Full Stability of Locally Optimal Solutions in Second-Order Cone Programs

Abstract

The paper presents complete characterizations of Lipschitzian full stability of locally optimal solutions to second-order cone programs (SOCPs) expressed entirely in terms of their initial data. These characterizations are obtained via appropriate versions of the quadratic growth and strong second-order sufficient conditions under the corresponding constraint qualifications. We also establish close relationships between full stability of local minimizers for SOCPs and strong regularity of the associated generalized equations at nondegenerate points. Our approach is mainly based on advanced tools of second-order variational analysis and generalized differentiation.

Graphical Derivatives and Stability Analysis for Parameterized Equilibria with Conic Constraints

“…This notion has been recognized as an important stability concept in optimization and has been completely characterized via various second-order conditions. We refer the reader to [12] and the recent papers [14,15,17,18,19] for such characterizations and their applications to broad classes of optimization and control problems. Now we are ready to formulate and prove the aforementioned proposition important in what follows.…”

“…Condition (3.6) can be treated as a proper extension of the classical strong second-order sufficient condition [24] to which (3.6) reduces in the case of Θ = R l − , i.e., in the case of standard equality and inequality constraints as in nonlinear programming. We refer the reader to [14,15,17,18,19] for constructive versions of (3.6) in other constraint systems. Note that (3.6) is satisfied when g is Θ-convex, i.e., the set (y, z) ∈ R m × R l g(y) − z ∈ Θ is convex.…”

Variational analysis of circular cone programs

“…These fundamental stability concepts were introduced in optimization theory by Rockafellar and his collaborators [15,16] and then have been intensively studied by many researchers, especially in the recent years, for various classes of optimization problems; see, e.g. [7,8,[16][17][18][19][20][21][22][23][24][25][26][27] and the references therein. The construction of the second-order subdifferential/generalized Hessian in the sense of Mordukhovich [28] (i.e.…”

“…the coderivative of the firstorder subgradient mapping) plays a crucial role in the characterization of tilt and full stability obtained in the literature. In this paper we establish, by using the obtained second-order calculations and the recent results of [25], complete characterizations of full and tilt stability for locally optimal solutions to mathematical programs with circular cone constraints expressed entirely in terms of the initial program data via certain second-order growth and strong sufficient optimality conditions under appropriate constraint qualifications.…”

“…Proof We combine (4.7) with calculations of N K n and its span obtained in [6,Lemma 25] and [25,Lemma 3.5 and (3.16)], respectively, to get the expressions…”