On the Exactness of Dantzig-Wolfe Relaxation for Rank Constrained Optimization Problems

This paper studies the rank constrained optimization problem (RCOP) that aims to minimize a linear objective function over intersecting a prespecified closed rank constrained domain set with m two-sided linear constraints. Replacing the domain set by its closed convex hull offers us a convex Dantzig-Wolfe Relaxation (DWR) of the RCOP. Our goal is to characterize necessary and sufficient conditions under which the DWR and RCOP are equivalent in the sense of extreme point, convex hull, and objective value. More precisely, we develop the first-known necessary and sufficient conditions about when the DWR feasible set matches that of RCOP for any m linear constraints from two perspectives: (i) extreme point exactness -- all extreme points in the DWR feasible set belong to that of the RCOP; and (ii) convex hull exactness -- the DWR feasible set is identical to the closed convex hull of RCOP feasible set. From the optimization view, we also investigate (iii) objective exactness -- the optimal values of the DWR and RCOP coincide for any $m$ linear constraints and a family of linear objective functions. We derive the first-known necessary and sufficient conditions of objective exactness when the DWR admits four favorable classes of linear objective functions, respectively. From the primal perspective, this paper presents how our proposed conditions refine and extend the existing exactness results in the quadratically constrained quadratic program (QCQP) and fair unsupervised learning.