Dualize, Split, Randomize: Toward Fast Nonsmooth Optimization Algorithms View Full Text


Ontology type: schema:ScholarlyArticle      Open Access: True


Article Info

DATE

2022-07-13

AUTHORS

Adil Salim, Laurent Condat, Konstantin Mishchenko, Peter Richtárik

ABSTRACT

We consider minimizing the sum of three convex functions, where the first one F is smooth, the second one is nonsmooth and proximable and the third one is the composition of a nonsmooth proximable function with a linear operator L. This template problem has many applications, for instance, in image processing and machine learning. First, we propose a new primal–dual algorithm, which we call PDDY, for this problem. It is constructed by applying Davis–Yin splitting to a monotone inclusion in a primal–dual product space, where the operators are monotone under a specific metric depending on L. We show that three existing algorithms (the two forms of the Condat–Vũ algorithm and the PD3O algorithm) have the same structure, so that PDDY is the fourth missing link in this self-consistent class of primal–dual algorithms. This representation eases the convergence analysis: it allows us to derive sublinear convergence rates in general, and linear convergence results in presence of strong convexity. Moreover, within our broad and flexible analysis framework, we propose new stochastic generalizations of the algorithms, in which a variance-reduced random estimate of the gradient of F is used, instead of the true gradient. Furthermore, we obtain, as a special case of PDDY, a linearly converging algorithm for the minimization of a strongly convex function F under a linear constraint; we discuss its important application to decentralized optimization. More... »

PAGES

1-29

References to SciGraph publications

  • 2017-06-14. A Three-Operator Splitting Scheme and its Optimization Applications in SET-VALUED AND VARIATIONAL ANALYSIS
  • 2018-03-02. A New Primal–Dual Algorithm for Minimizing the Sum of Three Functions with a Linear Operator in JOURNAL OF SCIENTIFIC COMPUTING
  • 2011-05-09. Proximal Splitting Methods in Signal Processing in FIXED-POINT ALGORITHMS FOR INVERSE PROBLEMS IN SCIENCE AND ENGINEERING
  • 2020-09-30. Projective splitting with forward steps in MATHEMATICAL PROGRAMMING
  • 2015-10-30. On the ergodic convergence rates of a first-order primal–dual algorithm in MATHEMATICAL PROGRAMMING
  • 2014-08-06. Recent Developments on Primal–Dual Splitting Methods with Applications to Convex Minimization in MATHEMATICS WITHOUT BOUNDARIES
  • 2018-08-16. On the equivalence of the primal-dual hybrid gradient method and Douglas–Rachford splitting in MATHEMATICAL PROGRAMMING
  • 2015-03-25. Coordinate descent algorithms in MATHEMATICAL PROGRAMMING
  • 2020-11-09. Single-forward-step projective splitting: exploiting cocoercivity in COMPUTATIONAL OPTIMIZATION AND APPLICATIONS
  • 2019-05-22. Uniqueness of DRS as the 2 operator resolvent-splitting and impossibility of 3 operator resolvent-splitting in MATHEMATICAL PROGRAMMING
  • 2011-11-29. A splitting algorithm for dual monotone inclusions involving cocoercive operators in ADVANCES IN COMPUTATIONAL MATHEMATICS
  • 2007-01-05. A family of projective splitting methods for the sum of two maximal monotone operators in MATHEMATICAL PROGRAMMING
  • 2010-12-21. A First-Order Primal-Dual Algorithm for Convex Problems with Applications to Imaging in JOURNAL OF MATHEMATICAL IMAGING AND VISION
  • 2020. First-order and Stochastic Optimization Methods for Machine Learning in NONE
  • 2017. Convex Analysis and Monotone Operator Theory in Hilbert Spaces in NONE
  • 2018. Lectures on Convex Optimization in NONE
  • 2016-07-05. Asynchronous block-iterative primal-dual decomposition methods for monotone inclusions in MATHEMATICAL PROGRAMMING
  • 1992-04. On the Douglas—Rachford splitting method and the proximal point algorithm for maximal monotone operators in MATHEMATICAL PROGRAMMING
  • 2012-12-29. A Primal–Dual Splitting Method for Convex Optimization Involving Lipschitzian, Proximable and Linear Composite Terms in JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS
  • 2011-08-27. Primal-Dual Splitting Algorithm for Solving Inclusions with Mixtures of Composite, Lipschitzian, and Parallel-Sum Type Monotone Operators in SET-VALUED AND VARIATIONAL ANALYSIS
  • Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1007/s10957-022-02061-8

    DOI

    http://dx.doi.org/10.1007/s10957-022-02061-8

    DIMENSIONS

    https://app.dimensions.ai/details/publication/pub.1149442510


    Indexing Status Check whether this publication has been indexed by Scopus and Web Of Science using the SN Indexing Status Tool
    Incoming Citations Browse incoming citations for this publication using opencitations.net

    JSON-LD is the canonical representation for SciGraph data.

