M. Ebrahim Sarabi

TL;DR: This work characterizes Lipschitzian full stability of locally optimal solutions to second-order cone programs (SOCPs) expressed entirely in terms of their initial data via appropriate versions of the quadratic growth and strong second- order sufficient conditions under the corresponding constraint qualifications.

TL;DR: In this article, it is shown that noncriticality is equivalent to a certain calmness property of a perturbed variational system and that critical multipliers can be ruled out by full stability of local minimizers in problems of composite optimization.

TL;DR: In this paper, the authors studied the twice epi-differentiablity of extended real-valued functions with an emphasis on functions satisfying a certain composite representation, and showed that these functions are twice differentiable.

Abstract: The paper is mainly devoted to systematic developments and applications of geometric aspects of second-order variational analysis that are revolved around the concept of parabolic regularity of sets. This concept has been known in variational analysis for more than two decades while being largely underinvestigated. We discover here that parabolic regularity is the key to derive new calculus rules and computation formulas for major second-order generalized differential constructions of variational analysis in connection with some properties of sets that go back to classical differential geometry and geometric measure theory. The established results of second-order variational analysis and generalized differentiation, being married to the developed calculus of parabolic regularity, allow us to obtain novel applications to both qualitative and quantitative/numerical aspects of constrained optimization including second-order optimality conditions, augmented Lagrangians, etc. under weak constraint qualifications.

TL;DR: In this article, a comprehensive study of composite models in variational analysis and optimization is presented, with the main attention paid to the new and rather large class of fully subamenable compositions, and the underlying theme of the study is a systematical replacement of conventional metric regularity and related requirements by much weaker metric subregulatity ones.