Second Order Optimality Conditions Based on Parabolic Second Order Tangent Sets

Abstract

Abstract. In this paper we discuss second order optimality conditions in optimization problems subject to abstract constraints. Our analysis is based on various concepts of second order tangent sets and parametric duality. We introduce a condition, called second order regularity, under which there is no gap between the corresponding second order necessary and second order sufficient conditions. We show that the second order regularity condition always holds in the case of semidefinite programming.

Second-Order Optimality and Beyond: Characterization and Evaluation Complexity in Convexly Constrained Nonlinear Optimization

“…Such an arc-based approach was used by several authors for first-and second-order conditions (see [4,10,24,33] for example). Note that, if s i 0 is the first nonzero s i in the sum in the right-hand side of (3.5) (if any), we may redefine α to be α s i 0 −1/i 0 without modifying the arc, so that we may assume, without loss of generality, that…”

“…For k ≤ j ≤ 4, these are given in Table 1. 3 It would be possible to generalize the approach of [10] and define the inner jth-order tangent set ( j > 1)…”

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].…”

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

“…There are essentially two cases when no-gap conditions were obtained: (i) the polyhedric framework, in the case when the Hessian of Lagrangian is a Legendre form, originating in the work by Haraux [15] and Mignot [31], applied to optimal control problems in e.g. Sokolowski [39] and Bonnans [5], and the extended polyhedricity framework in [9,Section 3.2.3]; this framework essentially covers the case of control constraints (and finitely many final state constraints); and (ii) the second-order regularity framework, introduced in [7] and [6], with applications to semi definite optimization. We refer to [9] for an overview of these theories.…”

“…However, only sufficient conditions without curvature terms were known. Two exceptions are a quite specific situation studied in [7] (with applications to some eigenvalue problems), and the case of finitely many contact points, when the problem can be reduced locally to finitely many inequality constraints in semi-infinite programming, see e.g. Hettich and Jongen [17].…”