- Error bounds for rank constrained optimization problems and applications Shujun Bi(bishjscut.edu.cn) Shaohua Pan(shhpanscut.edu.cn) Abstract: This paper is concerned with the rank constrained optimization problem whose feasible set is the intersection of the rank constraint set $\mathcal{R}=\!\big\{X\in\mathbb{X}\ |\ {\rm rank}(X)\le \kappa\big\}$ and a closed convex set $\Omega$. We establish the local (global) Lipschitzian type error bounds for estimating the distance from any $X\in \Omega$ ($X\in\mathbb{X}$) to the feasible set and the solution set, respectively, under the calmness of a multifunction associated to the feasible set at the origin, which is specially satisfied by three classes of common rank constrained optimization problems. As an application of the local Lipschitzian type error bounds, we show that the penalty problem yielded by moving the rank constraint into the objective is exact in the sense that its global optimal solution set coincides with that of the original problem when the penalty parameter is over a certain threshold. This particularly offers an affirmative answer to the open question whether the penalty problem (32) in \cite{GS-Major} is exact or not. As another application, we derive the error bounds of the iterates generated by a multi-stage convex relaxation approach to those three classes of rank constrained problems and show that the bounds are nonincreasing as the number of stages increases. Keywords: Rank constrained optimization, Error bounds, Calmness, Exact penalty Category 1: Applications -- OR and Management Sciences Citation: Operations Research Letters 44 (2016) 336–341 Download: [PDF]Entry Submitted: 05/21/2017Entry Accepted: 05/21/2017Entry Last Modified: 05/21/2017Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Optmization Society.