{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:30:45Z","timestamp":1760243445265,"version":"build-2065373602"},"reference-count":17,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2022,10,14]],"date-time":"2022-10-14T00:00:00Z","timestamp":1665705600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Taiwan Normal University and National Science and Technology Council, Taiwan"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The goal of this paper is to investigate the curves intersected by a vertical plane with the surfaces based on certain NCP functions. The convexity and differentiability of these curves are studied as well. In most cases, the inflection points of the curves cannot be expressed exactly. Therefore, we instead estimate the interval where the curves are convex under this situation. Then, with the help of differentiability and convexity, we obtain the local minimum or maximum of the curves accordingly. The study of these curves is very useful to binary quadratic programming.<\/jats:p>","DOI":"10.3390\/axioms11100557","type":"journal-article","created":{"date-parts":[[2022,10,16]],"date-time":"2022-10-16T21:10:10Z","timestamp":1665954610000},"page":"557","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Plane Section Curves on Surfaces of NCP Functions"],"prefix":"10.3390","volume":"11","author":[{"given":"Shun-Wei","family":"Li","sequence":"first","affiliation":[{"name":"Department of Mathematics, National Taiwan Normal University, Taipei 11677, Taiwan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6915-7490","authenticated-orcid":false,"given":"Yu-Lin","family":"Chang","sequence":"additional","affiliation":[{"name":"Department of Mathematics, National Taiwan Normal University, Taipei 11677, Taiwan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4596-9419","authenticated-orcid":false,"given":"Jein-Shan","family":"Chen","sequence":"additional","affiliation":[{"name":"Department of Mathematics, National Taiwan Normal University, Taipei 11677, Taiwan"}]}],"member":"1968","published-online":{"date-parts":[[2022,10,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"368","DOI":"10.1016\/j.neucom.2020.06.059","article-title":"A novel generalization of the natural residual function and a neural network approach for the NCP","volume":"413","author":"Alcantara","year":"2020","journal-title":"Neurocomputing"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"354","DOI":"10.1016\/j.orl.2015.04.007","article-title":"Symmetrization of generalized natural residual function for NCP","volume":"43","author":"Chang","year":"2015","journal-title":"Oper. 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Lett."},{"key":"ref_13","first-page":"31","article-title":"Geometric views of the generalized Fischer-Burmeister function and its induced merit function","volume":"237","author":"Tsai","year":"2014","journal-title":"Appl. Math. Comput."},{"key":"ref_14","first-page":"3975","article-title":"A continuation approach for solving binary quadratic program based on a class of NCP-functions","volume":"219","author":"Chen","year":"2012","journal-title":"Appl. Math. Comput."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1007\/s11590-015-0993-1","article-title":"Differentiability v.s. convexity for complementarity functions","volume":"11","author":"Huang","year":"2017","journal-title":"Optim. Lett."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Tuy, H. (2016). Convex Analysis and Global Optimization, Springer. [2nd ed.].","DOI":"10.1007\/978-3-319-31484-6"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Polyak, R.A. (2021). Introduction to Continuous Optimization, Springer. Springer Optimization and Its Applications.","DOI":"10.1007\/978-3-030-68713-7"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/10\/557\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:54:21Z","timestamp":1760144061000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/10\/557"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,10,14]]},"references-count":17,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2022,10]]}},"alternative-id":["axioms11100557"],"URL":"https:\/\/doi.org\/10.3390\/axioms11100557","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2022,10,14]]}}}