English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Solving large scale convex semidefinite programming (SDP) problems has longbeen a challenging task numerically. Fortunately, several powerful solversincluding SDPNAL, SDPNAL+ and QSDPNAL have recently been developed to solvelinear and convex quadratic SDP problems to high accuracy successfully. Thesesolvers are based on the augmented Lagrangian method (ALM) applied to the dualproblems with the subproblems being solved by semismooth Newton-CG methods.Noticeably, thanks to Rockafellar's general theory on the proximal pointalgorithms, the primal iteration sequence generated by the ALM enjoys anasymptotic Q-superlinear convergence rate under a second order sufficientcondition {for the primal problem}. This second order sufficient conditionimplies that the primal problem has a unique solution, which can be restrictivein many applications. For gaining more insightful interpretations on the highefficiency of these solvers, in this paper we conduct an asymptotic superlinearconvergence analysis of the ALM for convex SDP when the primal problem hasmultiple solutions (can be unbounded). Under a fairly mild second order growthcondition, we prove that the primal iteration sequence generated by the ALMconverges asymptotically Q-superlinearly, while the dual feasibility and thedual objective function value converge asymptotically R-superlinearly.Moreover, by studying the metric subregularity of the Karush-Kuhn-Tuckersolution mapping, we also provide sufficient conditions to guarantee theasymptotic R-superlinear convergence of the dual iterate.

On the Asymptotic Superlinear Convergence of the Augmented Lagrangian Method for Semidefinite Programming with Multiple Solutions