{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,7]],"date-time":"2026-05-07T07:34:55Z","timestamp":1778139295938,"version":"3.51.4"},"reference-count":48,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2025,2,7]],"date-time":"2025-02-07T00:00:00Z","timestamp":1738886400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,2,7]],"date-time":"2025-02-07T00:00:00Z","timestamp":1738886400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"Mines Paris - PSL"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Optim Theory Appl"],"published-print":{"date-parts":[[2025,3]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>This work proposes an implementable proximal-type method for a broad class of optimization problems involving nonsmooth and nonconvex objective and constraint functions. In contrast to existing methods that rely on an ad hoc model approximating the nonconvex functions, our approach can work with a nonconvex model constructed by the pointwise minimum of finitely many convex models. The latter can be chosen with reasonable flexibility to better fit the underlying functions\u2019 structure. We provide a unifying framework and analysis covering several subclasses of composite optimization problems and show that our method computes points satisfying certain necessary optimality conditions, which we will call model criticality. Depending on the specific model being used, our general concept of criticality boils down to standard necessary optimality conditions. Numerical experiments on some stochastic reliability-based optimization problems illustrate the practical performance of the method.<\/jats:p>","DOI":"10.1007\/s10957-024-02597-x","type":"journal-article","created":{"date-parts":[[2025,2,7]],"date-time":"2025-02-07T15:10:10Z","timestamp":1738941010000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A Proximal-Type Method for Nonsmooth and Nonconvex Constrained Minimization Problems"],"prefix":"10.1007","volume":"204","author":[{"ORCID":"https:\/\/orcid.org\/0009-0000-1805-2800","authenticated-orcid":false,"given":"Gregorio M.","family":"Sempere","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Welington","family":"de Oliveira","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Johannes O.","family":"Royset","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,2,7]]},"reference":[{"key":"2597_CR1","doi-asserted-by":"publisher","first-page":"683","DOI":"10.1007\/s10479-018-3091-9","volume":"292","author":"L Adam","year":"2018","unstructured":"Adam, L., Branda, M., Heitsch, H., Henrion, R.: Solving joint chance constrained problems using regularization and Benders\u2019 decomposition. Annals of Operations Research 292, 683\u2013709 (2018)","journal-title":"Annals of Operations Research"},{"issue":"1","key":"2597_CR2","doi-asserted-by":"publisher","first-page":"23","DOI":"10.1007\/s10479-004-5022-1","volume":"133","author":"LTH An","year":"2005","unstructured":"An, L.T.H., Tao, P.D.: The DC (difference of convex functions) programming and DCA revisited with DC models of real world nonconvex optimization problems. Annals of Operations Research 133(1), 23\u201346 (2005)","journal-title":"Annals of Operations Research"},{"issue":"3","key":"2597_CR3","doi-asserted-by":"publisher","first-page":"1516","DOI":"10.1137\/070685026","volume":"47","author":"P Apkarian","year":"2008","unstructured":"Apkarian, P., Noll, D., Rondepierre, A.: Mixed $$H_2\/H_{\\infty }$$ Control via Nonsmooth Optimization. SIAM Journal on Control and Optimization 47(3), 1516\u20131546 (2008)","journal-title":"SIAM Journal on Control and Optimization"},{"key":"2597_CR4","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-024-02142-8","volume-title":"Coderivative-based semi-Newton method in nonsmooth difference programming","author":"F Arag\u00f3n-Artacho","year":"2024","unstructured":"Arag\u00f3n-Artacho, F., Mordukhovich, B., P\u00e9rez-Aros, P.: Coderivative-based semi-Newton method in nonsmooth difference programming. Accepted in Math, Program (2024)"},{"key":"2597_CR5","unstructured":"Arag\u00f3n-Artacho, F., P\u00e9rez-Aros, P., Torregrosa-Bel\u00e9n, D.: The boosted double-proximal subgradient algorithm