Metric regularity, tangent sets, and second-order optimality conditions

Abstract

Abstract.A strong regularity theorem is proved, which shows that the usual constraint qualification conditions ensuring the regularity of the set-valued maps expressing feasibility in optimization problems, are in fact minimal assumptions. These results are then used to derive calculus rules for secondorder tangent sets, allowing us in turn to obtain a second-order (Lagrangian) necessary condition for optimality which completes the usual one of positive semidefiniteness on the Hessian of the Lagrangian function.

No-gap second-order optimality conditions for optimal control problems with a single state constraint and control

“…The above second-order necessary condition was improved by Cominetti in [12], by stating that for all convex set…”

“…This framework includes many optimal control problems. The theory of secondorder necessary optimality conditions involves a term taking into account the curvature of the convex set, see Kawasaki [21], Cominetti [12]. By contrast, second-order sufficient optimality conditions typically involve no such term; see e.g.…”

Linear and Nonlinear Semidefinite Programming

“…The optimality conditions of first and second order for NLSDP are widely characterized, see for instance [21,23,35,125,47]. An important effort research is recently devoted to the study and characterization of stability for solutions of nonlinear semidefinite programming (or in general conic) problems, see for instance [107,20,33,70,49,98,97].…”

Implicit multifunction theorems in complete metric spaces

“…This concept goes back to the surjectivity of a linear continuous mapping in the Banach Open Mapping Theorem and to its extension to nonlinear operators known as the Lyusternik & Graves Theorem ( [40], [27], see also [15]) and [21]). For a detailed account the reader is referred to the books or works of many researchers, [3], [5], [9], [10], [11], [12], [13], [17], [18], [20], [22], [29], [30], [32], [34], [36], [37], [40], [42], [43], [41], [44], [45], [46], [50], [51], [52], [57] and the references given therein for many theoretical results on the metric regularity as well as its various applications. Metric regularity or its equivalent notions (covering at a linear rate) [38] or Aubin property of the inverse [1] is now considered as a central concept in modern variational analysis (see the survey paper by Ioffe [34]).…”