A difference-of-convex functions approach for sparse PDE optimal control problems with nonconvex costs
We propose a local regularization of elliptic optimal control problems which involves the nonconvex L q quasi-norm penalization in the cost function. The proposed Huber type regularization allows us to formulate the PDE constrained optimization instance as a DC programming problem (difference of con...
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| Vydáno v: | Computational optimization and applications Ročník 74; číslo 1; s. 225 - 258 |
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| Médium: | Journal Article |
| Jazyk: | angličtina |
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01.09.2019
Springer Nature B.V |
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| ISSN: | 0926-6003, 1573-2894 |
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| Abstract | We propose a local regularization of elliptic optimal control problems which involves the nonconvex
L
q
quasi-norm penalization in the cost function. The proposed
Huber type
regularization allows us to formulate the PDE constrained optimization instance as a DC programming problem (difference of convex functions) that is useful to obtain necessary optimality conditions and tackle its numerical solution by applying the well known DC algorithm used in nonconvex optimization problems. By this procedure we approximate the original problem in terms of a consistent family of parameterized nonsmooth problems for which there are efficient numerical methods available. Finally, we present numerical experiments to illustrate our theory with different configurations associated to the parameters of the problem. |
|---|---|
| AbstractList | We propose a local regularization of elliptic optimal control problems which involves the nonconvex
L
q
quasi-norm penalization in the cost function. The proposed
Huber type
regularization allows us to formulate the PDE constrained optimization instance as a DC programming problem (difference of convex functions) that is useful to obtain necessary optimality conditions and tackle its numerical solution by applying the well known DC algorithm used in nonconvex optimization problems. By this procedure we approximate the original problem in terms of a consistent family of parameterized nonsmooth problems for which there are efficient numerical methods available. Finally, we present numerical experiments to illustrate our theory with different configurations associated to the parameters of the problem. We propose a local regularization of elliptic optimal control problems which involves the nonconvex \[L^q\] quasi-norm penalization in the cost function. The proposed Huber type regularization allows us to formulate the PDE constrained optimization instance as a DC programming problem (difference of convex functions) that is useful to obtain necessary optimality conditions and tackle its numerical solution by applying the well known DC algorithm used in nonconvex optimization problems. By this procedure we approximate the original problem in terms of a consistent family of parameterized nonsmooth problems for which there are efficient numerical methods available. Finally, we present numerical experiments to illustrate our theory with different configurations associated to the parameters of the problem. |
| Author | Merino, Pedro |
| Author_xml | – sequence: 1 givenname: Pedro orcidid: 0000-0002-8178-8834 surname: Merino fullname: Merino, Pedro email: pedro.merino@epn.edu.ec organization: Research Center of Mathematical Modeling (MODEMAT) and Department of Mathematics, Escuela Politécnica Nacional |
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| Cites_doi | 10.1007/s10589-007-9150-9 10.1137/120872395 10.1051/cocv/2015008 10.1007/s10589-017-9891-z 10.3934/mcrf.2017014 10.1016/0022-0396(83)90085-2 10.1007/978-1-4757-6019-4_13 10.1080/01630568908816302 10.1007/978-0-8176-4948-7 10.1023/A:1017535424813 10.1007/s40324-017-0121-5 10.1137/110854746 10.1137/120896529 10.1007/978-3-319-13395-9_3 |