    TIP: You can open this SciGraph record using an external JSON-LD service: JSON-LD Playground Google SDTT

    [
      {
        "@context": "https://springernature.github.io/scigraph/jsonld/sgcontext.json", 
        "about": [
          {
            "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/01", 
            "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
            "name": "Mathematical Sciences", 
            "type": "DefinedTerm"
          }, 
          {
            "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/0103", 
            "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
            "name": "Numerical and Computational Mathematics", 
            "type": "DefinedTerm"
          }
        ], 
        "author": [
          {
            "affiliation": {
              "alternateName": "Computer Science Program, Computer, Electrical and Mathematical Science and Engineering Division, King Abdullah University of Science and Technology (KAUST), 23955-6900, Thuwal, Kingdom of Saudi Arabia", 
              "id": "http://www.grid.ac/institutes/grid.45672.32", 
              "name": [
                "Computer Science Program, Computer, Electrical and Mathematical Science and Engineering Division, King Abdullah University of Science and Technology (KAUST), 23955-6900, Thuwal, Kingdom of Saudi Arabia"
              ], 
              "type": "Organization"
            }, 
            "familyName": "Salim", 
            "givenName": "Adil", 
            "id": "sg:person.07715126012.51", 
            "sameAs": [
              "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.07715126012.51"
            ], 
            "type": "Person"
          }, 
          {
            "affiliation": {
              "alternateName": "Computer Science Program, Computer, Electrical and Mathematical Science and Engineering Division, King Abdullah University of Science and Technology (KAUST), 23955-6900, Thuwal, Kingdom of Saudi Arabia", 
              "id": "http://www.grid.ac/institutes/grid.45672.32", 
              "name": [
                "Computer Science Program, Computer, Electrical and Mathematical Science and Engineering Division, King Abdullah University of Science and Technology (KAUST), 23955-6900, Thuwal, Kingdom of Saudi Arabia"
              ], 
              "type": "Organization"
            }, 
            "familyName": "Condat", 
            "givenName": "Laurent", 
            "id": "sg:person.01014365475.33", 
            "sameAs": [
              "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.01014365475.33"
            ], 
            "type": "Person"
          }, 
          {
            "affiliation": {
              "alternateName": "Computer Science Program, Computer, Electrical and Mathematical Science and Engineering Division, King Abdullah University of Science and Technology (KAUST), 23955-6900, Thuwal, Kingdom of Saudi Arabia", 
              "id": "http://www.grid.ac/institutes/grid.45672.32", 
              "name": [
                "Computer Science Program, Computer, Electrical and Mathematical Science and Engineering Division, King Abdullah University of Science and Technology (KAUST), 23955-6900, Thuwal, Kingdom of Saudi Arabia"
              ], 
              "type": "Organization"
            }, 
            "familyName": "Mishchenko", 
            "givenName": "Konstantin", 
            "id": "sg:person.016025561605.15", 
            "sameAs": [
              "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.016025561605.15"
            ], 
            "type": "Person"
          }, 
          {
            "affiliation": {
              "alternateName": "Computer Science Program, Computer, Electrical and Mathematical Science and Engineering Division, King Abdullah University of Science and Technology (KAUST), 23955-6900, Thuwal, Kingdom of Saudi Arabia", 
              "id": "http://www.grid.ac/institutes/grid.45672.32", 
              "name": [
                "Computer Science Program, Computer, Electrical and Mathematical Science and Engineering Division, King Abdullah University of Science and Technology (KAUST), 23955-6900, Thuwal, Kingdom of Saudi Arabia"
              ], 
              "type": "Organization"
            }, 
            "familyName": "Richt\u00e1rik", 
            "givenName": "Peter", 
            "id": "sg:person.016427176172.93", 
            "sameAs": [
              "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.016427176172.93"
            ], 
            "type": "Person"
          }
        ], 
