论文标题
通过投影梯度方法，关于全球求解非convex信任区域子问题
On globally solving nonconvex trust region subproblem via projected gradient method
论文作者
论文摘要
信任区子问题（TRS）是在欧几里得球上最大程度地减少可能的非凸二次函数。通常有两种情况（tr），即所谓的``简易案例''和``硬情况''。即使在``易于案例''中，当从非零度量可行的集合中任意选择初始点时，经典投影梯度方法（PG）生成的序列也可以以均方根局部速率收敛到鞍点。令我们惊讶的是，当应用（PG）解决（TRS）的便宜且可能是非凸的重新印象时，以{\ IT}的可行点初始化的生成序列几乎总是会汇聚到其全球最小化器中。对于``易于案例''的本地收敛率至少是线性的，而没有假设我们拥有``简易案例''所包含的信息。我们还考虑如何使用（PG）来解决全球求解相等约束（TRS）。
The trust region subproblem (TRS) is to minimize a possibly nonconvex quadratic function over a Euclidean ball. There are typically two cases for (TRS), the so-called ``easy case'' and ``hard case''. Even in the ``easy case'', the sequence generated by the classical projected gradient method (PG) may converge to a saddle point at a sublinear local rate, when the initial point is arbitrarily selected from a nonzero measure feasible set. To our surprise, when applying (PG) to solve a cheap and possibly nonconvex reformulation of (TRS), the generated sequence initialized with {\it any} feasible point almost always converges to its global minimizer. The local convergence rate is at least linear for the ``easy case'', without assuming that we have possessed the information that the ``easy case'' holds. We also consider how to use (PG) to globally solve equality-constrained (TRS).
