Variational Convexity of Functions and Variational Sufficiency in Optimization
Variational Convexity of Functions and Variational Sufficiency in Optimization
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DOI：
10.1137/22m1519250
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发表时间：
2022-08
期刊：
影响因子：
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通讯作者：
P. D. Khanh;B. Mordukhovich;Vo Thanh Phat
中科院分区：
文献类型：
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作者：
P. D. Khanh;B. Mordukhovich;Vo Thanh Phat
本文致力于研究，刻画和应用的变分凸性的功能，该属性已被最近介绍的Rockafellar连同其强对应。首先，我们证明了这些变分性质的扩展实值函数是等价的，分别是传统的（局部）凸性和强凸性的莫罗包络。然后，我们得到了新的特征的变分凸性和变分强凸性的一般功能通过其二阶次微分（广义Hessians），这是次梯度映射的余导数。我们还研究了这些概念与局部极小和倾斜稳定局部极小的关系。所得到的结果用于表征变分和强变分充分性的复合优化与非线性规划的应用相关的概念。
The paper is devoted to the study, characterizations, and applications of variational convexity of functions, the property that has been recently introduced by Rockafellar together with its strong counterpart. First we show that these variational properties of an extended-real-valued function are equivalent to, respectively, the conventional (local) convexity and strong convexity of its Moreau envelope. Then we derive new characterizations of both variational convexity and variational strong convexity of general functions via their second-order subdifferentials (generalized Hessians), which are coderivatives of subgradient mappings. We also study relationships of these notions with local minimizers and tilt-stable local minimizers. The obtained results are used for characterizing related notions of variational and strong variational sufficiency in composite optimization with applications to nonlinear programming.