Proximal Gradient Method for Solving Bilevel Optimization Problems

In this paper, we consider a bilevel optimization problem as a task of finding the optimum of the upper-level problem subject to the solution set of the split feasibility problem of fixed point problems and optimization problems. Based on proximal and gradient methods, we propose a strongly converge...

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Published inMathematical and computational applications Vol. 25; no. 4; p. 66
Main Authors Yimer, Seifu Endris, Kumam, Poom, Gebrie, Anteneh Getachew
Format Journal Article
LanguageEnglish
Published MDPI AG 04.10.2020
Subjects
bilevel problem
fixed point problem
gradient method
proximal method
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Abstract In this paper, we consider a bilevel optimization problem as a task of finding the optimum of the upper-level problem subject to the solution set of the split feasibility problem of fixed point problems and optimization problems. Based on proximal and gradient methods, we propose a strongly convergent iterative algorithm with an inertia effect solving the bilevel optimization problem under our consideration. Furthermore, we present a numerical example of our algorithm to illustrate its applicability.
AbstractList In this paper, we consider a bilevel optimization problem as a task of finding the optimum of the upper-level problem subject to the solution set of the split feasibility problem of fixed point problems and optimization problems. Based on proximal and gradient methods, we propose a strongly convergent iterative algorithm with an inertia effect solving the bilevel optimization problem under our consideration. Furthermore, we present a numerical example of our algorithm to illustrate its applicability.
Author Yimer, Seifu Endris
Gebrie, Anteneh Getachew
Kumam, Poom
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10.1088/0266-5611/18/2/310
10.1080/02331934.2015.1101599
10.1007/s10589-007-9066-4
10.1007/978-1-4419-9467-7
10.1137/16M105592X
10.1007/s10013-017-0256-9
10.1080/02331934.2015.1020943
10.1186/s13660-018-1898-1
10.3390/math7090841
10.1007/0-387-34221-4_2
10.1007/978-3-662-45827-3
10.1137/0314056
10.1088/0266-5611/28/8/085004
10.1002/aic.690420814
10.1007/BF02142692
10.1007/s11075-018-0583-2
10.1137/09076773X
10.1016/j.cor.2010.05.007
10.1112/S0024610702003332
10.22436/jnsa.011.02.03
10.1007/s11228-008-0102-z
10.1007/s11075-011-9490-5
10.1007/978-3-642-59073-3_11
10.22436/jnsa.009.04.10
10.1137/S105262340343467X
10.1016/j.na.2011.05.005
10.1016/0041-5553(64)90137-5
10.1080/10556788.2019.1619729
10.1007/s10957-011-9837-z
10.1088/1361-6420/aa6136
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References Censor (ref_9) 2012; 59
Combettes (ref_37) 2005; 6
(ref_36) 2008; 16
Anh (ref_18) 2016; 65
Anh (ref_17) 2019; 81
ref_34
Fampa (ref_27) 2008; 39
Xu (ref_35) 2002; 66
ref_19
Polyak (ref_33) 1964; 4
Wang (ref_14) 2012; 28
Martinet (ref_6) 1970; 4
ref_16
ref_38
ref_15
Chang (ref_24) 2016; 9
Qu (ref_12) 2005; 21
Rockafellar (ref_8) 1976; 14
Censor (ref_11) 1994; 8
Csetnek (ref_22) 2018; 46
Su (ref_1) 2010; 1
Byrne (ref_13) 2002; 18
Helou (ref_32) 2017; 33
Takahashi (ref_29) 2016; 65
Ceng (ref_3) 2011; 74
Solodov (ref_23) 2007; 14
ref_25
Mohideen (ref_26) 1996; 42
ref_20
Xu (ref_2) 2011; 150
Sabach (ref_21) 2017; 27
Hendrickx (ref_10) 2010; 31
Cabot (ref_31) 2005; 15
Martinet (ref_7) 1972; 274
ref_5
ref_4
Song (ref_30) 2018; 11
Calvete (ref_28) 2011; 38
References_xml – volume: 21
  start-page: 1655
  year: 2005
  ident: ref_12
  article-title: A note on the CQ algorithm for the split feasibility problem
  publication-title: Inverse Probl.
