{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:12:59Z","timestamp":1760145179521,"version":"build-2065373602"},"reference-count":23,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2024,6,27]],"date-time":"2024-06-27T00:00:00Z","timestamp":1719446400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Consejer\u00eda de Universidad, Investigaci\u00f3n e Innovaci\u00f3n de la Junta de Andaluc\u00eda","award":["ProyExcel00780","ProyExcel01036","TED2021-131704A-I00"],"award-info":[{"award-number":["ProyExcel00780","ProyExcel01036","TED2021-131704A-I00"]}]},{"DOI":"10.13039\/501100004837","name":"Ministerio de Ciencia e Innovaci\u00f3n","doi-asserted-by":"publisher","award":["ProyExcel00780","ProyExcel01036","TED2021-131704A-I00"],"award-info":[{"award-number":["ProyExcel00780","ProyExcel01036","TED2021-131704A-I00"]}],"id":[{"id":"10.13039\/501100004837","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Minimum norm problems consist of finding the distance of a closed subset of a normed space to the origin. Usually, the given closed subset is also asked to be convex, thus resulting in a convex minimum norm problem. There are plenty of techniques and algorithms to compute the distance of a closed convex set to the origin, which mostly exist in the Hilbert space setting. In this manuscript, we consider nonconvex minimum norm problems that arise from Bioengineering and reformulate them in such a way that the solution to their reformulation is already known. In particular, we tackle the problem of min\u2225x\u2225 subject to \u2225Rk(x)\u2225\u00a0\u2265\u00a0ak for k\u00a0=\u00a01,\u2026,l, where x\u2208X and Rk:X\u2192Y are continuous linear operators between real normed spaces X,Y, and ak\u00a0&gt;\u00a00 for k\u00a0=\u00a01,\u2026,l.\u00a0Notice that the region of constraints of the previous problem is neither convex nor balanced. However, it is additively symmetric, which is also the case for the objective function, due to the properties satisfied by norms, which makes possible the analytic resolution of such a nonconvex minimization. The recent literature shows that the design of optimal coils for electronics applications can be achieved by solving problems like this. However, in this work, we apply our analytical solutions to design an optimal coil for an electromagnetic sensor.<\/jats:p>","DOI":"10.3390\/sym16070809","type":"journal-article","created":{"date-parts":[[2024,6,27]],"date-time":"2024-06-27T13:26:43Z","timestamp":1719494803000},"page":"809","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Minimization over Nonconvex Sets"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3383-2634","authenticated-orcid":false,"given":"Jos\u00e9 Antonio","family":"Vilchez Membrilla","sequence":"first","affiliation":[{"name":"Department of Electronics, College of Engineering, University of C\u00e1diz, 11510 Puerto Real, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0005-9722-8945","authenticated-orcid":false,"given":"V\u00edctor","family":"Salas Moreno","sequence":"additional","affiliation":[{"name":"Department of Electronics, College of Engineering, University of C\u00e1diz, 11510 Puerto Real, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4537-9929","authenticated-orcid":false,"given":"Soledad","family":"Moreno-Pulido","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Engineering, University of C\u00e1diz, 11510 Puerto Real, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3763-8852","authenticated-orcid":false,"given":"Alberto","family":"S\u00e1nchez-Alzola","sequence":"additional","affiliation":[{"name":"Department of Statistics and Operation Research, College of Engineering, University of C\u00e1diz, 11510 Puerto Real, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0299-5358","authenticated-orcid":false,"given":"Clemente","family":"Cobos S\u00e1nchez","sequence":"additional","affiliation":[{"name":"Department of Electronics, College of Engineering, University of C\u00e1diz, 11510 Puerto Real, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6208-6071","authenticated-orcid":false,"given":"Francisco Javier","family":"Garc\u00eda-Pacheco","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Engineering, University of C\u00e1diz, 11510 Puerto Real, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,6,27]]},"reference":[{"key":"ref_1","first-page":"431","article-title":"Reflexivity, contraction functions and minimum-norm elements","volume":"42","author":"Aizpuru","year":"2005","journal-title":"Studia Sci. 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