关于具有谱约束的可行性问题
On feasibility problems with spectral constraints
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中文总结 AI 辅助
研究矩阵可行性问题，利用谱转移恒等式简化投影到特定集合C上的操作，将投影器嵌入交替投影循环，对线性和椭球约束的K进行实验，揭示了快速可行性、缓慢尾部收敛或信息性不可行平台三种情况。
中文摘要 AI 辅助
我们研究矩阵可行性问题“在K与C的交集中找到X”，其中K是一个闭凸矩阵集，C={X: σ(X)∈S}由有序奇异值上的凸约束S定义。利用经典谱转移恒等式，投影到C上可简化为奇异值分解加上一个小的二次规划。对于七个自然多面体S，我们将此投影器嵌入到一个普通的交替投影（AP）循环中。我们对K的两个具体族进行实验——线性约束（仿射子空间与逐元素框的交集）和椭球约束（非中心各向异性Frobenius椭球）——尽管该方法同样适用于更一般的凸约束。实验揭示了三种情况：快速可行性、缓慢的尾部收敛或信息性不可行平台。
英文摘要
We study matrix feasibility problems "find X in K intersect C" where K is a closed convex matrix set and C = {X : sigma(X) in S} is defined by a convex constraint S on the ordered singular values. Using the classical spectral transfer identity, projection onto C reduces to an SVD plus a small quadratic program. For seven natural polyhedral S we embed this projector in a plain alternating projection (AP) loop. We experiment with two concrete families of K - linear constraints (an affine subspace intersected with an entrywise box) and ellipsoidal constraints (a non-centered anisotropic Frobenius ellipsoid) - although the method applies equally to more general convex constraints. The experiments expose three regimes: rapid feasibility, slow tail convergence, or informative infeasibility plateaus.
