{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T19:34:17Z","timestamp":1787340857570,"version":"build-2736575974"},"reference-count":43,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Optim."],"published-print":{"date-parts":[[2018,1]]},"abstract":"<jats:p>Large-scale constrained convex optimization problems arise in several application domains. First-order methods are good candidates to tackle such problems due to their low iteration complexity and memory requirement. The level-set framework extends the applicability of first-order methods to tackle problems with complicated convex objectives and constraint sets. Current methods based on this framework either rely on the solution of challenging subproblems or do not guarantee a feasible solution, especially if the procedure is terminated before convergence. We develop a level-set method that finds an $\\epsilon$-relative optimal and feasible solution to a constrained convex optimization problem with a fairly general objective function and set of constraints, maintains a feasible solution at each iteration, and only relies on calls to first-order oracles. We establish the iteration complexity of our approach, also accounting for the smoothness and strong convexity of the objective function and constraints when these properties hold. The dependence of our complexity on $\\epsilon$ is similar to the analogous dependence in the unconstrained setting, which is not known to be true for level-set methods in the literature. Nevertheless, ensuring feasibility is not free. The iteration complexity of our method depends on a condition number, while existing level-set methods that do not guarantee feasibility can avoid such dependence. We numerically validate the usefulness of ensuring a feasible solution path by comparing our approach with an existing level set method on a Neyman--Pearson classification problem.<\/jats:p>","DOI":"10.1137\/17m1152334","type":"journal-article","created":{"date-parts":[[2018,12,11]],"date-time":"2018-12-11T14:11:15Z","timestamp":1544537475000},"page":"3290-3311","source":"Crossref","is-referenced-by-count":13,"title":["A Level-Set Method for Convex Optimization with a Feasible Solution Path"],"prefix":"10.1137","volume":"28","author":[{"given":"Qihang","family":"Lin","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Selvaprabu","family":"Nadarajah","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Negar","family":"Soheili","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2018,12,11]]},"reference":[{"key":"atypb1","unstructured":"Z. Allen-Zhu and E. Hazan,\n                      Optimal Black-Box Reductions Between Optimization Objectives\n                      , preprint, arXiv:1603.05642, 2016."},{"key":"atypb2","unstructured":"A. Y. Aravkin, J. V. Burke, D. Drusvyatskiy, M. P. Friedlander, and S. Roy,\n                      Level-Set Methods for Convex Optimization\n                      , preprint, arXiv:1602.01506, 2016."},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1137\/120899157"},{"key":"atypb4","unstructured":"K. Bache and M. Lichman,\n                      UCI Machine Learning Repository\n                      , 2013."},{"key":"atypb5","doi-asserted-by":"crossref","unstructured":"A. Bayandina,\n                      Adaptive Mirror Descent for Constrained Optimization\n                      , preprint, arXiv:1705.02029, 2017.","DOI":"10.1109\/CNSA.2017.7973938"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1137\/100800580"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1137\/080716542"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1137\/100818327"},{"key":"atypb9","unstructured":"J. A. Blackard, D. J. Dean, and C. W. Anderson,\n                      Covertype Data Set\n                      , in UCI Machine Learning Repository, 2013,https:\/\/archive.ics.uci.edu\/ml\/datasets\/covertype."},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1016\/S0377-2217(01)00179-5"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1561\/0100000005"},{"key":"atypb12","unstructured":"R.E. Fan and C.J. Lin,\n                      LIBSVM Data: Classification, Regression and Multi-Label\n                      , 2011,https:\/\/www.jmlr.org\/papers\/v9\/fan08a.html."},{"key":"atypb13","unstructured":"R. M. Freund and H. Lu,\n                      New Computational Guarantees for Solving Convex Optimization Problems with First Order Methods, Via a Function Growth Condition Measure\n                      , preprint, arxiv:1511.02974, 2015."