May  2021, 20(5): 1893-1906. doi: 10.3934/cpaa.2021051

Forward-backward approximation of nonlinear semigroups in finite and infinite horizon

1. 

University of Paris-Saclay, CentraleSuplec, OPIS, Inria Saclay 91190, Gif-sur-Yvette, France

2. 

Departamento de Ingeniería Matemática & , Centro de Modelamiento Matemático (CNRS IRL2807)

3. 

FCFM, Universidad de Chile, Beauchef 851, Santiago, Chile

4. 

Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence, University of Groningen, Bernoulliborg, Nijenborgh 9, 9747 AG Groningen, The Netherlands

* Corresponding author: Juan Peypouquet

Received  August 2020 Revised  January 2021 Published  May 2021 Early access  March 2021

Fund Project: Supported by FONDECYT Grant 1181179, CMM-Conicyt PIA AFB170001, ECOS-CONICYT Grant C18E04 and CONICYT-PFCHA/DOCTORADO NACIONAL/2016 21160994

This work is concerned with evolution equations and their forward-backward discretizations, and aims at building bridges between differential equations and variational analysis. Our first contribution is an estimation for the distance between iterates of sequences generated by forward-backward schemes, useful in the convergence and robustness analysis of iterative algorithms of widespread use in numerical optimization and variational inequalities. Our second contribution is the approximation, on a bounded time frame, of the solutions of evolution equations governed by accretive (monotone) operators with an additive structure, by trajectories constructed by interpolating forward-backward sequences. This provides a short, simple and self-contained proof of existence and regularity for such solutions; unifies and extends a number of classical results; and offers a guide for the development of numerical methods. Finally, our third contribution is a mathematical methodology that allows us to deduce the behavior, as the number of iterations tends to $ +\infty $, of sequences generated by forward-backward algorithms, based solely on the knowledge of the behavior, as time goes to $ +\infty $, of the solutions of differential inclusions, and viceversa.

Citation: Andrés Contreras, Juan Peypouquet. Forward-backward approximation of nonlinear semigroups in finite and infinite horizon. Communications on Pure and Applied Analysis, 2021, 20 (5) : 1893-1906. doi: 10.3934/cpaa.2021051
References:
[1]

F. Álvarez and J. Peypouquet, Asymptotic equivalence and Kobayashi-type estimates for nonautonomous monotone operators in Banach spaces, Discrete Contin. Dyn. Syst., 25 (2009), no. 4, 1109–1128. doi: 10.3934/dcds.2009.25.1109.

[2]

F. Álvarez and J. Peypouquet, Asymptotic almost-equivalence of Lipschitz evolution systems in Banach spaces, Nonlinear Anal., 73 (2010), no. 9, 3018–3033 doi: 10.1016/j.na.2010.06.070.

[3]

F. Álvarez and J. Peypouquet, A unified approach to the asymptotic almost-equivalence of evolution systems without Lipschitz conditions, Nonlinear Anal., 74 (2011), no. 11, 3440–3444 doi: 10.1016/j.na.2011.02.030.

[4]

J. B. Baillon, Un exemple concernant le comportement asymptotique de la solution du problème ${du/dt}+{\partial}\varphi(u){\ni}0$, J. Functional Analysis, 28 (1978), 369-376.  doi: 10.1016/0022-1236(78)90093-9.

[5]

P. Bénilan, Équations d'Évolution san un Espace de Banach Quelconque et Applications, Thése, Orsay, 1972.

[6]

H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Publishing Co., Amsterdam-London, American Elsevier Publishing Co., Inc., New York, 1973.

[7]

M. G. Crandall and A. Pazy, Semi-groups of nonlinear contractions and dissipative sets, J. Functional Analysis, 3 (1969), 376-418.  doi: 10.1016/0022-1236(69)90032-9.

[8]

A. Contreras and J. Peypouquet, Asymptotic equivalence of evolution equations governed by cocoercive operators and their forward discretizations, J. Optim. Theory Appl., 182 (2019), no. 1, 30–48. doi: 10.1007/s10957-018-1332-3.

[9]

O. Güler, On the convergence of the proximal point algorithm for convex minimization, SIAM J. Control Opt., 29 (1991), 403-419.  doi: 10.1137/0329022.

[10]

E. Hille, On the generation of semi-groups and the theory of conjugate functions. Kungl. Fysiografiska Sällskapets i Lund Förhandlingar, Proc. Roy. Physiog. Soc. Lund., 21, (1952). no. 14, 13 pp.

[11]

T. Kato, Nonlinear semigroups and evolution equations, J. Math Soc. Japan, 19 (1973), 508-520.  doi: 10.2969/jmsj/01940508.

[12]

T. Kato, On the Trotter-Lie product formula, Proc. Japan Acad., 50 (1974), 694-698.  doi: 10.3792/pja/1195518790.

[13]

Y. Kobayashi, Difference approximation of Cauchy problems for quasi-dissipative operators and generation of nonlinear semigroups, J. Math. Soc. Japan, 27 (1975), 640-665.  doi: 10.2969/jmsj/02740640.

[14]

K. KobayasiY. Kobayashi and S. Oharu, Nonlinear evolution operators in Banach spaces, Osaka J. Math., 21 (1984), 281-310. 

