Fully Hölderian Stable Minimum with Respect to Both Tilt and Parameter Perturbations
Xi Yin Zheng, Jiangxing Zhu, Kung Fu Ng · SIAM Journal on Optimization · 2018
When the objective function undergoes both a tilt perturbation and a general parameter perturbation, this paper considers the notions of a fully stable Hölder minimizer, a uniform Hölder growth condition, and a fully stable $(q,s)$-minimum, where the last notion reduces to the tilt-stable minimum by Levy, Poliquin, and Rockafellar [ SIAM J. Optim., 10 (2000), pp. 580--604] and the fully Hölder stable minimum by Mordukhovich and Nghia [ SIAM J. Optim., 24 (2014), pp. 1344--1381] as special cases by taking $(q,s)=(2,2)$ and $(q,s)=(2,1)$, respectively. Under weak-($\mathcal{B}\mathcal{C}\mathcal{Q}$) (a new constraint qualification), by using the techniques of variational analysis, we establish relationships among these notions and provide several characterizations for fully stable $(q,s)$-minima, which improve and generalize some existing results in the recent literature.