A note on the generalised Hessian of the least squares associated with systems of linear inequalities

TL;DR

This paper identifies an error in a previously published formula for the generalized Hessian related to least squares problems with linear inequalities and clarifies conditions under which the formula is valid.

Contribution

It corrects a known erroneous formula for the generalized Hessian and provides conditions ensuring its validity, clarifying the mathematical understanding of the problem.

Findings

Identifies an error in Mangasarian's formula for the generalized Hessian.

Provides sufficient conditions for the validity of the corrected formula.

Shows that Slater's condition guarantees certain elements belong to the generalized Hessian.

Abstract

The goal of this note is to point out an erroneous formula for the generalised Hessian of the least squares associated with a system of linear inequalities, that was given in the paper "A finite Newton method for classification" by O.L. Mangasarian (Optim. Methods Softw. 17: 913--929, 2002) and reproduced multiple times in other publications. We also provide sufficient contiditions for the validity of Mangasarian's formula and show that Slater's condition guarantees that some particular elements from the set defined by Mangasarian belong to the generalised Hessian of the corresponding function.