论文标题
部分可观测时空混沌系统的无模型预测
The Sparse(st) Optimization Problem: Reformulations, Optimality, Stationarity, and Numerical Results
论文作者
论文摘要
我们考虑非线性约束和目标函数的稀疏优化问题，该问题由一般平滑映射的总和和由$ \ ell_0 $ -quasi-norm定义的附加术语给出。该术语用于获得稀疏的解决方案，但由于其非概念性和非平滑度而难以处理（稀疏性改善术语甚至是不连续的）。本文的目的是将该计划的两种重新汇总为具有互补性限制的平滑非线性程序。我们表明，这些程序在本地和全球最小值方面是等效的，并引入了问题缩写的平稳性概念，事实证明，这与两个重新质量的问题的标准KKT条件相吻合。此外，研究了稀疏优化问题的合适约束资格以及二阶条件。然后使用这些来证明三种拉格朗日 - 纽顿型方法是局部快速收敛的。不同类别的测试问题的数值结果表明，这些方法可用于大大改善其他一些（全球收敛性）方法获得稀疏优化问题所获得的稀疏解决方案。
We consider the sparse optimization problem with nonlinear constraints and an objective function, which is given by the sum of a general smooth mapping and an additional term defined by the $ \ell_0 $-quasi-norm. This term is used to obtain sparse solutions, but difficult to handle due to its nonconvexity and nonsmoothness (the sparsity-improving term is even discontinuous). The aim of this paper is to present two reformulations of this program as a smooth nonlinear program with complementarity-type constraints. We show that these programs are equivalent in terms of local and global minima and introduce a problem-tailored stationarity concept, which turns out to coincide with the standard KKT conditions of the two reformulated problems. In addition, a suitable constraint qualification as well as second-order conditions for the sparse optimization problem are investigated. These are then used to show that three Lagrange-Newton-type methods are locally fast convergent. Numerical results on different classes of test problems indicate that these methods can be used to drastically improve sparse solutions obtained by some other (globally convergent) methods for sparse optimization problems.