        "citation": [
          {
            "id": "sg:pub.10.1007/s10107-015-0957-3", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1012909932", 
              "https://doi.org/10.1007/s10107-015-0957-3"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/978-1-4419-9569-8_10", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1052700561", 
              "https://doi.org/10.1007/978-1-4419-9569-8_10"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/978-1-4939-1124-0_3", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1037691419", 
              "https://doi.org/10.1007/978-1-4939-1124-0_3"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/978-3-319-48311-5", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1084688822", 
              "https://doi.org/10.1007/978-3-319-48311-5"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s10107-006-0070-8", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1021274319", 
              "https://doi.org/10.1007/s10107-006-0070-8"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s10107-018-1321-1", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1106172667", 
              "https://doi.org/10.1007/s10107-018-1321-1"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s11228-017-0421-z", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1086031536", 
              "https://doi.org/10.1007/s11228-017-0421-z"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s10107-016-1044-0", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1036311274", 
              "https://doi.org/10.1007/s10107-016-1044-0"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s11228-011-0191-y", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1037327416", 
              "https://doi.org/10.1007/s11228-011-0191-y"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s10589-020-00238-3", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1132459987", 
              "https://doi.org/10.1007/s10589-020-00238-3"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s10957-012-0245-9", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1027180321", 
              "https://doi.org/10.1007/s10957-012-0245-9"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s10444-011-9254-8", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1017372143", 
              "https://doi.org/10.1007/s10444-011-9254-8"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s10851-010-0251-1", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1010318529", 
              "https://doi.org/10.1007/s10851-010-0251-1"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf01581204", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1022153870", 
              "https://doi.org/10.1007/bf01581204"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s10107-015-0892-3", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1019991396", 
              "https://doi.org/10.1007/s10107-015-0892-3"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s10915-018-0680-3", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1101322880", 
              "https://doi.org/10.1007/s10915-018-0680-3"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s10107-020-01565-3", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1131335764", 
              "https://doi.org/10.1007/s10107-020-01565-3"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s10107-019-01403-1", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1115172565", 
              "https://doi.org/10.1007/s10107-019-01403-1"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/978-3-319-91578-4", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1110040882", 
              "https://doi.org/10.1007/978-3-319-91578-4"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/978-3-030-39568-1", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1127627565", 
              "https://doi.org/10.1007/978-3-030-39568-1"
            ], 
            "type": "CreativeWork"
          }
        ], 
        "datePublished": "2022-07-13", 
        "datePublishedReg": "2022-07-13", 
