Variable projection solves structured optimization problems by completely minimizing over a subset of the variables while iterating over the remaining variables. Over the last 30 years, the technique has been widely used, with empirical and theoretical results demonstrating both greater efficacy and greater stability compared to competing approaches. Classic examples have exploited closed-form projections and smoothness of the objective function. We extend the approach to problems that include non-smooth terms, and where the projection subproblems can only be solved inexactly by iterative methods. We propose an inexact adaptive algonrithm for solving such problems and analyze its computational complexity. Finally, we show how the theory can be used to design methods for selected problems occurring frequently in machine-learning and inverse problems.

翻译：变量预测解决了结构优化问题,在对剩余变量进行迭代的同时,将一组变量完全减少到最低程度,从而解决了结构优化问题。在过去30年中,该技术被广泛使用,其实验和理论结果表明,与竞争方法相比,效果更高,稳定性更高。典型的例子利用了封闭式预测和目标功能的顺利性。我们将这一方法扩大到包括非移动术语的问题,而且预测子问题只能通过迭代方法不精确地解决。我们提出了一种不精确的适应性 algorithm,用于解决这些问题并分析其计算复杂性。最后,我们展示了该理论如何用于设计在机器学习和反向问题中经常发生的选定问题的方法。