正则稀疏优化模型及算法研究综述

A Survey on Regularized Sparse Optimization Models and Algorithms

摘要: 稀疏优化在工业工程等实际问题中具有十分重要的作用。在过去的几十年里，很多实际问题可以归纳成正则稀疏优化模型并求解出欠定系统的稀疏解，因此其改进和算法设计得到广泛的研究。正则稀疏优化模型不仅可以将原问题降维，而且可以将不适定的问题转换为适定问题。关键问题是如何构造正则项，使得模型具有好的稀疏解的同时还有很好的泛化能力。本文，我们重点关注近30年来正则稀疏优化模型及算法的研究进展，总结归纳了最具代表性的几种正则稀疏优化模型及算法。最后，结合最新研究成果，针对损伤识别、故障诊断、超分辨率重建和电阻抗层析成像等实际问题，构造不同的新正则稀疏优化模型，并探讨其可以研究发展的方向以及更广阔的应用前景。

Abstract: Sparse optimization plays a very important role in practical problems such as industrial engineering. In the past decades, many practical problems can be generalized into regularized sparse optimization models to solve the sparse solutions of underdetermined systems. Therefore, the improvement of such models and the design of algorithms have been widely studied. The regularized sparse optimization model can reduce the dimension of the original problem and transform the ill-posed problem into a well-posed problem. The key point is how to choose and construct the regularization terms so that the model can obtain a good sparse solution with good generalization ability. In this paper, we focus on the research progress of regularized sparse optimization models and algorithms in the past 30 years, and summarize the most representative models and algorithms. In addition, according to the latest research results, different new regularized sparse optimization models are constructed to solve practical problems such as damage identification, fault diagnosis, super-resolution reconstruction and electrical impedance tomography, and the development direction and broader application prospect of these models are discussed.