        "description": "We consider minimizing the sum of three convex functions, where the first one F is smooth, the second one is nonsmooth and proximable and the third one is the composition of a nonsmooth proximable function with a linear operator L. This template problem has many applications, for instance, in image processing and machine learning. First, we propose a new primal\u2013dual algorithm, which we call PDDY, for this problem. It is constructed by applying Davis\u2013Yin splitting to a monotone inclusion in a primal\u2013dual product space, where the operators are monotone under a specific metric depending on L. We show that three existing algorithms (the two forms of the Condat\u2013V\u0169 algorithm and the PD3O algorithm) have the same structure, so that PDDY is the fourth missing link in this self-consistent class of primal\u2013dual algorithms. This representation eases the convergence analysis: it allows us to derive sublinear convergence rates in general, and linear convergence results in presence of strong convexity. Moreover, within our broad and flexible analysis framework, we propose new stochastic generalizations of the algorithms, in which a variance-reduced random estimate of the gradient of F is used, instead of the true gradient. Furthermore, we obtain, as a special case of PDDY, a linearly converging algorithm for the minimization of a strongly convex function F under a linear constraint; we discuss its important application to decentralized optimization.", 
        "genre": "article", 
        "id": "sg:pub.10.1007/s10957-022-02061-8", 
        "isAccessibleForFree": true, 
        "isPartOf": [
          {
            "id": "sg:journal.1044187", 
            "issn": [
              "0022-3239", 
              "1573-2878"
            ], 
            "name": "Journal of Optimization Theory and Applications", 
            "publisher": "Springer Nature", 
            "type": "Periodical"
          }
        ], 
        "keywords": [
          "primal-dual algorithm", 
          "linear convergence results", 
          "nonsmooth optimization algorithm", 
          "sublinear convergence rate", 
          "new primal-dual algorithm", 
          "convex function f", 
          "stochastic generalization", 
          "convergence analysis", 
          "monotone inclusions", 
          "convergence results", 
          "strong convexity", 
          "linear constraints", 
          "decentralized optimization", 
          "true gradient", 
          "convex functions", 
          "operator L.", 
          "convergence rate", 
          "product space", 
          "flexible analysis framework", 
          "optimization algorithm", 
          "random estimates", 
          "special case", 
          "function f", 
          "template problem", 
          "important applications", 
          "algorithm", 
          "second one", 
          "metric depending", 
          "same structure", 
          "machine learning", 
          "problem", 
          "analysis framework", 
          "third one", 
          "operators", 
          "generalization", 
          "convexity", 
          "minimization", 
          "optimization", 
          "space", 
          "constraints", 
          "image processing", 
          "sum", 
          "applications", 
          "function", 
          "gradient", 
          "class", 
          "one", 
          "representation", 
          "estimates", 
          "splitting", 
          "framework", 
          "instances", 
          "structure", 
          "depending", 
          "cases", 
          "results", 
          "link", 
          "split", 
          "analysis", 
          "processing", 
          "inclusion", 
          "missing link", 
          "learning", 
          "presence", 
          "rate", 
          "composition", 
          "L."
        ], 
        "name": "Dualize, Split, Randomize: Toward Fast Nonsmooth Optimization Algorithms", 
        "pagination": "1-29", 
        "productId": [
          {
            "name": "dimensions_id", 
            "type": "PropertyValue", 
            "value": [
              "pub.1149442510"
            ]
          }, 
          {
            "name": "doi", 
            "type": "PropertyValue", 
            "value": [
              "10.1007/s10957-022-02061-8"
            ]
          }
        ], 
        "sameAs": [
          "https://doi.org/10.1007/s10957-022-02061-8", 
          "https://app.dimensions.ai/details/publication/pub.1149442510"
        ], 
        "sdDataset": "articles", 
        "sdDatePublished": "2022-09-02T16:08", 
        "sdLicense": "https://scigraph.springernature.com/explorer/license/", 
        "sdPublisher": {
          "name": "Springer Nature - SN SciGraph project", 
          "type": "Organization"
        }, 
        "sdSource": "s3://com-springernature-scigraph/baseset/20220902/entities/gbq_results/article/article_937.jsonl", 
        "type": "ScholarlyArticle", 
        "url": "https://doi.org/10.1007/s10957-022-02061-8"
      }
    ]
     

