The paper investigates strategies for expansion of active set that can be employed by the MPRGP algorithm. The standard MPRGP expansion uses a projected line search in the free gradient direction with a fixed step length. Such a scheme is often too slow to identify the active set, requiring a large number of expansions. We propose to use adaptive step lengths based on the current gradient, which guarantees the decrease of the unconstrained cost function with different gradient-based search directions. Moreover, we also propose expanding the active set by projecting the optimal step for the unconstrained minimization. Numerical experiments demonstrate the benefits (up to 78% decrease in the number of Hessian multiplications) of our expansion step modifications on two benchmarks – contact problem of linear elasticity solved by TFETI and machine learning problems of SVM type, both implemented in PERMON toolbox.

本文研究了MPRGP算法可以采用的扩展活动集的策略。标准MPRGP扩展在自由梯度方向上使用固定步长的投影线搜索。这样的方案通常太慢以至于无法识别活动集，因此需要大量扩展。我们建议使用基于当前梯度的自适应步长，以确保在基于梯度的搜索方向不同的情况下，无约束成本函数的减少。此外，我们还建议通过投影无约束最小化的最佳步骤来扩展活动集。