An algorithm for projecting a point onto a level set of a quadratic function

Abstract

Here, we introduce the quadratic orthogonal projection (the problem of projecting a point onto a quadratic level set), as an extension of the linear orthogonal projection (the problem of projecting a point onto a hyperplane). As the latter problem is convex and has a closed formula solution, the former one belongs to a special class of non-convex problems. We propose an iterative algorithm for the quadratic orthogonal projection, and test it for distinct quadratic functions, showing its great potential in applications, such as in computer graphics, alternating projections, and orbit projections.

Keywords:

• Orthogonal projections
• quadratic level sets
• non-convex optimization

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 90C20
• 90C26

Acknowledgements

This research was carried out during visits of the first author to IMPA, in Rio de Janeiro, and to the Center for Mathematical Research (CRM), in the state of alert in Western Catalonia, whom appreciated the support received from the both outstanding institutes.

Disclosure statement

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

Additional information

Funding

This work was partially supported by Fundação de Apoio à Pesquisa do Distrito Federal (FAP-DF) by the grant 0193.001695/2017, PDE 05/2018 received by Wilfredo Sosa, and by National Council for Scientific and Technological Development (CNPq) by the grants 307679/2016-0,303170/2019-0 received by Fernanda MP Raupp.