We aim to find a solution \({\varvec{x}}\in {\mathbb {C}}^n\) to a system of quadratic equations of the form \(b_i=|{\varvec{a}}_i^*{\varvec{x}}|^2\), \(i=1,2,\ldots ,m\), e.g., the well-known NP-hard phase retrieval problem. As opposed to recently proposed state-of-the-art nonconvex methods, we revert to the semidefinite relaxation (SDR) PhaseLift convex formulation and propose a successive and incremental nonconvex optimization algorithm, termed as IncrePR, to indirectly minimize the resulting convex problem on the cone of positive semidefinite matrices. Our proposed method overcomes the excessive computational cost of typical SDP solvers as well as the need of a good initialization for typical nonconvex methods. For Gaussian measurements, which is usually needed for provable convergence of nonconvex methods, restart-IncrePR solving three consecutive PhaseLift problems outperforms state-of-the-art nonconvex gradient flow based solvers with a sharper phase transition of perfect recovery and typical convex solvers in terms of computational cost and storage. For more challenging structured (non-Gaussian) measurements often occurred in real applications, such as transmission matrix and oversampling Fourier transform, IncrePR with several consecutive repeats can be used to find a good initial guess. With further refinement by local nonconvex solvers, one can achieve a better solution than that obtained by applying nonconvex gradient flow based solvers directly when the number of measurements is relatively small. Extensive numerical tests are performed to demonstrate the effectiveness of the proposed method.

我们旨在在{\ mathbb {C}} ^ n \）中找到\（b_i = | {\ varvec {a}} _ i形式的二次方程组的解决方案\（{\ varvec {x}} \ ^ * {\ varvec {x}} | ^ 2 \），\（i = 1,2，\ ldots，m \），例如，众所周知的NP硬相位检索问题。与最近提出的最新非凸方法相反，我们恢复为半定松弛（SDR）PhaseLift凸公式，并提出了一种连续的和增量的非凸优化算法，称为IncrePR，以间接最小化在正半定矩阵的圆锥上产生的凸问题。我们提出的方法克服了典型SDP求解器的计算成本过高以及对典型非凸方法进行良好初始化的需求。对于高斯测量，这通常是可证明收敛的非凸方法所必需的，重新启动-IncrePR解决三个连续的PhaseLift问题的性能优于基于最新的非凸梯度流的求解器，其具有完美的相变和完美的典型相变求解器计算成本和存储量。对于更具挑战性的结构化（非高斯）测量，经常在实际应用中进行，例如传输矩阵和过采样傅立叶变换，IncrePR与几个连续的重复可以用来找到一个很好的初步猜测。通过局部非凸求解器的进一步改进，与在测量次数相对较少时直接应用基于非凸梯度流的求解器相比，可以实现更好的解决方案。进行了广泛的数值测试，以证明所提出的方法的有效性。