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| References | Flores-BazánFOettliWSimplified optimality conditions for minimizing the difference of vector-valued functionsJ. Optim. Theory Appl.20011083571586182867310.1023/A:10175354248130984.90053 Hiriart-UrrutyJ-BClarkeFHDem’yanovVFGiannessiFFrom convex optimization to nonconvex optimization. Necessary and sufficient conditions for global optimalityNonsmooth Optimization and Related Topics1989BerlinSpringer21923910.1007/978-1-4757-6019-4_13 Hiriart-UrrutyJ-BLemaréchalCFundamentals of Convex Analysis2012BerlinSpringer0998.49001 Dellacherie, C., Meyer, P.-A.: Probabilities and Potential. North Holland & Hermann, Mathematical Studies, vol. 29 (1975) CasasEA review on sparse solutions in optimal control of partial differential equationsSeMA J.2017743319344374594210.1007/s40324-017-0121-51391.35124 CasasEMateosMRoschAFinite element approximation of sparse parabolic control problemsAm. Inst. Math. Sci.20177339341736696631369.49006 WrightSNowozinSSraSOptimization for Machine Learning2012CambridgeMIT Press De Los ReyesJCLoayzaEMerinoPSecond-order orthan–based methods with enriched hessian information for sparse ℓ1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _1$$\end{document}-optimizationComput. Optim. Appl.2017672225258363657810.1007/s10589-017-9891-z1368.90183 AuchmutyGDuality for non-convex variational principlesJ. Differ. Equ.1983508014571786910.1016/0022-0396(83)90085-20533.49007 RamlauRZarzerCAOn the minimization of a Tikhonov functional with a non-convex sparsity constraintElectron. Trans. Numer. Anal.20123947650729993251287.65044 Ioffe, A.D., Tihomirov, V.M., Luderer, B.: Theorie der Extremalaufgaben. VEB Deutscher Verlag der Wissenschaften (1979) HinzeMPinnauRUlbrichMUlbrichSOptimization with PDE Constraints2008BerlinSpringer1167.49001 De los Reyes, J.C.: Theory of PDE-constrained optimization. In: Pardalos, P.M., Pintér, J.D., Robinson, S., Terlaky, T., Thai, M.T. (eds.) Numerical PDE-Constrained Optimization, pp. 25–41. Springer, Berlin (2015) HintermüllerMTaoWNonconvex TVq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$TV^{q}$$\end{document}-models in image restoration: analysis and a trust-region regularization-based superlinearly convergent solverSIAM J. Imaging Sci.20136313851415308099510.1137/1108547461281.65033 CasasEClasonCKunischKParabolic control problems in measure spaces with sparse solutionsSIAM J. Control Optim.20135112863303286610.1137/1208723951266.49037 ItoKKunischKLagrange multiplier approach to variational problems and applicationsSIAM Adv. Des. Control20145221251127510.1137/120896529 Ito, K., Kunisch, K.: Optimal control with Lp(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathit{L}^{p}({\Omega })$$\end{document}, p∈[0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p \in [0,1)$$\end{document}, control cost. SIAM J. Control Optim. (2008 RockafellarRTConvex Analysis2015PrincetonPrinceton University Press DinhTPLe ThiHANguyenNTLe ThiHARecent advances in DC programming and DCATransactions on Computational Intelligence XIII2014BerlinSpringer137 Dal MasoGAn Introduction to Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document}-Convergence2012BerlinSpringer AuchmutyGDuality algorithms for nonconvex variational principlesNumer. Func. Anal. Optim.1989103–421126498953410.1080/016305689088163020646.49023 CiarletPGLinear and nonlinear functional analysis with applicationsSIAM2013130472 StadlerGElliptic optimal control problems with L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^1$$\end{document}-control cost and applications for the placement of control devicesComput. Optim. Appl.2009442159181255684910.1007/s10589-007-9150-91185.49031 JahnJIntroduction to the Theory of Nonlinear Optimization20073BerlinSpringer1115.49001 AmbrosettiAProdiGA Primer of Nonlinear Analysis1995CambridgeCambridge University Press0818.47059 CasasEKunischKParabolic control problems in space-time measure spacesESAIM Control Optim. Calc. Var.2016222355370349177410.1051/cocv/20150081343.49036 FoucartSRauhutHA Mathematical Introduction to Compressive Sensing2013BaselBirkhäuser10.1007/978-0-8176-4948-71315.94002 101_CR12 RT Rockafellar (101_CR25) 2015 E Casas (101_CR5) 2016; 22 JC De Los Reyes (101_CR11) 2017; 67 K Ito (101_CR21) 2014; 52 101_CR10 R Ramlau (101_CR24) 2012; 39 S Wright (101_CR27) 2012 E Casas (101_CR6) 2017; 74 J-B Hiriart-Urruty (101_CR18) 1989 G Dal Maso (101_CR9) 2012 M Hintermüller (101_CR16) 2013; 6 TP Dinh (101_CR13) 2014 G Auchmuty (101_CR2) 1989; 10 M Hinze (101_CR17) 2008 G Auchmuty (101_CR3) 1983; 50 101_CR22 J Jahn (101_CR23) 2007 J-B Hiriart-Urruty (101_CR19) 2012 A Ambrosetti (101_CR1) 1995 PG Ciarlet (101_CR8) 2013; 130 101_CR20 E Casas (101_CR4) 2013; 51 G Stadler (101_CR26) 2009; 44 E Casas (101_CR7) 2017; 7 S Foucart (101_CR15) 2013 F Flores-Bazán (101_CR14) 2001; 108 |
