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.

The paper is devoted to a comprehensive study of composite models in variational analysis and optimization the importance of which for numerous theoretical, algorithmic, and applied issues of operations research is difficult to overstate. The underlying theme of our study is a systematical replacement of conventional metric regularity and related requirements by much weaker metric subregulatity ones that lead us to significantly stronger and completely new results of first-order and second-order variational analysis and optimization. In this way we develop extended calculus rules for first-order and secondorder generalized differential constructions with paying the main attention in second-order variational theory to the new and rather large class of fully subamenable compositions. Applications to optimization include deriving enhanced no-gap second-order optimality conditions in constrained composite models, complete characterizations of the uniqueness of Lagrange multipliers and strong metric subregularity of KKT systems in parametric optimization, etc.

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.

Summary Advanced distribution automation (ADA) is a vital basis of the smart grid, which leads to enhancement of reliability level. Moreover, demand response programs are highly proposed to be employed in the smart grid. In this paper, a cross‐sectional benefit of smart grid maturity model is proposed in which the emergency demand response programs (EDRPs) are considered in the ADA planning problem. In fact, the role of EDRPs is considered in the service restoration process that affects the ADA plan. The service restoration process possesses several uncertain parameters including the demanded power of customers during service restoration, which are considered in the proposed formulation for ADA planning. The objective function of the optimization problem includes various costs imposed to the distribution system operator due to contingency occurrence including the expected total customer interruption cost, total investment cost of automatic sectionalizing switches, circuit breakers, and fuses deployment, and the total expected cost of implementing EDRP programs in the service restoration. The presented ADA planning methodology is tested on the fourth bus number of Roy Bilinton test system (RBTS4) to investigate its effectiveness.

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This paper is devoted to studying the first-order variational analysis of non-convex and nondifferentiable functions that may not be subdifferentially regular. To achieve this goal, we entirely rely on two concepts of directional derivatives known as subderivative and semi-derivative. We establish the exact chain and sum rules for this class of functions via these directional derivatives. These calculus rules provide an implementable auto-differentiation process such as back-propagation in composite functions. The latter calculus rules can be used to identify the directional stationary points defined by the subderivative. We show that the distance function of a geometrically derivable constraint set is semi-differentiable, which opens the door for designing first-order algorithms for non-Clarke regular constrained optimization problems. We propose a first-order algorithm to find a directional stationary point of non-Clarke regular and perhaps non-Lipschitz functions. We introduce a descent property under which we establish the non-asymptotic convergence of our method with rate O(ε −2 ), akin to gradient descent for smooth minimization. We show that the latter descent property holds for free in some interesting non-amenable composite functions, in particular, it holds for the Moreau envelope of any bounded-below function.

The paper is devoted to a comprehensive study of composite models in variational analysis and optimization the importance of which for numerous theoretical, algorithmic, and applied issues of operations research is difficult to overstate. The underlying theme of our study is a systematical replacement of conventional metric regularity and related requirements by much weaker metric subregulatity ones that lead us to significantly stronger and completely new results of first-order and second-order variational analysis and optimization. In this way, we develop extended calculus rules for first-order and second-order generalized differential constructions while paying the main attention in second-order variational theory to the new and rather large class of fully subamenable compositions. Applications to optimization include deriving enhanced no-gap second-order optimality conditions in constrained composite models, complete characterizations of the uniqueness of Lagrange multipliers, strong metric subregularity of Karush-Kuhn-Tucker systems in parametric optimization, and so on.