AI 中文总结
研究紧凸集上开域凸优化问题，采用域感知阿米尔乔投影梯度法，在边界爆破条件下有相关性质保证收敛等，还将此框架用于可控性评分，分析了可行性、唯一性等，通过示例展示了对最优分配的影响。
AI 中文摘要
我们研究在紧凸集上的凸优化，此时目标函数仅在开域上光滑且凸。在边界爆破条件下，每个可行初始化产生一个与目标域补集分离的紧不变子水平集且存在最优解。对于域感知的阿米尔乔投影梯度法，安全邻域分析建立了明确的目标评估、有限回溯、充分下降以及接受步长的特定运行但与迭代无关的正下界。这些性质产生显式次线性目标和平稳性保证以及全迭代序列的收敛性。正曲率限制在可行位移方向上进一步保证唯一性和线性收敛。我们将该框架应用于具有规定输入方向和紧凸分配约束的可控性评分。可行性由可进行正分配的输入方向的可控性精确表征，而格拉姆映射的受限单射性保证最优分配的唯一性并提供显式强凸性界。一个有向网络示例说明了候选排除如何以依赖于准则和视界的方式保持或破坏可行性并改变最优分配。
英文摘要
We study convex optimization over a compact convex set when the objective is smooth and convex only on an open domain. Under a boundary-blow-up condition, every feasible initialization yields a compact invariant sublevel set separated from the complement of the objective domain, and an optimal solution exists. For a domain-aware Armijo projected-gradient method, a safe-neighborhood analysis establishes well-defined objective evaluations, finite backtracking, sufficient decrease, and a run-specific but iteration-independent positive lower bound on the accepted step sizes. These properties yield explicit sublinear objective and stationarity guarantees, together with convergence of the full iterate sequence. Positive curvature restricted to feasible displacement directions further guarantees uniqueness and linear convergence. We apply the framework to controllability scoring with prescribed input directions and compact convex allocation constraints. Feasibility is characterized exactly by controllability of the input directions eligible for positive allocation, while restricted injectivity of the Gramian map guarantees uniqueness of the optimal allocation and provides explicit strong-convexity bounds. A directed-network example illustrates how candidate exclusion can preserve or destroy feasibility and alter the optimal allocation in a criterion- and horizon-dependent manner.