| References_xml | – reference: CasasEMateosMRoschAFinite element approximation of sparse parabolic control problemsAm. Inst. Math. Sci.20177339341736696631369.49006 – reference: Ioffe, A.D., Tihomirov, V.M., Luderer, B.: Theorie der Extremalaufgaben. VEB Deutscher Verlag der Wissenschaften (1979) – reference: HinzeMPinnauRUlbrichMUlbrichSOptimization with PDE Constraints2008BerlinSpringer1167.49001 – reference: CasasEA review on sparse solutions in optimal control of partial differential equationsSeMA J.2017743319344374594210.1007/s40324-017-0121-51391.35124 – reference: Ito, K., Kunisch, K.: Optimal control with Lp(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathit{L}^{p}({\Omega })$$\end{document}, p∈[0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p \in [0,1)$$\end{document}, control cost. SIAM J. Control Optim. (2008) – reference: AmbrosettiAProdiGA Primer of Nonlinear Analysis1995CambridgeCambridge University Press0818.47059 – reference: JahnJIntroduction to the Theory of Nonlinear Optimization20073BerlinSpringer1115.49001 – reference: FoucartSRauhutHA Mathematical Introduction to Compressive Sensing2013BaselBirkhäuser10.1007/978-0-8176-4948-71315.94002 – reference: WrightSNowozinSSraSOptimization for Machine Learning2012CambridgeMIT Press – reference: Hiriart-UrrutyJ-BLemaréchalCFundamentals of Convex Analysis2012BerlinSpringer0998.49001 – reference: CasasEKunischKParabolic control problems in space-time measure spacesESAIM Control Optim. Calc. Var.2016222355370349177410.1051/cocv/20150081343.49036 – reference: RamlauRZarzerCAOn the minimization of a Tikhonov functional with a non-convex sparsity constraintElectron. Trans. Numer. Anal.20123947650729993251287.65044 – reference: DinhTPLe ThiHANguyenNTLe ThiHARecent advances in DC programming and DCATransactions on Computational Intelligence XIII2014BerlinSpringer137 – reference: Dellacherie, C., Meyer, P.-A.: Probabilities and Potential. North Holland & Hermann, Mathematical Studies, vol. 29 (1975) – reference: ItoKKunischKLagrange multiplier approach to variational problems and applicationsSIAM Adv. Des. Control20145221251127510.1137/120896529 – reference: CiarletPGLinear and nonlinear functional analysis with applicationsSIAM2013130472 – reference: De los Reyes, J.C.: Theory of PDE-constrained optimization. In: Pardalos, P.M., Pintér, J.D., Robinson, S., Terlaky, T., Thai, M.T. (eds.) Numerical PDE-Constrained Optimization, pp. 25–41. Springer, Berlin (2015) – reference: Hiriart-UrrutyJ-BClarkeFHDem’yanovVFGiannessiFFrom convex optimization to nonconvex optimization. Necessary and sufficient conditions for global optimalityNonsmooth Optimization and Related Topics1989BerlinSpringer21923910.1007/978-1-4757-6019-4_13 – reference: De Los ReyesJCLoayzaEMerinoPSecond-order orthan–based methods with enriched hessian information for sparse ℓ1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _1$$\end{document}-optimizationComput. Optim. Appl.2017672225258363657810.1007/s10589-017-9891-z1368.90183 – reference: Flores-BazánFOettliWSimplified optimality conditions for minimizing the difference of vector-valued functionsJ. Optim. Theory Appl.20011083571586182867310.1023/A:10175354248130984.90053 – reference: HintermüllerMTaoWNonconvex TVq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$TV^{q}$$\end{document}-models in image restoration: analysis and a trust-region regularization-based superlinearly convergent solverSIAM J. Imaging Sci.20136313851415308099510.1137/1108547461281.65033 – reference: RockafellarRTConvex Analysis2015PrincetonPrinceton University Press – reference: AuchmutyGDuality algorithms for nonconvex variational principlesNumer. Func. Anal. Optim.1989103–421126498953410.1080/016305689088163020646.49023 – reference: AuchmutyGDuality for non-convex variational principlesJ. Differ. Equ.1983508014571786910.1016/0022-0396(83)90085-20533.49007 – reference: StadlerGElliptic optimal control problems with L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^1$$\end{document}-control cost and applications for the placement of control devicesComput. Optim. Appl.2009442159181255684910.1007/s10589-007-9150-91185.49031 – reference: Dal MasoGAn Introduction to Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document}-Convergence2012BerlinSpringer – reference: CasasEClasonCKunischKParabolic control problems in measure spaces with sparse solutionsSIAM J. 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| Snippet | We propose a local regularization of elliptic optimal control problems which involves the nonconvex
L
q
quasi-norm penalization in the cost function. The... We propose a local regularization of elliptic optimal control problems which involves the nonconvex \[L^q\] quasi-norm penalization in the cost function. The... |
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| SubjectTerms | Algorithms Convex analysis Convex and Discrete Geometry Economic models Management Science Mathematics Mathematics and Statistics Nonlinear programming Numerical methods Operations Research Operations Research/Decision Theory Optimal control Optimization Regularization Statistics |
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| Title | A difference-of-convex functions approach for sparse PDE optimal control problems with nonconvex costs |
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