for nonconvex optimization. Submitted in July 2023 (2023)"},{"issue":"1","key":"2597_CR6","doi-asserted-by":"publisher","first-page":"89","DOI":"10.1137\/21M1465445","volume":"33","author":"F Atenas","year":"2023","unstructured":"Atenas, F., Sagastizabal, C., Silva, P.J., Solodov, M.: A unified analysis of descent sequences in weakly convex optimization, including convergence rates for bundle methods. SIAM Journal on Optimization 33(1), 89\u2013115 (2023)","journal-title":"SIAM Journal on Optimization"},{"issue":"3","key":"2597_CR7","doi-asserted-by":"publisher","first-page":"260","DOI":"10.1007\/BF01584377","volume":"33","author":"JV Burke","year":"1985","unstructured":"Burke, J.V.: Descent methods for composite nondifferentiable optimization problems. Mathematical Programming 33(3), 260\u2013279 (1985)","journal-title":"Mathematical Programming"},{"issue":"2","key":"2597_CR8","doi-asserted-by":"publisher","first-page":"179","DOI":"10.1007\/BF01585997","volume":"71","author":"JV Burke","year":"1995","unstructured":"Burke, J.V., Ferris, M.C.: A Gauss-Newton method for convex composite optimization. Mathematical Programming 71(2), 179\u2013194 (1995)","journal-title":"Mathematical Programming"},{"key":"2597_CR9","doi-asserted-by":"publisher","DOI":"10.1016\/j.ress.2023.109314","volume":"236","author":"J-E Byun","year":"2023","unstructured":"Byun, J.-E., de Oliveira, W., Royset, J.O.: S-BORM: Reliability-based optimization of general systems using buffered optimization and reliability method. Reliability Engineering & System Safety 236, 109314 (2023)","journal-title":"Reliability Engineering & System Safety"},{"key":"2597_CR10","unstructured":"Clarke, F.: Optimisation and Nonsmooth Analysis. Classics in Applied Mathematics. Society for Industrial and Applied Mathematics. ISBN 978-0-89871-256-8 (1990)"},{"issue":"2","key":"2597_CR11","doi-asserted-by":"publisher","first-page":"261","DOI":"10.1007\/BF01585170","volume":"62","author":"R Correa","year":"1993","unstructured":"Correa, R., Lemar\u00e9chal, C.: Convergence of some algorithms for convex minimization. Mathematical Programming 62(2), 261\u2013275 (1993)","journal-title":"Mathematical Programming"},{"issue":"3","key":"2597_CR12","doi-asserted-by":"publisher","first-page":"1149","DOI":"10.1007\/s11228-022-00639-y","volume":"30","author":"Y Cui","year":"2022","unstructured":"Cui, Y., Liu, J., Pang, J.-S.: Nonconvex and nonsmooth approaches for affine chance-constrained stochastic programs. Set-Valued and Variational Analysis 30(3), 1149\u20131211 (2022)","journal-title":"Set-Valued and Variational Analysis"},{"key":"2597_CR13","doi-asserted-by":"crossref","unstructured":"Cui, Y., Pang, J.-S.: Modern Nonconvex Nondifferentiable Optimization. SIAM (2022)","DOI":"10.1137\/1.9781611976748"},{"issue":"4","key":"2597_CR14","doi-asserted-by":"publisher","first-page":"3344","DOI":"10.1137\/18M117337X","volume":"28","author":"Y Cui","year":"2018","unstructured":"Cui, Y., Pang, J.-S., Sen, B.: Composite difference-max programs for modern statistical estimation problems. SIAM Journal on Optimization 28(4), 3344\u20133374 (2018)","journal-title":"SIAM Journal on Optimization"},{"issue":"2","key":"2597_CR15","doi-asserted-by":"publisher","first-page":"523","DOI":"10.1007\/s10898-019-00755-4","volume":"75","author":"W de Oliveira","year":"2019","unstructured":"de Oliveira, W.: Proximal bundle methods for nonsmooth DC programming. Journal of Global Optimization 75(2), 523\u2013563 (2019)","journal-title":"Journal of Global Optimization"},{"key":"2597_CR16","doi-asserted-by":"crossref","unstructured":"de Oliveira, W., Solodov, M.: Bundle Methods for Inexact Data, pages 417\u2013459. Springer International Publishing, Cham (2020)","DOI":"10.1007\/978-3-030-34910-3_12"},{"issue":"2","key":"2597_CR17","doi-asserted-by":"publisher","first-page":"424","DOI":"10.1137\/21M1428601","volume":"33","author":"M Diaz","year":"2023","unstructured":"Diaz, M., Grimmer, B.: Optimal convergence rates for the proximal bundle method. SIAM Journal on Optimization 33(2), 424\u2013454 (2023)","journal-title":"SIAM Journal on Optimization"},{"key":"2597_CR18","doi-asserted-by":"publisher","unstructured":"Drusvyatskiy, D., Ioffe, A., Lewis, A.: Nonsmooth optimization using Taylor-like models. Math. Program. 