The problem of minimizing a continuously differentiable convex function over an intersection of closed convex sets is ubiquitous in applied mathematics. It is particularly interesting when it is easy to project onto each separate set, but nontrivial to project onto their intersection. Algorithms based on Newton’s method such as the interior point method are viable for small to medium-scale problems. However, modern applications in statistics, engineering, and machine learning are posing problems with potentially tens of thousands of parameters or more. We revisit this convex programming problem and propose an algorithm that scales well with dimensionality. Our proposal is an instance of a sequential unconstrained minimization technique and revolves around three ideas: the majorization-minimization principle, the classical penalty method for constrained optimization, and quasi-Newton acceleration of fixed-point algorithms. The performance of our distance majorization algorithms is illustrated in several applications.

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距离专业化及其应用

在闭凸集的交集上最小化连续可微凸函数的问题在应用数学中无处不在。当投影到每个单独的集合上很容易，但投影到它们的交集上却很重要时，这一点特别有趣。基于牛顿法的算法（如内点法）适用于中小型问题。然而，统计、工程和机器学习中的现代应用程序可能会带来数以万计或更多参数的问题。我们重新审视了这个凸规划问题，并提出了一种可以很好地扩展维度的算法。我们的提议是顺序无约束最小化技术的一个实例，围绕三个想法：majorization-minimization 原则，约束优化的经典惩罚方法，以及定点算法的拟牛顿加速。我们的距离优化算法的性能在几个应用程序中得到了说明。