发布时间: 2022-06-23 15:53:00
题 目：Slater Condition for Tangent Derivatives
报 告 人：郑喜印教授 云南大学
时 间：2022-06-22 09:30-10:30
郑喜印，云南大学东陆骨干教授， 博士生导师， 长期从事泛函分析、变分分析和非光滑优化理论的交叉研究，发表论文90余篇，其中有30余篇发表在"SIAM Journal on Optimization"、"Mathematical Programming"和"Mathematics of Operations Research"等优化顶级刊物，获云南省自然科学奖一等奖一项和二等奖二项。
Noting that the usual Slater condition is not applicable in nonconvex optimization,we introduce the Slater condition for the Bouligand and Clarke tangent derivatives of a general (not necessarily convex) vector-valued function F with respect to a closed convex cone. Under the Slater condition of the Clarke tangent derivative of a vector-valued function F, it is proved that the normal cone of the sublevel set of F at a reference point can be formulated by the subdi erential of F at the same reference point. Without any assumption, it is proved that the Slater condition of the Bouligand tangent derivative is always stable when the objective function undergoes small calm perturbations. Based on this, we prove that the Slater condition of the Bouligand tangent derivative of a vector-valued function F is sufficient for F to have a stable error bound when F undergoes small calm Clarke-regular perturbations. In the composite-convexity case, it is also proved to be necessary. Moreover, we give some quantitative formulae for error bound moduli. Our results are new even in the case of scalar-valued functions.