无界可行域上的条件梯度法
Conditional gradient methods on unbounded feasible regions
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中文总结 AI 辅助
研究无界可行域上条件梯度法的收敛性，提出两种紧致限制原则，一种用含初始目标子水平集的紧致凸集，另一种通过迭代中紧致凸集交集更新限制，在特定条件下目标值收敛到全局最优，还给出相关构造性限制及数值实验结果。
中文摘要 AI 辅助
当在可行域上进行线性最小化比投影便宜得多时，条件梯度法很有吸引力。然而，其经典收敛理论是针对紧致可行集制定的，而许多自然凸可行域是封闭且无界的。本文研究了一种简单的紧致限制原则，用于将条件梯度步应用于无界可行域。第一种方案使用一个包含初始目标子水平集的紧致凸集。第二种方案通过迭代过程中生成的紧致凸集的交集来更新限制。在这两种情况下，线性最小化预言机仅在紧致子集上求解，但只要紧致限制包含相应的目标子水平集，所得目标值就会收敛到原始问题的全局最优值。对于光滑凸目标，我们得到了目标的标准超线性收敛速度。我们还记录了基于强凸性、精确子水平集和上图帽的构造性限制，并包括了在曲率假设下具有次线性收敛速度的非光滑条件次梯度扩展。数值实验说明了无界可行域上固定和动态限制的行为。
英文摘要
The conditional gradient method is attractive when linear minimization over the feasible region is substantially cheaper than projection. Its classical convergence theory, however, is formulated for compact feasible sets, whereas many natural convex feasible regions are closed and unbounded. This paper studies a simple compact-restriction principle for applying conditional gradient steps to unbounded feasible regions. The first scheme uses one compact convex set containing the initial objective sublevel set. The second scheme updates the restriction by intersecting compact convex sets generated along the iterations. In both cases the linear minimization oracle is solved only over compact subsets, but the resulting objective values converge to the global optimum of the original problem, provided the compact restrictions contain the corresponding objective sublevel sets. For smooth convex objectives we obtain the standard superlinear convergence rate of the objective. We also record constructive restrictions based on strong convexity, exact sublevel sets, and epigraph caps, and include a nonsmooth conditional subgradient extension with a sublinear convergence rate under a curvature assumption. Numerical experiments illustrate the behaviour of the fixed and dynamic restrictions on unbounded feasible regions.
