD隶属度结论。