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Comp."],"abstract":"<p>\n                    The augmented Lagrangian method (ALM) is classic for canonical convex programming problems with linear constraints, and it finds many applications in various scientific computing areas. A major advantage of the ALM is that the step for updating the dual variable can be further relaxed with a step size in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-parenthesis 0 comma 2 right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">(0,2)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , and this advantage can easily lead to numerical acceleration for the ALM. When a separable convex programming problem is discussed and a corresponding splitting version of the classic ALM is considered, convergence may not be guaranteed and thus it is seemingly impossible that a step size in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-parenthesis 0 comma 2 right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">(0,2)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    can be carried on to the relaxation step for updating the dual variable. We show that for a parallel splitting version of the ALM, a step size in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-parenthesis 0 comma 2 right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">(0,2)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    can be maintained for further relaxing both the primal and dual variables if the relaxation step is simply corrected by a rank-two matrix. Hence, a rank-two relaxed parallel splitting version of the ALM with a step size in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-parenthesis 0 comma 2 right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">(0,2)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is proposed for separable convex programming problems. We validate that the new algorithm can numerically outperform existing algorithms of the same kind significantly by testing some applications.\n                  <\/p>","DOI":"10.1090\/mcom\/3822","type":"journal-article","created":{"date-parts":[[2023,1,11]],"date-time":"2023-01-11T16:13:03Z","timestamp":1673453583000},"page":"1633-1663","source":"Crossref","is-referenced-by-count":4,"title":["A rank-two relaxed parallel splitting version of the augmented Lagrangian method with step size in (0,2) for separable convex programming"],"prefix":"10.1090","volume":"92","author":[{"given":"Bingsheng","family":"He","sequence":"first","affiliation":[]},{"given":"Feng","family":"Ma","sequence":"additional","affiliation":[]},{"given":"Shengjie","family":"Xu","sequence":"additional","affiliation":[]},{"given":"Xiaoming","family":"Yuan","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2023,2,28]]},"reference":[{"issue":"4","key":"1","doi-asserted-by":"publisher","first-page":"1286","DOI":"10.1137\/060654797","article-title":"On augmented Lagrangian methods with general lower-level constraints","volume":"18","author":"Andreani, R.","year":"2007","journal-title":"SIAM J. 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