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Optim."],"published-print":{"date-parts":[[2018,1]]},"abstract":"<jats:p>We revisit the foundations of gauge duality and demonstrate that it can be explained using a modern approach to duality based on a perturbation framework. We therefore put gauge duality and Fenchel--Rockafellar duality on equal footing, including explaining gauge dual variables as sensitivity measures, and show how to recover primal solutions from those of the gauge dual. This vantage point allows a direct proof that optimal solutions of the Fenchel--Rockafellar dual of the gauge dual are precisely the primal solutions rescaled by the optimal value. We extend the gauge duality framework to the setting in which the functional components are general nonnegative convex functions, including problems with piecewise linear quadratic functions and constraints that arise from generalized linear models used in regression.<\/jats:p>","DOI":"10.1137\/17m1119020","type":"journal-article","created":{"date-parts":[[2018,8,30]],"date-time":"2018-08-30T15:34:43Z","timestamp":1535643283000},"page":"2406-2434","source":"Crossref","is-referenced-by-count":13,"title":["Foundations of Gauge and Perspective Duality"],"prefix":"10.1137","volume":"28","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1875-1801","authenticated-orcid":true,"given":"A. Y.","family":"Aravkin","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"J. V.","family":"Burke","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5245-0458","authenticated-orcid":true,"given":"D.","family":"Drusvyatskiy","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0222-5222","authenticated-orcid":true,"given":"M. P.","family":"Friedlander","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"K. J.","family":"MacPhee","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2018,8,30]]},"reference":[{"key":"atypb1","unstructured":"A. Y. Aravkin, J. V. Burke, D. Drusvyatskiy, M. P. Friedlander, and S. Roy,\n                      Level-Set Methods for Convex Optimization\n                      , arXiv:1602.01506, 2016."},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1137\/120899157"},{"key":"atypb3","doi-asserted-by":"crossref","unstructured":"A. Y. Aravkin, J. V. Burke, and G. Pillonetto,\n                      Linear system identification using stable spline kernels and PLQ penalties\n                      , in Proceedings of the 52nd IEEE Conference on Decision and Control, 2013, pp. 5168-5173.","DOI":"10.1109\/CDC.2013.6760701"},{"key":"atypb4","first-page":"1705","volume":"6","author":"Banerjee A.","year":"2005","journal-title":"J. Mach. Learn. Res."},{"key":"atypb5","doi-asserted-by":"crossref","unstructured":"S. Boyd and L. Vandenberghe,\n                      Convex Optimization\n                      , Cambridge University Press, Cambridge, 2004.","DOI":"10.1017\/CBO9780511804441"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2005.862083"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.20124"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1007\/s10851-010-0251-1"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1007\/s11228-017-0407-x"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2016.12.021"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2006.871582"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1287\/moor.2014.0672"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1007\/BF02591851"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1137\/15M1034283"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1137\/130940785"},{"key":"atypb16","unstructured":"Gurobi Optimizer Reference Manual\n                      , Gurobi Optimization, 2015."},{"key":"atypb17","doi-asserted-by":"crossref","unstructured":"P. Huber,\n                      Robust Statistics\n                      , Wiley, New York, 1981.","DOI":"10.1002\/0471725250"},{"key":"atypb18","doi-asserted-by":"crossref","unstructured":"J. A. Nelder and R. J. Baker,\n                      Generalized linear models\n                      , Encyclopedia of Statistical Sciences, Wiley, 1972.","DOI":"10.2307\/2344614"},{"key":"atypb19","doi-asserted-by":"crossref","unstructured":"Y. Nesterov and A. Nemirovskii,\n                      Interior-Point Polynomial Algorithms in Convex Programming\n                      , Stud. Appl. Numer. Math. 13, SIAM, Philadelphia, 1994.","DOI":"10.1137\/1.9781611970791"},{"key":"atypb20","unstructured":"R. T. Rockafellar,\n                      Convex Analysis\n                      , Princeton University Press, Princeton, NJ, 1972."},{"key":"atypb21","doi-asserted-by":"crossref","unstructured":"R. T. Rockafellar,\n                      Conjugate Duality and Optimization\n                      , SIAM, Philadelphia, 1974.","DOI":"10.1137\/1.9781611970524"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1988-0936806-9"},{"key":"atypb23","unstructured":"R. T. Rockafellar and R. J.B. Wets,\n                      Variational Analysis\n                      , Grundkehren Math. Wiss. 317, Springer, New York, 2009."},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-015-0952-8"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1007\/s13675-015-0048-5"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2005.864420"},{"key":"atypb27","doi-asserted-by":"crossref","unstructured":"M. Udell, K. Mohan, D. Zeng, J. Hong, S. Diamond, and S. Boyd,\n                      Convex optimization in Julia\n                      , in Proceedings of the SC14 Workshop on High Performance Technical Computing in Dynamic Languages, 2014.","DOI":"10.1109\/HPTCDL.2014.5"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-9868.2005.00503.x"}],"container-title":["SIAM Journal on Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/17M1119020","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T19:19:21Z","timestamp":1787339961000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/17M1119020"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,1]]},"references-count":28,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2018,1]]}},"alternative-id":["10.1137\/17M1119020"],"URL":"https:\/\/doi.org\/10.1137\/17m1119020","relation":{},"ISSN":["1052-6234","1095-7189"],"issn-type":[{"value":"1052-6234","type":"print"},{"value":"1095-7189","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,1]]}}}