Isolated calmness of solution mappings and exact recovery conditions for nuclear norm optimization problems

Abstract

ABSTRACT We derive some conditions for a feasible point to be the isolated optimal solution for three classes of nuclear norm optimization problems by using the isolated calmness of two set-valued mappings: one is the level set mapping constructed by these problems themselves, and the other is the multiplier set mapping of their dual problems. Among others, the conditions from the dual angle are shown to be weaker than those from the primal angle, while the conditions from the primal angle are almost necessary and require the same number of measurements as the sufficient and necessary one does when the estimation mechanism in [Chandrasekaran et al. The convex geometry of linear inverse problems. Found Comput Math. 2012;12:805–849; Amelunxen et al. Living on the edge: phase transitions in convex programs with random data. Inf Inference. 2014;3:224–294] is used. In particular, we show that the deterministic exact recovery conditions in [Candès and Recht. Exact matrix completion via convex optimization. Found Comput Math. 2009;9:717–772; Chandrasekaran et al. Rank-sparsity inchoherence for matrix decomposition. SIAM J Optim. 2011;21:572–596] are stronger than the constraint nondegeneracy of the dual problems, whereas our conditions from the dual view are equivalent to the strict Robinson constraint qualification for them.