Second-order optimality conditions for non-convex set-constrained optimization problems

Abstract

In this paper we study second-order optimality conditions for non-convex setconstrained optimization problems. For a convex set-constrained optimization problem, it is well-known that second-order optimality conditions involve the support function of the second-order tangent set. In this paper we propose two approaches for establishing secondorder optimality conditions for the non-convex case. In the first approach we extend the concept of the support function so that it is applicable to general non-convex set-constrained problems, whereas in the second approach we introduce the notion of the directional regular tangent cone and apply classical results of convex duality theory. Besides the second-order optimality conditions, the novelty of our approach lies in the systematic introduction and use, respectively, of directional versions of well-known concepts from variational analysis.

Combined approach with second-order optimality conditions for bilevel programming problems

“…Since (GCP) is equivalent to (2.8), by [14, Theorem 2.1] (or Theorem 3.1 in [39]), we get the result for the M-stationary point; by Corollary 6 in [20] and the expression for the limiting normal cone of the complementarity set [36, Proposition 3.7], we get the result for the S-stationary point. Alternatively, if the set C is a polyhedral set, then we can also use the [26,Theorem 3.8] to derive the desired result.…”

Get Your Foes Fooled: Proximal Gradient Split Learning for Defense Against Model Inversion Attacks on IoMT Data

“…The notation ∥∥ * refer to the nuclear norm of the matrix and λ represents the weighting parameter, accordingly. The said equation is also referred to as robust principle component analysis [31], [32] which is commonly used for image recovery. Equation 2 is also considered to be a special case of a general optimization problem that can be represented in the form shown in equation 3.…”

Second-order optimality conditions for SDCMPCC and application to rank optimization problems

“…We not only characterize the second-order tangent set to the SDCC set, but also investigate the second-order necessary and sufficient optimality conditions under suitable metric subregularity CQs. Just when this work is finished, we learned that Gfrerer et al [13] under weaker conditions established second-order optimality conditions for nonconvex set-constrained optimization problems, which covers the problem (2) as a special case. However, the second-order necessary optimality condition obtained there is weaker than ours (see Corollary 5.2), and moreover, it seems much more difficult to characterize the lower generalized support function of the second-order tangent set to Ω.…”

“…Applying Corollary 5.1 withΣ(d) = {V } for each V ∈ T 2 Ω (Θ(x * ); Θ ′ (x * )d))and the singleton of M(x * ) yields the second part. ✷ Recently, Gfrerer et al derived a dual form of second-order necessary conditions by the lower generalized support function under the directional metric subregularity (see[13, Theorem 2]). Although the condition (42) is stronger than the subregularity assumption, the conclusion of Corollary 5.2 implies the result of[13, Theorem 2].Corollary 5.2 implies that a no gap second-order sufficient optimality condition issup (ξ,Γ)∈M(x) d, ∇ 2 xx L(x, ξ, Γ)d − σ (ξ, Γ) | T 2 K×Ω (Υ(x); Υ ′ (x)d) > 0 ∀d ∈ C(x).…”