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An inexact regularized proximal Newton method is proposed by an approximation to the Hessian of <jats:italic>f<\/jats:italic> involving the <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\varrho $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03f1<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>th power of the KKT residual. For <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\varrho =0$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u03f1<\/mml:mi>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mn>0<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, we justify the global convergence of the iterate sequence for the KL objective function and its R-linear convergence rate for the KL objective function of exponent 1\/2. For <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\varrho \\in (0,1)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u03f1<\/mml:mi>\n                    <mml:mo>\u2208<\/mml:mo>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mn>0<\/mml:mn>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mn>1<\/mml:mn>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, by assuming that cluster points satisfy a locally H\u00f6lderian error bound of order <jats:italic>q<\/jats:italic> on a second-order stationary point set and a local error bound of order <jats:inline-formula><jats:alternatives><jats:tex-math>$$q&gt;1\\!+\\!\\varrho $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>q<\/mml:mi>\n                    <mml:mo>&gt;<\/mml:mo>\n                    <mml:mn>1<\/mml:mn>\n                    <mml:mspace\/>\n                    <mml:mo>+<\/mml:mo>\n                    <mml:mspace\/>\n                    <mml:mi>\u03f1<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> on the common stationary point set, respectively, we establish the global convergence of the iterate sequence and its superlinear convergence rate with order depending on <jats:italic>q<\/jats:italic> and <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\varrho $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03f1<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. A dual semismooth Newton augmented Lagrangian method is also developed for seeking an inexact minimizer of subproblems. Numerical comparisons with two state-of-the-art methods on <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\ell _1$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>\u2113<\/mml:mi>\n                    <mml:mn>1<\/mml:mn>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-regularized Student\u2019s <jats:italic>t<\/jats:italic>-regressions, group penalized Student\u2019s <jats:italic>t<\/jats:italic>-regressions, and nonconvex image restoration confirm the efficiency of the proposed method.<\/jats:p>","DOI":"10.1007\/s10589-024-00560-0","type":"journal-article","created":{"date-parts":[[2024,2,20]],"date-time":"2024-02-20T06:02:05Z","timestamp":1708408925000},"page":"603-641","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["An inexact regularized proximal Newton method for nonconvex and nonsmooth optimization"],"prefix":"10.1007","volume":"88","author":[{"given":"Ruyu","family":"Liu","sequence":"first","affiliation":[]},{"given":"Shaohua","family":"Pan","sequence":"additional","affiliation":[]},{"given":"Yuqia","family":"Wu","sequence":"additional","affiliation":[]},{"given":"Xiaoqi","family":"Yang","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,2,20]]},"reference":[{"issue":"1","key":"560_CR1","doi-asserted-by":"publisher","first-page":"267","DOI":"10.1111\/j.2517-6161.1996.tb02080.x","volume":"58","author":"R Tibshirani","year":"1996","unstructured":"Tibshirani, R.: Regression shrinkage and selection via the lasso. 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