AI 中文总结
该研究在无凸性或线性结构约束下，利用方向导数和切锥刻画一般函数的全局与局部误差界，建立等价条件、关联稳定性性质并得到有限维下的精确局部结果。
AI 中文摘要
我们针对一般函数，在不施加凸性或线性结构的条件下，利用方向导数和切锥建立了全局与局部误差界的新刻画。首先，针对非负下半连续函数的全局误差界，我们建立了若干等价条件，这些等价关系适用于一般的、可能非凸且非光滑的函数。我们进一步将误差界与扰动稳定性、子水平集的Hausdorff稳定性以及逆子水平集估计关联起来。转向方向导数，我们在切锥上引入了最小单位球方向导数φ(x)，并阐明了其与全局斜率的精确关系。对于Lipschitz连续函数，我们证明了当x∉S₀时，sup φ(x) < 0是误差界成立的充分条件；对于凸集上的凸函数，该条件也是必要的。在有限维空间中，我们得到了精确的局部结果：若在解x̄处φ(x̄) > 0，则局部误差界成立，且最优常数恰好为1/φ(x̄)；若φ(x̄) = 0且解集的切锥外存在合适方向，则局部误差界不成立。我们还推导了方向导数与解集切锥距离之间的一般估计式。
英文摘要
We develop new characterizations of both global and local error bounds for general functions, using directional derivatives and tangent cones without imposing convexity or linear structure. We first establish several equivalent conditions for the global error bound of a nonnegative lower semicontinuous function. These equivalences hold for general, possibly nonconvex and nonsmooth functions. We further link the error bound with perturbation stability, Hausdorff stability of sublevel sets, and an inverse-sublevel-set estimate. Turning to directional derivatives, we introduce the minimal unit-sphere directional derivative \(φ(x)\) on the tangent cone and clarify its exact relation with the global slope. For Lipschitz continuous functions we prove that \(\sup_{x\notin S_0} φ(x) < 0\) is sufficient for an error bound, and for convex functions on convex sets this condition is also necessary, In finite dimensions we obtain sharp local results: if \(φ(\bar{x}) > 0\) at a solution \(\bar{x}\), then a local error bound holds and the optimal constant is exactly \(1/φ(\bar{x})\); if \(φ(\bar{x}) = 0\) and a suitable direction exists outside the tangent cone of the solution set, the local error bound fails. A general estimate relating the directional derivative to the distance from the tangent cone of the solution set is also derived.
CommentsThis paper is a decomposition of a long paper (over 60 pages) entitled Directional Derivatives and Error Bounds of Merit Functions in Vector Optimization (arXiv:2607.18781) abstracting the merit function $θ$ into a general function to enhance the generality of the theory