185, 357\u2013383 (2021). https:\/\/doi.org\/10.1007\/s10107-019-01432-w","DOI":"10.1007\/s10107-019-01432-w"},{"key":"2597_CR19","doi-asserted-by":"crossref","unstructured":"Frangioni, A.: Standard Bundle Methods: Untrusted Models and Duality, pages 61\u2013116. Springer International Publishing, Cham (2020)","DOI":"10.1007\/978-3-030-34910-3_3"},{"key":"2597_CR20","doi-asserted-by":"publisher","first-page":"427","DOI":"10.1007\/s00186-016-0564-y","volume":"84","author":"C Gotzes","year":"2016","unstructured":"Gotzes, C., Heitsch, H., Henrion, R.: On the quantification of nomination feasibility in stationary gas networks with random load. Mathematical Methods of Operational Research 84, 427\u2013457 (2016)","journal-title":"Mathematical Methods of Operational Research"},{"issue":"1","key":"2597_CR21","doi-asserted-by":"publisher","first-page":"112","DOI":"10.1287\/moor.17.1.112","volume":"17","author":"J Higle","year":"1992","unstructured":"Higle, J., Sen, S.: On the convergence of algorithms with implications for stochastic and nondifferentiable optimization. Mathematics of operations research 17(1), 112\u2013131 (1992)","journal-title":"Mathematics of operations research"},{"issue":"3","key":"2597_CR22","doi-asserted-by":"publisher","first-page":"509","DOI":"10.1007\/s10898-011-9765-3","volume":"52","author":"HA Le Thi","year":"2012","unstructured":"Le Thi, H.A., Pham Dinh, T., Ngai, H.V.: Exact penalty and error bounds in DC programming. Journal of Global Optimization 52(3), 509\u2013535 (2012)","journal-title":"Journal of Global Optimization"},{"issue":"1","key":"2597_CR23","doi-asserted-by":"publisher","first-page":"501","DOI":"10.1007\/s10107-015-0943-9","volume":"158","author":"AS Lewis","year":"2016","unstructured":"Lewis, A.S., Wright, S.J.: A proximal method for composite minimization. Mathematical Programming 158(1), 501\u2013546 (2016)","journal-title":"Mathematical Programming"},{"issue":"3","key":"2597_CR24","doi-asserted-by":"publisher","first-page":"2530","DOI":"10.1137\/19M1276819","volume":"30","author":"J Liu","year":"2020","unstructured":"Liu, J., Cui, Y., Pang, J.-S., Sen, S.: Two-stage stochastic programming with linearly bi-parameterized quadratic recourse. SIAM Journal on Optimization 30(3), 2530\u20132558 (2020)","journal-title":"SIAM Journal on Optimization"},{"issue":"2","key":"2597_CR25","doi-asserted-by":"publisher","first-page":"1077","DOI":"10.1137\/15M1042644","volume":"28","author":"A Mafusalov","year":"2018","unstructured":"Mafusalov, A., Uryasev, S.: Buffered probability of exceedance: Mathematical properties and optimization. SIAM Journal on Optimization 28(2), 1077\u20131103 (2018)","journal-title":"SIAM Journal on Optimization"},{"issue":"3","key":"2597_CR26","doi-asserted-by":"publisher","first-page":"403","DOI":"10.1007\/s10898-018-0651-0","volume":"72","author":"O Montonen","year":"2018","unstructured":"Montonen, O., Joki, K.: Bundle-based descent method for nonsmooth multiobjective DC optimization with inequality constraints. Journal of Global Optimization 72(3), 403\u2013429 (2018)","journal-title":"Journal of Global Optimization"},{"issue":"3","key":"2597_CR27","doi-asserted-by":"publisher","first-page":"687","DOI":"10.1287\/moor.1070.0262","volume":"32","author":"JS Pang","year":"2007","unstructured":"Pang, J.S.: Partially B-regular optimization and equilibrium problems. Mathematics of Operations Research 32(3), 687\u2013699 (2007)","journal-title":"Mathematics of Operations Research"},{"issue":"1","key":"2597_CR28","doi-asserted-by":"publisher","first-page":"95","DOI":"10.1287\/moor.2016.0795","volume":"42","author":"J-S Pang","year":"2017","unstructured":"Pang, J.