We consider a parametric family of quadratically constrained quadratic programs and their associated semidefinite programming (SDP) relaxations. Given a nominal value of the parameter at which the SDP relaxation is exact, we study conditions (and quantitative bounds) under which the relaxation will continue to be exact as the parameter moves in a neighborhood around the nominal value. Our framework captures a wide array of statistical estimation problems including tensor principal component analysis, rotation synchronization, orthogonal Procrustes, camera triangulation and resectioning, essential matrix estimation, system identification, and approximate GCD. Our results can also be used to analyze the stability of SOS relaxations of general polynomial optimization problems.

我们考虑二次约束二次规划的参数族及其相关的半定规划（SDP）松弛。给定 SDP 松弛精确的参数标称值，我们研究了在参数在标称值附近移动时松弛将继续精确的条件（和定量界限）。我们的框架捕获了广泛的统计估计问题，包括张量主成分分析、旋转同步、正交 Procrustes、相机三角测量和后方交会、基本矩阵估计、系统识别和近似 GCD。我们的结果也可用于分析一般多项式优化问题的 SOS 松弛的稳定性。