    Download the RDF metadata as:  json-ld nt turtle xml License info

    HOW TO GET THIS DATA PROGRAMMATICALLY:

    JSON-LD is a popular format for linked data which is fully compatible with JSON.

    curl -H 'Accept: application/ld+json' 'https://scigraph.springernature.com/pub.10.1007/s10957-022-02061-8'

    N-Triples is a line-based linked data format ideal for batch operations.

    curl -H 'Accept: application/n-triples' 'https://scigraph.springernature.com/pub.10.1007/s10957-022-02061-8'

    Turtle is a human-readable linked data format.

    curl -H 'Accept: text/turtle' 'https://scigraph.springernature.com/pub.10.1007/s10957-022-02061-8'

    RDF/XML is a standard XML format for linked data.

    curl -H 'Accept: application/rdf+xml' 'https://scigraph.springernature.com/pub.10.1007/s10957-022-02061-8'


     

    This table displays all metadata directly associated to this object as RDF triples.

    219 TRIPLES      21 PREDICATES      109 URIs      81 LITERALS      4 BLANK NODES

    Subject Predicate Object
    1 sg:pub.10.1007/s10957-022-02061-8 schema:about anzsrc-for:01
    2 anzsrc-for:0103
    3 schema:author N512ffff60be741c2bd78851ed85dca4a
    4 schema:citation sg:pub.10.1007/978-1-4419-9569-8_10
    5 sg:pub.10.1007/978-1-4939-1124-0_3
    6 sg:pub.10.1007/978-3-030-39568-1
    7 sg:pub.10.1007/978-3-319-48311-5
    8 sg:pub.10.1007/978-3-319-91578-4
    9 sg:pub.10.1007/bf01581204
    10 sg:pub.10.1007/s10107-006-0070-8
    11 sg:pub.10.1007/s10107-015-0892-3
    12 sg:pub.10.1007/s10107-015-0957-3
    13 sg:pub.10.1007/s10107-016-1044-0
    14 sg:pub.10.1007/s10107-018-1321-1
    15 sg:pub.10.1007/s10107-019-01403-1
    16 sg:pub.10.1007/s10107-020-01565-3
    17 sg:pub.10.1007/s10444-011-9254-8
    18 sg:pub.10.1007/s10589-020-00238-3
    19 sg:pub.10.1007/s10851-010-0251-1
    20 sg:pub.10.1007/s10915-018-0680-3
    21 sg:pub.10.1007/s10957-012-0245-9
    22 sg:pub.10.1007/s11228-011-0191-y
    23 sg:pub.10.1007/s11228-017-0421-z
    24 schema:datePublished 2022-07-13
    25 schema:datePublishedReg 2022-07-13
    26 schema:description We consider minimizing the sum of three convex functions, where the first one F is smooth, the second one is nonsmooth and proximable and the third one is the composition of a nonsmooth proximable function with a linear operator L. This template problem has many applications, for instance, in image processing and machine learning. First, we propose a new primal–dual algorithm, which we call PDDY, for this problem. It is constructed by applying Davis–Yin splitting to a monotone inclusion in a primal–dual product space, where the operators are monotone under a specific metric depending on L. We show that three existing algorithms (the two forms of the Condat–Vũ algorithm and the PD3O algorithm) have the same structure, so that PDDY is the fourth missing link in this self-consistent class of primal–dual algorithms. This representation eases the convergence analysis: it allows us to derive sublinear convergence rates in general, and linear convergence results in presence of strong convexity. Moreover, within our broad and flexible analysis framework, we propose new stochastic generalizations of the algorithms, in which a variance-reduced random estimate of the gradient of F is used, instead of the true gradient. Furthermore, we obtain, as a special case of PDDY, a linearly converging algorithm for the minimization of a strongly convex function F under a linear constraint; we discuss its important application to decentralized optimization.
    27 schema:genre article
    28 schema:isAccessibleForFree true
    29 schema:isPartOf sg:journal.1044187
    30 schema:keywords L.
    31 algorithm
    32 analysis
    33 analysis framework
    34 applications
    35 cases
    36 class
    37 composition
    38 constraints
    39 convergence analysis
    40 convergence rate
    41 convergence results
    42 convex function f
    43 convex functions
    44 convexity
    45 decentralized optimization
    46 depending
    47 estimates
    48 flexible analysis framework
    49 framework
    50 function
    51 function f
    52 generalization
    53 gradient
    54 image processing
    55 important applications
    56 inclusion
    57 instances
    58 learning
    59 linear constraints
