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

Abstract

The goal of this note is to point out an erroneous formula for the generalized 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 Mangasarian (Optim. Methods Softw. 17 (2002), pp. 913–929) 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 generalized Hessian of the corresponding function.

Keywords:

• Linear inequalities
• generalized Jacobian
• generalized Hessian
• Clarke subidfferential

2020 Mathematics Subject Classifications:

• 49J52
• 49J53
• 90C56

Acknowledgments

The author is sincerely grateful to professor V.N. Malozemov for drawing the author's attention to paper [10], during a careful examination of which the author made the observations presented in this note.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The research was supported by the Ministry of Science and Higher Education of the Russian Federation (project no. 124041500008-1).

Notes on contributors

M.V. Dolgopolik

M.V. Dolgopolik received a specialist (M. Sc.) degree in Applied Mathematics from St. Petersburg State University in 2012, Candidate of Science (Ph.D.) degree in Discrete Mathematics and Mathematical Cybernetics in 2015, and Doctor of Sciences (Habilitation) degree in Real, Complex, and Functional Analysis in 2022 from the same university. Since 2015, he is a senior researcher at the Institute for Problems in Mechanical Engineering of the Russian Academy of Sciences, Saint Petersburg, Russia. His current research interest include nonsmooth and DC optimization, nonsmooth and variational analysis, calculus of variations, and optimal control.