{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,13]],"date-time":"2026-01-13T20:43:22Z","timestamp":1768337002906,"version":"3.49.0"},"reference-count":25,"publisher":"MathDoc\/Centre Mersenne","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"abstract":"<jats:p>\n                    Frank and Wolfe\u2019s celebrated conditional gradient method is a well-known tool for solving smooth optimization problems for which minimizing a linear function over the feasible set is computationally cheap. However, when the objective function is nonsmooth, the method may fail to compute a stationary point. In this work, we show that the Frank\u2013Wolfe algorithm can be employed to compute Clarke-stationary points for nonconvex and nonsmooth optimization problems consisting of minimizing upper-\n                    <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <mml:msup>\n                        <mml:mi>C<\/mml:mi>\n                        <mml:mrow>\n                          <mml:mn>1<\/mml:mn>\n                          <mml:mo>,<\/mml:mo>\n                          <mml:mi>\u03b1<\/mml:mi>\n                        <\/mml:mrow>\n                      <\/mml:msup>\n                    <\/mml:math>\n                    functions over convex and compact sets. Furthermore, under more restrictive assumptions, we propose a new algorithm variant with stronger stationarity guarantees, namely directional stationarity and even local optimality.\n                  <\/jats:p>","DOI":"10.5802\/ojmo.21","type":"journal-article","created":{"date-parts":[[2023,1,24]],"date-time":"2023-01-24T12:27:29Z","timestamp":1674563249000},"page":"1-10","source":"Crossref","is-referenced-by-count":6,"title":["Short Paper - A note on the Frank\u2013Wolfe algorithm for a class of nonconvex and nonsmooth optimization problems"],"prefix":"10.5802","volume":"4","author":[{"given":"Welington","family":"de Oliveira","sequence":"first","affiliation":[]}],"member":"3842","published-online":{"date-parts":[[2023,1,24]]},"reference":[{"issue":"1","key":"key2025101710010986423_1","doi-asserted-by":"publisher","first-page":"115","DOI":"10.1137\/130941961","article-title":"Duality between subgradient and conditional gradient methods","volume":"26","author":"Bach, Francis","year":"2015","unstructured":"[1] Bach, Francis Duality between subgradient and conditional gradient methods, SIAM J. Optim., Volume 26 (2015) no. 1, pp. 115-129","journal-title":"SIAM J. Optim."},{"key":"key2025101710010986423_2","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611974997","volume":"25","author":"Beck, Amir","year":"2017","unstructured":"[2] Beck, Amir First-Order Methods in Optimization, MOS-SIAM Series on Optimization, 25, Society for Industrial and Applied Mathematics, 2017","journal-title":"First-Order Methods in Optimization"},{"issue":"1","key":"key2025101710010986423_3","doi-asserted-by":"publisher","first-page":"56","DOI":"10.1137\/18M1217760","article-title":"On the convergence to stationary points of deterministic and randomized feasible descent directions methods","volume":"30","author":"Beck, Amir","year":"2020","unstructured":"[3] Beck, Amir; Hallak, Nadav On the convergence to stationary points of deterministic and randomized feasible descent directions methods, SIAM J. Optim., Volume 30 (2020) no. 1, pp. 56-79","journal-title":"SIAM J. Optim."},{"issue":"2","key":"key2025101710010986423_4","doi-asserted-by":"publisher","first-page":"616","DOI":"10.1137\/15M1035793","article-title":"The alternating descent conditional gradient method for sparse inverse problems","volume":"27","author":"Boyd, Nicholas","year":"2017","unstructured":"[4] Boyd, Nicholas; Schiebinger, Geoffrey; Recht, Benjamin The alternating descent conditional gradient method for sparse inverse problems, SIAM J. Optim., Volume 27 (2017) no. 2, pp. 616-639","journal-title":"SIAM J. 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Convex Anal., Volume 12 (2005) no. 2, pp. 315-329","journal-title":"J. Convex Anal."},{"issue":"4","key":"key2025101710010986423_8","doi-asserted-by":"publisher","first-page":"679","DOI":"10.1007\/s11228-020-00566-w","article-title":"The ABC of DC programming","volume":"28","author":"de Oliveira, Welington","year":"2020","unstructured":"[8] de Oliveira, Welington The ABC of DC programming, Set-Valued Var. Anal., Volume 28 (2020) no. 4, pp. 679-706","journal-title":"Set-Valued Var. Anal."},{"issue":"1","key":"key2025101710010986423_9","article-title":"The sliding Frank\u2013Wolfe algorithm and its application to super-resolution microscopy","volume":"36","author":"Denoyelle, Quentin","year":"2020","unstructured":"[9] Denoyelle, Quentin; Duval, Vincent; Peyr\u00e9, Gabriel; Soubies, Emmanuel. The sliding Frank\u2013Wolfe algorithm and its application to super-resolution microscopy, Inverse Probl., Volume 36 (2020) no. 1, 014001, 42 pages","journal-title":"Inverse Probl."},{"issue":"2","key":"key2025101710010986423_10","doi-asserted-by":"publisher","first-page":"674","DOI":"10.1137\/110831659","article-title":"Randomized smoothing for stochastic optimization","volume":"22","author":"Duchi, John C.","year":"2012","unstructured":"[10] Duchi, John C.; Bartlett, Peter L.; Wainwright, Martin J. Randomized smoothing for stochastic optimization, SIAM J. Optim., Volume 22 (2012) no. 2, pp. 674-701","journal-title":"SIAM J. Optim."},{"issue":"1-2","key":"key2025101710010986423_11","doi-asserted-by":"publisher","first-page":"95","DOI":"10.1002\/nav.3800030109","article-title":"An algorithm for quadratic programming","volume":"3","author":"Frank, Marguerite","year":"1956","unstructured":"[11] Frank, Marguerite; Wolfe, Philip An algorithm for quadratic programming, Nav. Res. Logist. 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Extension of the Frank\u2013Wolfe algorithm to concave nondifferentiable objective functions, J. Optim. Theory Appl., Volume 78 (1993) no. 2, pp. 283-301","journal-title":"J. Optim. 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