},{"key":"atypb14","unstructured":"B. Grimmer,\n                      Radial Subgradient Method\n                      , preprint, arXiv:1703.09280v2, 2017."},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-014-0778-9"},{"key":"atypb16","first-page":"121","author":"Juditsky A.","year":"2011","journal-title":"MA"},{"key":"atypb17","first-page":"341","volume":"6","author":"Keerthi S. S.","year":"2005","journal-title":"J. Mach. Learn. Res."},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1007\/BF01585554"},{"key":"atypb19","unstructured":"S. Lacoste-Julien, M. Schmidt, and F. Bach,\n                      A Simpler Approach to Obtaining an $\\mathcal{O}(1\/t)$ Convergence Rate for the Projected Stochastic Subgradient Method\n                      , preprint, arXiv:1212.2002, 2012."},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-008-0261-6"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-013-0737-x"},{"key":"atypb22","unstructured":"G. Lan and Z. Zhou,\n                      Algorithms for Stochastic Optimization with Expectation Constraints\n                      , preprint, arXiv:1604.03887, 2016."},{"key":"atypb23","first-page":"95","author":"Lemar\u00e9chal C.","year":"1975","journal-title":"Berlin"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1007\/BF01585555"},{"key":"atypb25","first-page":"361","volume":"5","author":"Lewis D. D.","year":"2004","journal-title":"J. Mach. Learn. Res."},{"key":"atypb26","first-page":"3118","author":"Lin Q.","year":"2018","journal-title":"Proceedings of the International Conference on Machine Learning"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1137\/S1052623403425629"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1137\/070704277"},{"key":"atypb29","doi-asserted-by":"crossref","unstructured":"Y. Nesterov,\n                      Introductory lectures on convex optimization: A basic course\n                      , Kluwer Academic Publishers, the Netherlands, 2004.","DOI":"10.1007\/978-1-4419-8853-9"},{"key":"atypb30","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-004-0552-5"},{"key":"atypb31","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-007-0149-x"},{"key":"atypb32","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-012-0629-5"},{"key":"atypb33","doi-asserted-by":"publisher","DOI":"10.1016\/j.ejor.2015.01.029"},{"key":"atypb34","doi-asserted-by":"publisher","DOI":"10.1007\/s10957-010-9759-1"},{"key":"atypb35","doi-asserted-by":"publisher","DOI":"10.1137\/15M1027371"},{"key":"atypb36","first-page":"2831","volume":"12","author":"Rigollet P.","year":"2011","journal-title":"J. Mach. Learn. Res."},{"key":"atypb37","doi-asserted-by":"publisher","DOI":"10.1002\/wics.1376"},{"key":"atypb39","doi-asserted-by":"publisher","DOI":"10.1137\/080714488"},{"key":"atypb40","doi-asserted-by":"publisher","DOI":"10.1137\/100785028"},{"key":"atypb41","first-page":"145","author":"Wolfe P.","year":"1975","journal-title":"New York"},{"key":"atypb42","unstructured":"Y. Xu,\n                      First-Order Methods for Constrained Convex Programming Based on Linearized Augmented Lagrangian Function\n                      , preprint, arXiv:1711.08020, 2017."},{"key":"atypb43","unstructured":"Y. Xu,\n                      Global Convergence Rates of Augmented Lagrangian Methods for Constrained Convex Programming\n                      , preprint, arXiv:1711.05812, 2017."},{"key":"atypb44","unstructured":"Y. Xu,\n                      Primal-Dual Stochastic Gradient Method for Convex Programs with Many Functional Constraints\n                      , preprint, arXiv:802.02724, 2018."}],"container-title":["SIAM Journal on Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/17M1152334","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:52:54Z","timestamp":1787338374000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/17M1152334"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,1]]},"references-count":43,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2018,1]]}},"alternative-id":["10.1137\/17M1152334"],"URL":"https:\/\/doi.org\/10.1137\/17m1152334","relation":{},"ISSN":["1052-6234","1095-7189"],"issn-type":[{"value":"1052-6234","type":"print"},{"value":"1095-7189","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,1]]}}}