On condition numbers for least squares with quadric inequality constraint

Abstract

In this paper, we will study normwise, mixed and componentwise condition numbers for the linear mapping of the solution for general least squares with quadric inequality constraint (GLSQI) and its standard form (LSQI). We will introduce the mappings from the data space to the interested data space, and the Fréchet derivative of the introduced mapping can deduced through matrix differential techniques. Based on condition number theory, we derive the explicit expressions of normwise, mixed and componentwise condition numbers for the linear function of the solution for GLSQI and LSQI. Also, easier computable upper bounds for mixed and componentwise condition numbers are given. Numerical example shows that the mixed and componentwise condition numbers can tell us the true conditioning of the problem when its data is sparse or badly scaled. Compared with normwise condition numbers, the mixed and componentwise condition number can give sharp perturbation bounds.