-S., Razaviyayn, M., Alvarado, A.: Computing B-stationary points of nonsmooth DC programs. Mathematics of Operations Research 42(1), 95\u2013118 (2017)","journal-title":"Mathematics of Operations Research"},{"key":"2597_CR29","series-title":"Handbooks in Operations Research and Management Science","doi-asserted-by":"publisher","first-page":"267","DOI":"10.1016\/S0927-0507(03)10005-9","volume-title":"Stochastic Programming","author":"A Pr\u00e9kopa","year":"2003","unstructured":"Pr\u00e9kopa, A.: Probabilistic programming. In: Ruszczy\u0144ski, A., Shapiro, A. (eds.) Stochastic Programming. Handbooks in Operations Research and Management Science, vol. 10, pp. 267\u2013351. Amsterdam, Elsevier (2003)"},{"issue":"5","key":"2597_CR30","doi-asserted-by":"publisher","first-page":"499","DOI":"10.1016\/j.ress.2010.01.001","volume":"95","author":"R Rockafellar","year":"2010","unstructured":"Rockafellar, R., Royset, J.: On buffered failure probability in design and optimization of structures. Reliability Engineering & System Safety 95(5), 499\u2013510 (2010)","journal-title":"Reliability Engineering & System Safety"},{"key":"2597_CR31","doi-asserted-by":"publisher","first-page":"63","DOI":"10.1007\/BFb0121126","volume":"28","author":"RT Rockafellar","year":"1986","unstructured":"Rockafellar, R.T., Wets, R.J.-B.: A Lagrangian finite generation technique for solving linear-quadratic problems in stochastic programming. Mathematical Programming Study 28, 63\u201393 (1986)","journal-title":"Mathematical Programming Study"},{"key":"2597_CR32","unstructured":"Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis, volume 317 of Grundlehren der mathematischen Wissenschaften. Springer Verlag Berlin, 3rd edition (2009)"},{"key":"2597_CR33","doi-asserted-by":"crossref","unstructured":"Royset, J., Wets, R.-B.: An optimization primer. Springer Series in Operations Research and Financial Engineering. Springer Cham, 1 edition (2022)","DOI":"10.1007\/978-3-030-76275-9_1"},{"issue":"1","key":"2597_CR34","doi-asserted-by":"publisher","first-page":"339","DOI":"10.1007\/s10107-022-01909-1","volume":"201","author":"JO Royset","year":"2023","unstructured":"Royset, J.O.: Consistent approximations in composite optimization. Mathematical Programming 201(1), 339\u2013372 (2023)","journal-title":"Mathematical Programming"},{"issue":"1","key":"2597_CR35","doi-asserted-by":"publisher","first-page":"187","DOI":"10.1007\/s10107-012-0570-7","volume":"134","author":"C Sagastiz\u00e1bal","year":"2012","unstructured":"Sagastiz\u00e1bal, C.: Divide to conquer: Decomposition methods for energy optimization. Mathematical Programming 134(1), 187\u2013222 (2012)","journal-title":"Mathematical Programming"},{"issue":"1","key":"2597_CR36","doi-asserted-by":"publisher","first-page":"189","DOI":"10.1007\/s10107-012-0600-5","volume":"140","author":"C Sagastiz\u00e1bal","year":"2013","unstructured":"Sagastiz\u00e1bal, C.: Composite proximal bundle method. Mathematical Programming 140(1), 189\u2013233 (2013)","journal-title":"Mathematical Programming"},{"issue":"1","key":"2597_CR37","doi-asserted-by":"publisher","first-page":"146","DOI":"10.1137\/040603875","volume":"16","author":"C Sagastiz\u00e1bal","year":"2005","unstructured":"Sagastiz\u00e1bal, C., Solodov, M.: An infeasible bundle method for nonsmooth convex constrained optimization without a penalty function or a filter. SIAM Journal on Optimization 16(1), 146\u2013169 (2005)","journal-title":"SIAM Journal on Optimization"},{"key":"2597_CR38","doi-asserted-by":"crossref","unstructured":"Shapiro, A., Dentcheva, D., Ruszczy\u0144ski, A.: Lectures on Stochastic Programming: Modeling and Theory. MPS-SIAM Series on Optimization. SIAM - Society for Industrial and Applied Mathematics and Mathematical Programming Society, Philadelphia (2009)","DOI":"10.1137\/1.9780898718751"},{"issue":"6","key":"2597_CR39","doi-asserted-by":"publisher","first-page":"627","DOI":"10.1061\/(ASCE)0733-9399(2003)129:6(627)","volume":"129","author":"J Song","year":"2003","unstructured":"Song, J., Kiureghian, A.D.: Bounds on system reliability by linear programming. Journal of Engineering Mechanics 129(6), 627\u2013636 (2003)","journal-title":"Journal of Engineering Mechanics"},{"key":"2597_CR40","doi-asserted-by":"crossref","unstructured":"Strekalovsky, A., Minarchenko, I.: On local search in d.c. optimization. In 2017 Constructive Nonsmooth Analysis and Related Topics (dedicated to the memory of V.F. Demyanov) (CNSA), pages 1\u20134 (2017)","DOI":"10.1109\/CNSA.2017.7974022"},{"key":"2597_CR41","first-page":"36","volume-title":"Difference-of-convex