  doi: 10.1088/0266-5611/21/5/009
– volume: 18
  start-page: 441
  year: 2002
  ident: ref_13
  article-title: Iterative oblique projection onto convex sets and the split feasibility problem
  publication-title: Inverse Probl.
  doi: 10.1088/0266-5611/18/2/310
– volume: 65
  start-page: 1229
  year: 2016
  ident: ref_18
  article-title: A projection-fixed point method for a class of bilevel variational inequalities with split fixed point constraints
  publication-title: Optimization
  doi: 10.1080/02331934.2015.1101599
– volume: 39
  start-page: 121
  year: 2008
  ident: ref_27
  article-title: Bilevel optimization applied to strategic pricing in competitive electricity markets
  publication-title: Comput. Optim. Appl.
  doi: 10.1007/s10589-007-9066-4
– ident: ref_5
  doi: 10.1007/978-1-4419-9467-7
– volume: 27
  start-page: 640
  year: 2017
  ident: ref_21
  article-title: A first order method for solving convex bilevel optimization problems
  publication-title: SIAM J. Optimiz.
  doi: 10.1137/16M105592X
– volume: 46
  start-page: 53
  year: 2018
  ident: ref_22
  article-title: An inertial proximal-gradient penalization scheme for constrained convex optimization problems
  publication-title: Vietnam J. Math.
  doi: 10.1007/s10013-017-0256-9
– volume: 65
  start-page: 281
  year: 2016
  ident: ref_29
  article-title: The split common null point problem and the shrinking projection method in Banach spaces
  publication-title: Optimization
  doi: 10.1080/02331934.2015.1020943
– volume: 14
  start-page: 227
  year: 2007
  ident: ref_23
  article-title: An explicit descent method for bilevel convex optimization
  publication-title: J. Convex Anal.
– volume: 274
  start-page: 163
  year: 1972
  ident: ref_7
  article-title: Détermination approchée d’un point fixe d’une application pseudo-contractante
  publication-title: CR Acad. Sci. Paris
– ident: ref_19
  doi: 10.1186/s13660-018-1898-1
– ident: ref_15
  doi: 10.3390/math7090841
– ident: ref_25
  doi: 10.1007/0-387-34221-4_2
– ident: ref_16
  doi: 10.1007/978-3-662-45827-3
– volume: 14
  start-page: 877
  year: 1976
  ident: ref_8
  article-title: Monotone operators and the proximal point algorithm
  publication-title: SIAM J. Control Optim.
  doi: 10.1137/0314056
– volume: 28
  start-page: 085004
  year: 2012
  ident: ref_14
  article-title: Solving the split feasibility problem without prior knowledge of matrix norms
  publication-title: Inverse Probl.
  doi: 10.1088/0266-5611/28/8/085004
– volume: 42
  start-page: 2251
  year: 1996
  ident: ref_26
  article-title: Optimal design of dynamic systems under uncertainty
  publication-title: AIChE J.
  doi: 10.1002/aic.690420814
– volume: 8
  start-page: 221
  year: 1994
  ident: ref_11
  article-title: A multiprojection algorithm using Bregman projections in a product space
  publication-title: Numer. Algorithms
  doi: 10.1007/BF02142692
– volume: 81
  start-page: 1067
  year: 2019
  ident: ref_17
  article-title: Linesearch methods for bilevel split pseudomonotone variational inequality problems
  publication-title: Numer. Algorithms
  doi: 10.1007/s11075-018-0583-2
– volume: 31
  start-page: 2802
  year: 2010
  ident: ref_10
  article-title: Matrix p-Norms Are NP-Hard to Approximate If p ≠ 1,2,∞
  publication-title: SIAM J. Matrix Anal. A
  doi: 10.1137/09076773X
– volume: 38
  start-page: 320
  year: 2011
  ident: ref_28
  article-title: Bilevel model for production–distribution planning solved by using ant colony optimization
  publication-title: Comput. Oper. Res.