    60 linear convergence results
    61 link
    62 machine learning
    63 metric depending
    64 minimization
    65 missing link
    66 monotone inclusions
    67 new primal-dual algorithm
    68 nonsmooth optimization algorithm
    69 one
    70 operator L.
    71 operators
    72 optimization
    73 optimization algorithm
    74 presence
    75 primal-dual algorithm
    76 problem
    77 processing
    78 product space
    79 random estimates
    80 rate
    81 representation
    82 results
    83 same structure
    84 second one
    85 space
    86 special case
    87 split
    88 splitting
    89 stochastic generalization
    90 strong convexity
    91 structure
    92 sublinear convergence rate
    93 sum
    94 template problem
    95 third one
    96 true gradient
    97 schema:name Dualize, Split, Randomize: Toward Fast Nonsmooth Optimization Algorithms
    98 schema:pagination 1-29
    99 schema:productId N01afbd990adb4d3888f7aefd728570a8
    100 N97b8cdb11fe14fc4a7a0dbf738c12e9d
    101 schema:sameAs https://app.dimensions.ai/details/publication/pub.1149442510
    102 https://doi.org/10.1007/s10957-022-02061-8
    103 schema:sdDatePublished 2022-09-02T16:08
    104 schema:sdLicense https://scigraph.springernature.com/explorer/license/
    105 schema:sdPublisher Nb2da416aec77441aa96cec8c61f1de3b
    106 schema:url https://doi.org/10.1007/s10957-022-02061-8
    107 sgo:license sg:explorer/license/
    108 sgo:sdDataset articles
    109 rdf:type schema:ScholarlyArticle
    110 N01afbd990adb4d3888f7aefd728570a8 schema:name doi
    111 schema:value 10.1007/s10957-022-02061-8
    112 rdf:type schema:PropertyValue
    113 N3eb6ded6da6743b9876f1448093e6547 rdf:first sg:person.01014365475.33
    114 rdf:rest Ndf82028849754f91a8d214ecd481c4bf
    115 N512ffff60be741c2bd78851ed85dca4a rdf:first sg:person.07715126012.51
    116 rdf:rest N3eb6ded6da6743b9876f1448093e6547
    117 N97b8cdb11fe14fc4a7a0dbf738c12e9d schema:name dimensions_id
    118 schema:value pub.1149442510
    119 rdf:type schema:PropertyValue
    120 Nb2da416aec77441aa96cec8c61f1de3b schema:name Springer Nature - SN SciGraph project
    121 rdf:type schema:Organization
    122 Nd909d1d95716468a844ee50378fdca97 rdf:first sg:person.016427176172.93
    123 rdf:rest rdf:nil
    124 Ndf82028849754f91a8d214ecd481c4bf rdf:first sg:person.016025561605.15
    125 rdf:rest Nd909d1d95716468a844ee50378fdca97
    126 anzsrc-for:01 schema:inDefinedTermSet anzsrc-for:
    127 schema:name Mathematical Sciences
    128 rdf:type schema:DefinedTerm
    129 anzsrc-for:0103 schema:inDefinedTermSet anzsrc-for:
    130 schema:name Numerical and Computational Mathematics
    131 rdf:type schema:DefinedTerm
    132 sg:journal.1044187 schema:issn 0022-3239
    133 1573-2878
    134 schema:name Journal of Optimization Theory and Applications
    135 schema:publisher Springer Nature
    136 rdf:type schema:Periodical
    137 sg:person.01014365475.33 schema:affiliation grid-institutes:grid.45672.32
    138 schema:familyName Condat
    139 schema:givenName Laurent
    140 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.01014365475.33
    141 rdf:type schema:Person
    142 sg:person.016025561605.15 schema:affiliation grid-institutes:grid.45672.32
    143 schema:familyName Mishchenko
    144 schema:givenName Konstantin
    145 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.016025561605.15
    146 rdf:type schema:Person
    147 sg:person.016427176172.93 schema:affiliation grid-institutes:grid.45672.32
    148 schema:familyName Richtárik
    149 schema:givenName Peter
    150 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.016427176172.93
    151 rdf:type schema:Person
    152 sg:person.07715126012.51 schema:affiliation grid-institutes:grid.45672.32
    153 schema:familyName Salim
    154 schema:givenName Adil
    155 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.07715126012.51
    156 rdf:type schema:Person
    157 sg:pub.10.1007/978-1-4419-9569-8_10 schema:sameAs https://app.dimensions.ai/details/publication/pub.1052700561
    158 https://doi.org/10.1007/978-1-4419-9569-8_10
    159 rdf:type schema:CreativeWork
    160 sg:pub.10.1007/978-1-4939-1124-0_3 schema:sameAs https://app.dimensions.ai/details/publication/pub.1037691419
    161 https://doi.org/10.1007/978-1-4939-1124-0_3
