A characterization of the distance to infeasibility under block-structured perturbations

We discuss several generalizations of the classical Eckart and Young identity: We show that a natural extension of this identity holds for rectangular matrices defining conic systems of constraints, and for perturbations restricted to a particular block structure, such as those determined by a sparsity pattern. Our results extend and unify the classical Eckart and Young identity, Renegar’s characterization of the distance to infeasibility [Math. Program. 70 (1995) 279], Rohn’s characterization of the componentwise distance to singularity [Linear Algebra Appl. 126 (1989) 39], and Cheung and Cucker’s characterization of the normalized distance to ill-posedness [Math. Program. 91 (2001) 163].