稀疏相位检索问题的光滑化方法研究

Research on Smoothing Method for Sparse Phase Retrieval Problems

摘要: 对稀疏相位检索问题，本文建立了稀疏最小一乘优化模型。首先，将不连续的稀疏正则函数松弛为连续的Capped-L1正则函数。利用方向导数和方向稳定点建立了该松弛问题的一阶最优性条件。进一步，分析了方向稳定点的下界性质，并建立了松弛问题与原始问题全局解的等价性。最后，本文建议采用光滑化方法求解该问题。构造了目标函数的光滑化函数，并建立了光滑问题与松弛问题解的一致性，为使用光滑化方法求解该问题提供了理论保证。

Abstract: For sparse phase retrieval problems, this paper constructed a class of sparse least absolute deviation optimization model. First, we used the continuous Capped-L1 regularization function to relax the discontinuous sparsity function. By virtue of directional derivatives and the directional stationary points, we provided the first- order optimality condition for the relaxed optimization problem. Furthermore, we derived the lower bound property of the directional stationary points, based on which we proved the equivalence between the original problem and the relaxed problem in the sense of global solutions. Finally, we advised using smoothing methods to solve the relaxed problem. We proposed a class of smooth function to approximate the objective function, and established the consistency of the solutions of the smoothing problem and the relaxed problem. Our result provides a theoretical basis for using smoothing methods to solve sparse phase retrieval problems.