Approximate mean value theorem for upper subderivatives
Non-Linear Analysis
Partially finite convex programming, part I: quasi relative interiors and duality theory
Mathematical Programming: Series A and B
Metric regularity for strongly compactly Lipschitzian mappings
Nonlinear Analysis: Theory, Methods & Applications
Stablity of Set-Valued Mappings In Infinite Dimensions: Point Criteria and Applications
SIAM Journal on Control and Optimization
Clarke-Ledyaev mean value inequalities in smooth Banach spaces
Nonlinear Analysis: Theory, Methods & Applications
Nonsmooth Constrained Optimization and Multidirectional Mean Value Inequalities
SIAM Journal on Optimization
Links between directional derivatives through multidirectional mean value inequalities
Mathematical Programming: Series A and B - Nonlinear convex optimization and variational inequalities
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In this work we introduce for extended real valued functions, defined on a Banach space $X$, the concept of $K$ directionally Lipschitzian behavior, where $K$ is a bounded subset of $X$. For different types of sets $K$ (e.g., zero, singleton, or compact), the $K$ directionally Lipschitzian behavior recovers well-known concepts in variational analysis (locally Lipschitzian, directionally Lipschitzian, or compactly epi-Lipschitzian properties, respectively). Characterizations of this notion are provided in terms of the lower Dini subderivatives. We also adapt the concept for sets and establish characterizations of the mentioned behavior in terms of the Bouligand tangent cones. The special case of convex functions and sets is also studied.