{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:21:51Z","timestamp":1760149311679,"version":"build-2065373602"},"reference-count":18,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2023,7,25]],"date-time":"2023-07-25T00:00:00Z","timestamp":1690243200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Ministry of Education and Science","award":["144\/23\/B"],"award-info":[{"award-number":["144\/23\/B"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>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.<\/jats:p>","DOI":"10.3390\/e25081112","type":"journal-article","created":{"date-parts":[[2023,7,26]],"date-time":"2023-07-26T00:45:02Z","timestamp":1690332302000},"page":"1112","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On the Finite Complexity of Solutions in a Degenerate System of Quadratic Equations: Exact Formula"],"prefix":"10.3390","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6247-9780","authenticated-orcid":false,"given":"Olga","family":"Brezhneva","sequence":"first","affiliation":[{"name":"Department of Mathematics, Miami University, Oxford, OH 45056, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6091-6884","authenticated-orcid":false,"given":"Agnieszka","family":"Prusi\u0144ska","sequence":"additional","affiliation":[{"name":"Faculty of Exact and Natural Sciences, Siedlce University, ul. 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Appl."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"515","DOI":"10.1016\/j.camwa.2013.11.009","article-title":"On R-linear convergence of semi-monotonic inexact augmented Lagrangians for bound and equality constrained quadratic programming problems with application","volume":"67","author":"Dostal","year":"2014","journal-title":"Comput. Math. Appl."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"90","DOI":"10.1016\/0041-5553(88)90116-4","article-title":"Methods for solving degenerate problems","volume":"28","author":"Belash","year":"1988","journal-title":"USSR Comput. Math. Math. Phys."},{"key":"ref_14","unstructured":"Bertsekas, D.P. (1999). Nonlinear Programming, Athena Scientific."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Alekseev, V.M., Tikhomirov, V.M., and Fomin, S.V. (1987). Optimal Control, Consultants Bureau.","DOI":"10.1007\/978-1-4615-7551-1"},{"key":"ref_16","unstructured":"Szczepanik, E., and Tret\u2019yakov, A.A. (2020). p-Regularity Theory and Methods of Solving Nonlinear Optimization Problems, Uniwersytet Przyrodniczo-Humanistyczny w Siedlcach. (In Polish)."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Nocedal, J., and Wright, S.J. (1999). Numerical Optimization, Springer.","DOI":"10.1007\/b98874"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1137\/S1052623496305882","article-title":"On the accurate identification of active constraints","volume":"9","author":"Facchinei","year":"1998","journal-title":"SIAM J. 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