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New optimization modeling techniques and convex relaxations for hard nonconvex problems have shown that the loss of strict feasibility is a more pronounced phenomenon than has previously been realized. In this text, we describe various reasons for the loss of strict feasibility, whether due to poor modeling choices or (more interestingly) rich underlying structure, and discuss ways to cope with it and, in many pronounced cases, how to use it as an advantage. In large part, we emphasize the facial reduction preprocessing technique due to its mathematical elegance, geometric transparency, and computational potential.<\/jats:p>","DOI":"10.1561\/2400000011","type":"journal-article","created":{"date-parts":[[2017,12,20]],"date-time":"2017-12-20T06:10:55Z","timestamp":1513750255000},"page":"77-170","source":"Crossref","is-referenced-by-count":51,"title":["The Many Faces of Degeneracy in Conic Optimization"],"prefix":"10.1108","volume":"3","author":[{"given":"Dmitriy","family":"Drusvyatskiy","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Washington"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Henry","family":"Wolkowicz","sequence":"additional","affiliation":[{"name":"Faculty of Mathematics, University of Waterloo"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"140","published-online":{"date-parts":[[2017,12,20]]},"reference":[{"issue":"1-3","key":"2026033014421269100_ref001","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1016\/S0024-3795(00)00066-5","article-title":"Graph rigidity via Euclidean distance matrices","volume":"310","author":"Alfakih","year":"2000","journal-title":"Linear Algebra Appl"},{"issue":"1","key":"2026033014421269100_ref002","doi-asserted-by":"crossref","first-page":"53","DOI":"10.4171\/pm\/1881","article-title":"Euclidean distance matrices, semidefinite programming, and sensor network localization","volume":"68","author":"Alfakih","year":"2011","journal-title":"Portug. 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