{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:39:30Z","timestamp":1760146770876,"version":"build-2065373602"},"reference-count":28,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2024,12,9]],"date-time":"2024-12-09T00:00:00Z","timestamp":1733702400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>This paper analyses the solution of a specific quadratic sub-problem, along with its possible applications, within both constrained and unconstrained Nonlinear Programming frameworks. We give evidence that this sub\u2013problem may appear in a number of Linesearch Based Methods (LBM) schemes, and to some extent it reveals a close analogy with the solution of trust\u2013region sub\u2013problems. Namely, we refer to a two-dimensional structured quadratic problem, where five linear inequality constraints are included. Finally, we detail how to compute an exact global solution of our two-dimensional quadratic sub-problem, exploiting first order Karush-Khun-Tucker (KKT) conditions.<\/jats:p>","DOI":"10.3390\/a17120563","type":"journal-article","created":{"date-parts":[[2024,12,9]],"date-time":"2024-12-09T10:11:47Z","timestamp":1733739107000},"page":"563","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Issues on a 2\u2013Dimensional Quadratic Sub\u2013Problem and Its Applications in Nonlinear Programming: Trust\u2013Region Methods (TRMs) and Linesearch Based Methods (LBMs)"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4721-8114","authenticated-orcid":false,"given":"Giovanni","family":"Fasano","sequence":"first","affiliation":[{"name":"Department of Management, Venice School of Management, Ca\u2019 Foscari University, S. Giobbe, Cannaregio 873, 30121 Venice, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Christian","family":"Piermarini","sequence":"additional","affiliation":[{"name":"Department of Computer, Control and Management Engineering, Sapienza University of Rome, Via Ariosto 25, 00185 Rome, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9858-3616","authenticated-orcid":false,"given":"Massimo","family":"Roma","sequence":"additional","affiliation":[{"name":"Department of Computer, Control and Management Engineering, Sapienza University of Rome, Via Ariosto 25, 00185 Rome, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,12,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Ben-Tal, A., and Nemirovski, A. (2001). Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications. 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