  doi: 10.1016/j.cor.2010.05.007
– ident: ref_4
– volume: 66
  start-page: 240
  year: 2002
  ident: ref_35
  article-title: Iterative algorithms for nonlinear operators
  publication-title: J. Lond. Math. Soc.
  doi: 10.1112/S0024610702003332
– volume: 11
  start-page: 198
  year: 2018
  ident: ref_30
  article-title: Iterative methods for fixed point problems and generalized split feasibility problems in Banach spaces
  publication-title: J. Nonlinear Sci. Appl.
  doi: 10.22436/jnsa.011.02.03
– volume: 16
  start-page: 899
  year: 2008
  ident: ref_36
  article-title: Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization
  publication-title: Set Valued Anal.
  doi: 10.1007/s11228-008-0102-z
– volume: 59
  start-page: 301
  year: 2012
  ident: ref_9
  article-title: Algorithms for the split variational inequality problem
  publication-title: Numer. Algorithms
  doi: 10.1007/s11075-011-9490-5
– volume: 6
  start-page: 117
  year: 2005
  ident: ref_37
  article-title: Equilibrium programming in Hilbert spaces
  publication-title: J. Nonlinear Convex Anal.
– ident: ref_34
  doi: 10.1007/978-3-642-59073-3_11
– volume: 4
  start-page: 154
  year: 1970
  ident: ref_6
  article-title: Brève communication. Régularisation d’inéquations variationnelles par approximations successives
  publication-title: Revue Française d’Informatique et de Recherche Opérationnelle Série Rouge
– ident: ref_38
– volume: 9
  start-page: 1515
  year: 2016
  ident: ref_24
  article-title: Feasible iterative algorithms and strong convergence theorems for bi-level fixed point problems
  publication-title: J. Nonlinear Sci. Appl.
  doi: 10.22436/jnsa.009.04.10
– volume: 1
  start-page: 35
  year: 2010
  ident: ref_1
  article-title: Remarks on the gradient-projection algorithm
  publication-title: J. Nonlinear Anal. Optim.
– volume: 15
  start-page: 555
  year: 2005
  ident: ref_31
  article-title: Proximal point algorithm controlled by a slowly vanishing term: Applications to hierarchical minimization
  publication-title: SIAM J. Optim.
  doi: 10.1137/S105262340343467X
– volume: 74
  start-page: 5286
  year: 2011
  ident: ref_3
  article-title: Some iterative methods for finding fixed points and for solving constrained convex minimization problems
  publication-title: Nonlinear Anal. Theor.
  doi: 10.1016/j.na.2011.05.005
– volume: 4
  start-page: 1
  year: 1964
  ident: ref_33
  article-title: Some methods of speeding up the convergence of iteration methods
  publication-title: USSR Comput. Math. Math. Phys.
  doi: 10.1016/0041-5553(64)90137-5
– ident: ref_20
  doi: 10.1080/10556788.2019.1619729
– volume: 150
  start-page: 360
  year: 2011
  ident: ref_2
  article-title: Averaged mappings and the gradient-projection algorithm
  publication-title: J. Optim. Theory Appl.
  doi: 10.1007/s10957-011-9837-z
– volume: 33
  start-page: 5
  year: 2017
  ident: ref_32
  article-title: ϵ-subgradient algorithms for bilevel convex optimization
  publication-title: Inverse Probl.
  doi: 10.1088/1361-6420/aa6136
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StartPage 66
SubjectTerms bilevel problem
fixed point problem
gradient method
proximal method
Title Proximal Gradient Method for Solving Bilevel Optimization Problems
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