ABSTRACT
This paper focuses on perturbation analysis of the conic optimization problem, which is defined as the optimization problem over the epigraph of the weighted
norm. The motivation for studying such problem comes from recent interest in the
regularized (possibly non-convex) optimization problems arising in a wide variety of fields such as compressive sensing, signal processing and statistical learning. This paper first derives some important geometrical properties of relevant closed convex cone, including the tangent cone, the normal cone and the critical cone. We then show that under the Robinson's constraint qualification, the following conditions are equivalent: the constraint nondegeneracy and the strong second order sufficient optimality condition, the strong regularity of the Karush–Kuhn–Tucker (KKT) point, and the nonsingularity of Clarke's generalized Jacobian of the KKT system, and others. We further provide an important characterization of the isolated calmness for the
conic optimization problem, namely, under the Robinson's constraint qualification, the isolated calmness of the KKT solution mapping holds if and only if the strict constraint qualification and the second order sufficient condition hold at a locally optimal solution. These characterizations provide theoretical results to design and analyse efficient algorithms for the
conic optimization problem.
Disclosure statement
No potential conflict of interest was reported by the authors.
ORCID
Yong-Jin Liu http://orcid.org/0000-0002-6586-2862