A complete characterization of the robust isolated calmness of nuclear norm regularized convex optimization problems
نویسندگان
چکیده
In this paper, we provide a complete characterization of the robust isolated calmness of the KarushKuhn-Tucker (KKT) solution mapping for convex constrained optimization problems regularized by the nuclear norm function. This study is motivated by the recent work in [8], where the authors show that under the Robinson constraint qualification at a local optimal solution, the KKT solution mapping for a wide class of conic programming problems is robustly isolated calm if and only if both the second order sufficient condition (SOSC) and the strict Robinson constraint qualification (SRCQ) are satisfied. Based on the variational properties of the nuclear norm function and its conjugate, we establish the equivalence between the primal/dual SOSC and the dual/primal SRCQ. The derived results lead to several equivalent characterizations of the robust isolated calmness of the KKT solution mapping and add insights to the existing literature on the stability of nuclear norm regularized convex optimization problems.
منابع مشابه
Regularization Paths with Guarantees for Convex Semidefinite Optimization
We devise a simple algorithm for computing an approximate solution path for parameterized semidefinite convex optimization problems that is guaranteed to be ε-close to the exact solution path. As a consequence, we can compute the entire regularization path for many regularized matrix completion and factorization approaches, as well as nuclear norm or weighted nuclear norm regularized convex opt...
متن کاملAn accelerated proximal gradient algorithm for nuclear norm regularized least squares problems
The affine rank minimization problem, which consists of finding a matrix of minimum rank subject to linear equality constraints, has been proposed in many areas of engineering and science. A specific rank minimization problem is the matrix completion problem, in which we wish to recover a (low-rank) data matrix from incomplete samples of its entries. A recent convex relaxation of the rank minim...
متن کاملAn accelerated proximal gradient algorithm for nuclear norm regularized linear least squares problems
The affine rank minimization problem, which consists of finding a matrix of minimum rank subject to linear equality constraints, has been proposed in many areas of engineering and science. A specific rank minimization problem is the matrix completion problem, in which we wish to recover a (low-rank) data matrix from incomplete samples of its entries. A recent convex relaxation of the rank minim...
متن کاملSVD-free Convex-Concave Approaches for Nuclear Norm Regularization
Minimizing a convex function of matrices regularized by the nuclear norm arises in many applications such as collaborative filtering and multi-task learning. In this paper, we study the general setting where the convex function could be non-smooth. When the size of the data matrix, denoted bym×n, is very large, existing optimization methods are inefficient because in each iteration, they need t...
متن کاملGeometry of Factored Nuclear Norm Regularization
This work investigates the geometry of a nonconvex reformulation of minimizing a general convex loss function f(X) regularized by the matrix nuclear norm ‖X‖∗. Nuclear-norm regularized matrix inverse problems are at the heart of many applications in machine learning, signal processing, and control. The statistical performance of nuclear norm regularization has been studied extensively in litera...
متن کامل