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However, many functions arising in practical applications such as low rank matrix factorization or deep neural network problems do not have a Lipschitz continuous gradient. This led to the development of a generalized notion known as the <jats:italic>L<\/jats:italic>-smad property, which is based on generalized proximity measures called Bregman distances. However, the <jats:italic>L<\/jats:italic>-smad property cannot handle nonsmooth functions, for example, simple nonsmooth functions like <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\vert x^4-1 \\vert $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mrow>\n                      <mml:mo>|<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:msup>\n                      <mml:mi>x<\/mml:mi>\n                      <mml:mn>4<\/mml:mn>\n                    <\/mml:msup>\n                    <mml:mrow>\n                      <mml:mo>-<\/mml:mo>\n                      <mml:mn>1<\/mml:mn>\n                      <mml:mo>|<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and also many practical composite problems are out of scope. We fix this issue by proposing the MAP property, which generalizes the <jats:italic>L<\/jats:italic>-smad property and is also valid for a large class of structured nonconvex nonsmooth composite problems. Based on the proposed MAP property, we propose a globally convergent algorithm called Model BPG, that unifies several existing algorithms. The convergence analysis is based on a new Lyapunov function. We also numerically illustrate the superior performance of Model BPG on standard phase retrieval problems and Poisson linear inverse problems, when compared to a state of the art optimization method that is valid for generic nonconvex nonsmooth optimization problems.<\/jats:p>","DOI":"10.1007\/s10898-021-01114-y","type":"journal-article","created":{"date-parts":[[2021,12,1]],"date-time":"2021-12-01T10:02:47Z","timestamp":1638352967000},"page":"753-781","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Global convergence of model function based Bregman proximal minimization algorithms"],"prefix":"10.1007","volume":"83","author":[{"given":"Mahesh Chandra","family":"Mukkamala","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jalal","family":"Fadili","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Peter","family":"Ochs","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2021,12,1]]},"reference":[{"issue":"46","key":"1114_CR1","doi-asserted-by":"publisher","first-page":"22924","DOI":"10.1073\/pnas.1908018116","volume":"116","author":"H Asi","year":"2019","unstructured":"Asi, H., Duchi, J.C.: The importance of better models in stochastic optimization. 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