An update-and-stabilize framework for the minimum-norm-point problem
提出一个结合积极集法和一阶方法的通用算法框架，用于求解多面体上的最小范数点问题，并给出多项式迭代界，在经典非负最小二乘算法上获得收敛保证。
Abstract We consider the minimum-norm-point (MNP) problem over polyhedra, a well-studied problem that encompasses linear programming. We present a general algorithmic framework that combines two fundamental approaches for this problem: active set methods and first order methods. Our algorithm performs first order update steps, followed by iterations that aim to ‘stabilize’ the current iterate with additional projections, i.e., find a locally optimal solution whilst keeping the current tight inequalities. Such steps have been previously used in active set methods for the nonnegative least squares (NNLS) problem. We bound on the number of iterations polynomially in the dimension and in the associated circuit imbalance measure. In particular, the algorithm is strongly polynomial for network flow instances. Classical NNLS algorithms such as the Lawson–Hanson algorithm are special instantiations of our framework; as a consequence, we obtain convergence bounds for these algorithms. Our preliminary computational experiments show promising practical performance.