{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,3]],"date-time":"2026-10-03T01:30:50Z","timestamp":1790991050308,"version":"4.1.0"},"reference-count":28,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","funder":[{"DOI":"10.13039\/501100005041","name":"Universit\u00e9 Catholique de Louvain","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100005041","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["FA9550-15-1-0276"],"award-info":[{"award-number":["FA9550-15-1-0276"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Optim."],"published-print":{"date-parts":[[2018,1]]},"abstract":"<jats:p>The usual approach to developing and analyzing first-order methods for smooth convex optimization assumes that the gradient of the objective function is uniformly smooth with some Lipschitz constant $L$. However, in many settings the differentiable convex function $f(\\cdot)$ is not uniformly smooth---for example, in $D$-optimal design where $f(x):=-\\ln \\det(HXH^T)$ and $X:= \\mbox{{\\bf D}iag} (x)$, or even the univariate setting with $f(x) := -\\ln(x) + x^2$. In this paper we develop a notion of \u201crelative smoothness\u201d and relative strong convexity that is determined relative to a user-specified \u201creference function\u201d $h(\\cdot)$ (that should be computationally tractable for algorithms), and we show that many differentiable convex functions are relatively smooth with respect to a correspondingly fairly simple reference function $h(\\cdot)$. We extend two standard algorithms---the primal gradient scheme and the dual averaging scheme---to our new setting, with associated computational guarantees. We apply our new approach to develop a new first-order method for the $D$-optimal design problem, with associated computational complexity analysis. Some of our results have a certain overlap with the recent work [H. H. Bauschke, J. Bolte, and M. Teboulle, Math. Oper. Res., 42 (2017), pp. 330--348].<\/jats:p>","DOI":"10.1137\/16m1099546","type":"journal-article","created":{"date-parts":[[2018,2,6]],"date-time":"2018-02-06T12:33:02Z","timestamp":1517920382000},"page":"333-354","source":"Crossref","is-referenced-by-count":212,"title":["Relatively Smooth Convex Optimization by First-Order Methods, and Applications"],"prefix":"10.1137","volume":"28","author":[{"given":"Haihao","family":"Lu","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Robert M.","family":"Freund","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yurii","family":"Nesterov","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2018,2,6]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1080\/10556780701589669"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1007\/BF02206828"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1287\/moor.25.3.365.12212"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177697374"},{"key":"atypb5","unstructured":"M. Avriel,\n                      Nonlinear Programming: Analysis and Methods\n                      , Prentice-Hall, Englewood Cliffs, NJ, 1976."},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1287\/moor.2016.0817"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1137\/06067198X"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1023\/A:1020713207683"},{"key":"atypb9","first-page":"187","author":"John F.","year":"1948","journal-title":"New York"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1287\/moor.21.2.307"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1007\/BF01582144"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.4153\/CJM-1960-030-4"},{"key":"atypb13","first-page":"126","author":"Knorr E. M.","year":"2001","journal-title":"New York"},{"key":"atypb14","unstructured":"A. Nemirovsky and D. B. Yudin,\n                      Problem Complexity and Method Efficiency in Optimization\n                      , Wiley, New York, 1983."},{"key":"atypb15","doi-asserted-by":"crossref","unstructured":"Y. Nesterov,\n                      Introductory Lectures on Convex Optimization: A Basic Course\n                      , Kluwer Academic, Boston, 2003.","DOI":"10.1007\/978-1-4419-8853-9"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-004-0552-5"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-007-0149-x"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-012-0629-5"},{"key":"atypb19","unstructured":"B. Polyak,\n                      Introduction to Optimization\n                      , Optimization Software, New York, 1987."},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1287\/opre.1040.0115"},{"key":"atypb21","doi-asserted-by":"crossref","unstructured":"M. J. Todd,\n                      Minimum-Volume Ellipsoids: Theory and Algorithms\n                      , MOS-SIAM Ser. Optim. 23, SIAM, Philadelphia, 2016.","DOI":"10.1137\/1.9781611974386"},{"key":"atypb22","unstructured":"P. Tseng,\n                      On Accelerated Proximal Gradient Methods for Convex-Concave Optimization\n                      , Technical report, MIT, Cambridge, MA, 2008."},{"key":"atypb23","first-page":"338","author":"Vaidya P. M.","year":"1989","journal-title":"Los Angeles"},{"key":"atypb24","first-page":"19","author":"Ye Y.","year":"1992","journal-title":"NJ"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1137\/050622560"},{"key":"atypb26","doi-asserted-by":"crossref","unstructured":"C. Zalinescu,\n                      Convex Analysis in General Vector Spaces\n                      , World Scientific, Singapore, 2002.","DOI":"10.1142\/5021"},{"key":"atypb27","unstructured":"Y. Zhang,\n                      An Interior-Point Algorithm for the Maximum-Volume Ellipsoid Problem\n                      , Department of Computational and Applied Mathematics, Technical report TR98-15, Rice University, Houston, TX, 1998."},{"key":"atypb28","unstructured":"Y. Zhou, Y. Liang, and L. Shen,\n                      A Unified Approach to Proximal Algorithms using Bregman Distance\n                      , Technical report, Syracuse University, Syracuse, NY, 2016."}],"container-title":["SIAM Journal on Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/16M1099546","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:15:13Z","timestamp":1787336113000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/16M1099546"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,1]]},"references-count":28,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2018,1]]}},"alternative-id":["10.1137\/16M1099546"],"URL":"https:\/\/doi.org\/10.1137\/16m1099546","relation":{},"ISSN":["1052-6234","1095-7189"],"issn-type":[{"value":"1052-6234","type":"print"},{"value":"1095-7189","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,1]]}}}