On the Finite Complexity of Solutions in a Degenerate System of Quadratic Equations: Exact Formula

Author:

Brezhneva Olga1ORCID,Prusińska Agnieszka2ORCID,Tret’yakov Alexey A.23ORCID

Affiliation:

1. Department of Mathematics, Miami University, Oxford, OH 45056, USA

2. Faculty of Exact and Natural Sciences, Siedlce University, ul. Konarskiego 2, 08-110 Siedlce, Poland

3. Systems Research Institute, Polish Academy of Sciences, ul. Newelska 6, 01-447 Warszawa, Poland

Abstract

The paper describes an application of the p-regularity theory to Quadratic Programming (QP) and nonlinear equations with quadratic mappings. In the first part of the paper, a special structure of the nonlinear equation and a construction of the 2-factor operator are used to obtain an exact formula for a solution to the nonlinear equation. In the second part of the paper, the QP problem is reduced to a system of linear equations using the 2-factor operator. The solution to this system represents a local minimizer of the QP problem along with its corresponding Lagrange multiplier. An explicit formula for the solution of the linear system is provided. Additionally, the paper outlines a procedure for identifying active constraints, which plays a crucial role in constructing the linear system.

Funder

Ministry of Education and Science

Publisher

MDPI AG

Subject

General Physics and Astronomy

Reference18 articles.

1. When a system of real quadratic equations has a solution;Barvinok;Adv. Math.,2022

2. Bereznev, V.A. (2007). Theoretical and Applied Problems of Nonlinear Analysis, Russian Academy of Sciences, Computing Center.

3. Improved two-step Newton’s method for computing simple multiple zeros of polynomial systems;Li;Numer. Algorithm,2022

4. Approximating roots by quadratic iteration;Poirier;Proyecciones J. Math.,2023

5. Factor–analysis of nonlinear mappings: p–regularity theory;Marsden;Commun. Pure Appl. Anal.,2003

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