We consider the setup of a constrained optimization problem with two agents $E_1$ and $E_2$ who jointly wish to learn the optimal solution set while keeping their feasible sets $\mathcal{P}_1$ and $\mathcal{P}_2$ private from each other. The objective function $f$ is globally known and each feasible set is a collection of points from a global alphabet. We adopt a sequential symmetric private information retrieval (SPIR) framework where one of the agents (say $E_1$) privately checks in $\mathcal{P}_2$, the presence of candidate solutions of the problem constrained to $\mathcal{P}_1$ only, while learning no further information on $\mathcal{P}_2$ than the solution alone. Further, we extract an information theoretically private threshold PSI (ThPSI) protocol from our scheme and characterize its download cost. We show that, compared to privately acquiring the feasible set $\mathcal{P}_1\cap \mathcal{P}_2$ using an SPIR-based private set intersection (PSI) protocol, and finding the optimum, our scheme is better as it incurs less information leakage and less download cost than the former. Over all possible uniform mappings of $f$ to a fixed range of values, our scheme outperforms the former with a high probability.

翻译：我们考虑一个带有两个代理$E_1$和$E_2$的约束优化问题设置，这两个代理希望共同学习最优解集，同时保持各自可行集$\mathcal{P}_1$和$\mathcal{P}_2$的隐私性。目标函数$f$是全局已知的，每个可行集是由全局字母表中的点构成的集合。我们采用一种顺序对称私有信息检索（SPIR）框架，其中一个代理（例如$E_1$）私下检查候选解在$\mathcal{P}_2$中的存在性（这些候选解仅受限于$\mathcal{P}_1$的问题约束），同时除解本身外不学习关于$\mathcal{P}_2$的任何进一步信息。此外，我们从该方案中提取了一个信息论意义上的私有阈值PSI（ThPSI）协议，并刻画了其下载代价。我们证明，与使用基于SPIR的私有集合交集（PSI）协议来私密获取可行集$\mathcal{P}_1\cap \mathcal{P}_2$并寻找最优解相比，我们的方案更具优势，因为它产生的信息泄露更少，且下载代价更低。在$f$到固定值域的所有均匀映射中，我们的方案以高概率优于前者。