approach to chance-constrained optimal power flow modelling the DSO power modulation lever for distribution networks","author":"K Syrtseva","year":"2023","unstructured":"Syrtseva, K., de Oliveira, W., Demassey, S., Morais, H., Javal, P., Swaminathan, B.: Difference-of-convex approach to chance-constrained optimal power flow modelling the DSO power modulation lever for distribution networks, p. 36. Sustainable Energy, Grids and Networks (2023)"},{"key":"2597_CR42","unstructured":"Syrtseva, K., de\u00a0Oliveira, W., Demassey, S., van Ackooij, W.: Minimizing the difference of convex and weakly convex functions via bundle method. Pacific Journal of Optimization. 20(4), 699\u2013741 (2024)"},{"key":"2597_CR43","doi-asserted-by":"publisher","first-page":"177","DOI":"10.1016\/j.cor.2016.08.002","volume":"77","author":"W van Ackooij","year":"2017","unstructured":"van Ackooij, W., Berge, V., de Oliveira, W., Sagastiz\u00e1bal, C.: Probabilistic optimization via approximate p-efficient points and bundle methods. Computers & Operations Research 77, 177\u2013193 (2017)","journal-title":"Computers & Operations Research"},{"issue":"4","key":"2597_CR44","doi-asserted-by":"publisher","first-page":"890","DOI":"10.1080\/10556788.2019.1595619","volume":"34","author":"W van Ackooij","year":"2019","unstructured":"van Ackooij, W., de Oliveira, W.: Non-smooth DC-constrained optimization: constraint qualification and minimizing methodologies. Optimization Methods and Software 34(4), 890\u2013920 (2019)","journal-title":"Optimization Methods and Software"},{"issue":"3","key":"2597_CR45","doi-asserted-by":"publisher","first-page":"509","DOI":"10.1007\/s11590-019-01477-y","volume":"14","author":"W van Ackooij","year":"2020","unstructured":"van Ackooij, W., de Oliveira, W.: Some brief observations in minimizing the sum of locally Lipschitzian functions. Optimization Letters 14(3), 509\u2013520 (2020)","journal-title":"Optimization Letters"},{"key":"2597_CR46","doi-asserted-by":"crossref","unstructured":"van Ackooij, W., de Oliveira, W.: Addendum to the paper \u2018nonsmooth DC-constrained optimization: constraint qualification and minimizing methodologies\u2019. Optimization Methods and Software, pages 2241\u20132250 (2022)","DOI":"10.1080\/10556788.2022.2063861"},{"issue":"2","key":"2597_CR47","doi-asserted-by":"publisher","first-page":"451","DOI":"10.1007\/s10589-020-00241-8","volume":"78","author":"W van Ackooij","year":"2021","unstructured":"van Ackooij, W., Demassey, S., Javal, P., Morais, H., de Oliveira, W., Swaminathan, B.: A bundle method for nonsmooth DC programming with application to chance-constrained problems. Computational Optimization and Applications 78(2), 451\u2013490 (2021)","journal-title":"Computational Optimization and Applications"},{"key":"2597_CR48","first-page":"509","volume":"15","author":"W van Ackooij","year":"2014","unstructured":"van Ackooij, W., Henrion, R., M\u00f6ller, A., Zorgati, R.: Joint chance constrained programming for hydro reservoir management. Optimization and Engineering 15, 509\u2013531 (2014)","journal-title":"Optimization and Engineering"}],"container-title":["Journal of Optimization Theory and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-024-02597-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10957-024-02597-x\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-024-02597-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,3,4]],"date-time":"2025-03-04T07:27:52Z","timestamp":1741073272000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10957-024-02597-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,2,7]]},"references-count":48,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2025,3]]}},"alternative-id":["2597"],"URL":"https:\/\/doi.org\/10.1007\/s10957-024-02597-x","relation":{},"ISSN":["0022-3239","1573-2878"],"issn-type":[{"value":"0022-3239","type":"print"},{"value":"1573-2878","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,2,7]]},"assertion":[{"value":"12 April 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 December 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 February 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"54"}}