SIAM Journal on Control and Optimization, Volume 58, Issue 2, Page 1183-1206, January 2020.
We study the following optimization problem over a dynamical system that consists of several linear subsystems: Given a finite set of $n\times n$ matrices and an $n$-dimensional vector, find a sequence of $K$ matrices, each chosen from the given set of matrices, to maximize a convex function over the product of the $K$ matrices and the given vector. This simple problem has many applications in operations research and control, yet a moderate-sized instance is challenging to solve to optimality for state-of-the-art optimization software. We propose a simple exact algorithm for this problem. Our algorithm runs in polynomial time when the given set of matrices has the oligo-vertex property, a concept we introduce in this paper for a finite set of matrices. We derive several sufficient conditions for a set of matrices to have the oligo-vertex property. Numerical results demonstrate the clear advantage of our algorithm in solving large-sized instances of the problem over one state-of-the-art global optimization solver. We also propose several open questions on the oligo-vertex property and discuss its potential connection with the finiteness property of a set of matrices, which may be of independent interest.

中文翻译：

---

线性切换系统的最佳切换顺序

SIAM控制与优化杂志，第58卷，第2期，第1183-1206页，2020年1月。
我们研究一个由几个线性子系统组成的动力系统的以下优化问题：给定一个有限的$ n \ timesn $个矩阵和一个$ n $维向量，找到一个$ K $矩阵的序列，每个序列都从给定矩阵集，以使凸函数在$ K $矩阵与给定向量的乘积上最大化。这个简单的问题在运筹学和控制中有许多应用，但是中等规模的实例对于解决最先进的优化软件的最优性提出了挑战。我们针对此问题提出了一种简单的精确算法。当给定的矩阵集具有oligo-vertex属性时，我们的算法将在多项式时间内运行，这是本文针对有限矩阵集引入的一种概念。我们导出了一组具有oligo-vertex属性的矩阵的充分条件。数值结果表明，与一种最新的全局优化求解器相比，我们的算法在解决问题的大型实例方面具有明显的优势。我们还提出了关于寡角顶点属性的几个开放问题，并讨论了它与一组矩阵的有限性之间的潜在联系，这可能是独立引起关注的。