    162 rdf:type schema:CreativeWork
    163 sg:pub.10.1007/978-3-030-39568-1 schema:sameAs https://app.dimensions.ai/details/publication/pub.1127627565
    164 https://doi.org/10.1007/978-3-030-39568-1
    165 rdf:type schema:CreativeWork
    166 sg:pub.10.1007/978-3-319-48311-5 schema:sameAs https://app.dimensions.ai/details/publication/pub.1084688822
    167 https://doi.org/10.1007/978-3-319-48311-5
    168 rdf:type schema:CreativeWork
    169 sg:pub.10.1007/978-3-319-91578-4 schema:sameAs https://app.dimensions.ai/details/publication/pub.1110040882
    170 https://doi.org/10.1007/978-3-319-91578-4
    171 rdf:type schema:CreativeWork
    172 sg:pub.10.1007/bf01581204 schema:sameAs https://app.dimensions.ai/details/publication/pub.1022153870
    173 https://doi.org/10.1007/bf01581204
    174 rdf:type schema:CreativeWork
    175 sg:pub.10.1007/s10107-006-0070-8 schema:sameAs https://app.dimensions.ai/details/publication/pub.1021274319
    176 https://doi.org/10.1007/s10107-006-0070-8
    177 rdf:type schema:CreativeWork
    178 sg:pub.10.1007/s10107-015-0892-3 schema:sameAs https://app.dimensions.ai/details/publication/pub.1019991396
    179 https://doi.org/10.1007/s10107-015-0892-3
    180 rdf:type schema:CreativeWork
    181 sg:pub.10.1007/s10107-015-0957-3 schema:sameAs https://app.dimensions.ai/details/publication/pub.1012909932
    182 https://doi.org/10.1007/s10107-015-0957-3
    183 rdf:type schema:CreativeWork
    184 sg:pub.10.1007/s10107-016-1044-0 schema:sameAs https://app.dimensions.ai/details/publication/pub.1036311274
    185 https://doi.org/10.1007/s10107-016-1044-0
    186 rdf:type schema:CreativeWork
    187 sg:pub.10.1007/s10107-018-1321-1 schema:sameAs https://app.dimensions.ai/details/publication/pub.1106172667
    188 https://doi.org/10.1007/s10107-018-1321-1
    189 rdf:type schema:CreativeWork
    190 sg:pub.10.1007/s10107-019-01403-1 schema:sameAs https://app.dimensions.ai/details/publication/pub.1115172565
    191 https://doi.org/10.1007/s10107-019-01403-1
    192 rdf:type schema:CreativeWork
    193 sg:pub.10.1007/s10107-020-01565-3 schema:sameAs https://app.dimensions.ai/details/publication/pub.1131335764
    194 https://doi.org/10.1007/s10107-020-01565-3
    195 rdf:type schema:CreativeWork
    196 sg:pub.10.1007/s10444-011-9254-8 schema:sameAs https://app.dimensions.ai/details/publication/pub.1017372143
    197 https://doi.org/10.1007/s10444-011-9254-8
    198 rdf:type schema:CreativeWork
    199 sg:pub.10.1007/s10589-020-00238-3 schema:sameAs https://app.dimensions.ai/details/publication/pub.1132459987
    200 https://doi.org/10.1007/s10589-020-00238-3
    201 rdf:type schema:CreativeWork
    202 sg:pub.10.1007/s10851-010-0251-1 schema:sameAs https://app.dimensions.ai/details/publication/pub.1010318529
    203 https://doi.org/10.1007/s10851-010-0251-1
    204 rdf:type schema:CreativeWork
    205 sg:pub.10.1007/s10915-018-0680-3 schema:sameAs https://app.dimensions.ai/details/publication/pub.1101322880
    206 https://doi.org/10.1007/s10915-018-0680-3
    207 rdf:type schema:CreativeWork
    208 sg:pub.10.1007/s10957-012-0245-9 schema:sameAs https://app.dimensions.ai/details/publication/pub.1027180321
    209 https://doi.org/10.1007/s10957-012-0245-9
    210 rdf:type schema:CreativeWork
    211 sg:pub.10.1007/s11228-011-0191-y schema:sameAs https://app.dimensions.ai/details/publication/pub.1037327416
    212 https://doi.org/10.1007/s11228-011-0191-y
    213 rdf:type schema:CreativeWork
    214 sg:pub.10.1007/s11228-017-0421-z schema:sameAs https://app.dimensions.ai/details/publication/pub.1086031536
    215 https://doi.org/10.1007/s11228-017-0421-z
    216 rdf:type schema:CreativeWork
    217 grid-institutes:grid.45672.32 schema:alternateName Computer Science Program, Computer, Electrical and Mathematical Science and Engineering Division, King Abdullah University of Science and Technology (KAUST), 23955-6900, Thuwal, Kingdom of Saudi Arabia
    218 schema:name Computer Science Program, Computer, Electrical and Mathematical Science and Engineering Division, King Abdullah University of Science and Technology (KAUST), 23955-6900, Thuwal, Kingdom of Saudi Arabia
    219 rdf:type schema:Organization
     




    Preview window. Press ESC to close (or click here)


    ...