非负张量分解的优化模型与算法研究

Nonnegative tensor factorization arises frequently from many fields of modern science and engineering, where nonnegativity, sparsity and low rank are favorable and important characteristics. On the other hand, these cause the resulted models nonconvex, which is the intrinsic difficulty for theoretical analysis and algorithm design. Due to their wonderful properties such as "large to small", "difficult to easy", and "divide-and-conquer", decomposition methods are among the fundamental methods for solving large scale optimization problems. This project aims to study large nonconvex optimization problems, whose main contents are: (1) to design several decomposition methods for some concrete application models from nonnegative tensor factorization by fully utilizing the good properties of each component function such as smoothness, and so on, to compensate the difficulty caused by the nonconvexity; (2) to propose several accelerated versions to make the methods more efficient by using the accelerated techniques such as inertial; (3) to design inexact methods with new accuracy criteria, and ensure their global convergence, stability and fastness; (4) to establish some new and practical results of error bounds, KL inequality, and analyze connections between them, and analyze convergence and rate of convergence of the methods by using these results; (5) to tailor the algorithms for some practical problems with similar properties accordingly. The study focuses on algorithm design, as well as on theoretical analysis and applications. The results can help us find the real optimal solutions to the nonconvex models arising directly from the applications.

非负张量分解在现代科学和工程中有广泛的应用，其中非负、稀疏、低秩是实际问题内在的可利用的重要特征，而非凸性是理论分析和算法设计的难点。分解算法独特的化大为小、化难为易、分而治之的特点使其成为处理大规模复杂优化问题的首选算法。本项目针对这类非凸优化问题开展研究，内容有：充分利用模型的其它性质（如光滑性等）弥补非凸性所带来的困难，设计针对特定模型的高效算法；引入惯性加速技巧，设计算法加速策略，提高其效率；设计多种实用的非精确准则，使算法更稳健实用；通过建立新的误差界、KL不等式，以及它们之间的关联，分析算法的收敛性和收敛率；对算法进行适当的“裁剪”，为实际应用问题设计量身定制的算法。本项目以算法设计为主，兼有理论分析、应用研究。通过本项目的研究，扩大分解算法的适用范围，将只能应用于凸问题的算法推广到非负张量分解等大规模非凸优化问题，为求解问题的实际数学模型提供可能。

现代科学和工程中有大量的非凸优化问题，非负张量分解就是其中的一个特殊的例子，其中非负、稀疏、低秩是实际问题内在的可利用的重要特征，而非凸性是理论分析和算法设计的难点。分解算法独特的化大为小、化难为易、分而治之的特点使其成为处理大规模复杂优化问题的首选算法。本项目从凸优化理论出发，针对这类非凸优化问题开展研究，内容有：充分利用模型的其它性质（如光滑性等）弥补非凸性所带来的困难，设计针对特定模型的高效算法；引入惯性加速技巧，不定正则项，设计算法加速策略，提高其效率；设计多种实用的非精确准则，使算法更稳健实用；通过建立新的误差界、KL不等式，以及它们之间的关联，分析算法的收敛性和收敛率；对特殊的非凸问题，如DC规划、伪凸随机优化问题进行研究，设计高效定制算法。本项目以算法设计为主，兼有理论分析、应用研究。通过本项目的研究，扩大分解算法的适用范围，将只能应用于凸问题的算法推广到大规模非凸优化问题，为求解问题的实际数学模型提供可能。