[15]

G. Lumer and R. S. Phillips, Dissipative operators in a Banach space, Pacific J. Math., 11 (1961), 679-698.  doi: 10.2140/pjm.1961.11.679.

[16]

B. Martinet, Régularisation d'inéquations variationnelles par approximations successives, Rev. Française Informat. Recherche Opérationnelle, 4 (1970), 154–158.

[17]

I. Miyadera and K. Kobayasi, On the asymptotic behavior of almost-orbits of nonlinear contractions in Banach spaces, Nonlinear Anal., 6 (1982), 349-365.  doi: 10.1016/0362-546X(82)90021-9.

[18]

O. Nevanlinna and S. Reich, Strong convergence of contraction semigroups and of iterative methods for accretive operators in Banach spaces, Israel J. Math., 32 (1979), 44-58.  doi: 10.1007/BF02761184.

[19]

G. B. Passty, Preservation of the asymptotic behavior of a nonlinear contraction semigroup by backward differencing, Houston J. Math., 7 (1981), 103-110. 

[20]

J. Peypouquet, Convex Optimization in Normed Spaces. Theory, Methods and Examples, Springer, Cham, 2015. doi: 10.1007/978-3-319-13710-0.

[21]

J. Peypouquet and S. Sorin, Evolution equations for maximal monotone operators: Asymptotic analysis in continuous and discrete time, J. Convex Anal., 17 (2010), 1113-1163. 

[22]

S. Rasmussen, Nonlinear semigroups, evolution equations and product integral representations, Various Publication Series, Vol. 20, Aarhus Universitet, (1971/72).

[23]

R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), 877-898.  doi: 10.1137/0314056.

[24]

H. F. Trotter, On the product of semi-groups of operators, Proc. Amer. Math. Soc., 10 (1959), 545-551.  doi: 10.1090/S0002-9939-1959-0108732-6.

[25]

P. Tseng, A modified forward-backward splitting method for maximal monotone mappings, SIAM J. Control Optim., 38 (2000), 431-446.  doi: 10.1137/S0363012998338806.

[26]

T. Sugimoto and M. Koizumi, On the asymptotic behaviour of a nonlinear contraction semigroup and the resolvente iteration, Proc. Japan Acad. Ser. A. Math. Sci., 59 (1983), no. 6,238–240. doi: 10.3792/pjaa.59.238.

[27]

G. Vigeral, Evolution equations in discrete and continuous time for nonexpansive opreators in Banach spaces, ESAIM, Control Optim. Calc. Var., 16 (2010), 809-832.  doi: 10.1051/cocv/2009026.

[28]

H. K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal., 16 (1991), 1127-1138.  doi: 10.1016/0362-546X(91)90200-K.

[29]

K. Yosida, On the differentiability and the representation of one-parameter semi-group of linear operators, J. Math. Soc. Japan, 1 (1948), 15-21.  doi: 10.2969/jmsj/00110015.

show all references

References:
[1]

F. Álvarez and J. Peypouquet, Asymptotic equivalence and Kobayashi-type estimates for nonautonomous monotone operators in Banach spaces, Discrete Contin. Dyn. Syst., 25 (2009), no. 4, 1109–1128. doi: 10.3934/dcds.2009.25.1109.

[2]

F. Álvarez and J. Peypouquet, Asymptotic almost-equivalence of Lipschitz evolution systems in Banach spaces, Nonlinear Anal., 73 (2010), no. 9, 3018–3033 doi: 10.1016/j.na.2010.06.070.

[3]

F. Álvarez and J. Peypouquet, A unified approach to the asymptotic almost-equivalence of evolution systems without Lipschitz conditions, Nonlinear Anal., 74 (2011), no. 11, 3440–3444 doi: 10.1016/j.na.2011.02.030.

[4]

J. B. Baillon, Un exemple concernant le comportement asymptotique de la solution du problème ${du/dt}+{\partial}\varphi(u){\ni}0$, J. Functional Analysis, 28 (1978), 369-376.  doi: 10.1016/0022-1236(78)90093-9.

[5]

P. Bénilan, Équations d'Évolution san un Espace de Banach Quelconque et Applications, Thése, Orsay, 1972.

[6]

H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Publishing Co., Amsterdam-London, American Elsevier Publishing Co., Inc., New York, 1973.

[7]

M. G. Crandall and A. Pazy, Semi-groups of nonlinear contractions and dissipative sets, J. Functional Analysis, 3 (1969), 376-418.  doi: 10.1016/0022-1236(69)90032-9.

[8]

A. Contreras and J. Peypouquet, Asymptotic equivalence of evolution equations governed by cocoercive operators and their forward discretizations, J. Optim. Theory Appl., 182 (2019), no. 1, 30–48. doi: 10.1007/s10957-018-1332-3.

[9]

O. Güler, On the convergence of the proximal point algorithm for convex minimization, SIAM J. Control Opt., 29 (1991), 403-419.  doi: 10.1137/0329022.

[10]

E. Hille, On the generation of semi-groups and the theory of conjugate functions. Kungl. Fysiografiska Sällskapets i Lund Förhandlingar, Proc. Roy. Physiog. Soc. Lund., 21, (1952). no. 14, 13 pp.

[11]

T. Kato, Nonlinear semigroups and evolution equations, J. Math Soc. Japan, 19 (1973), 508-520.  doi: 10.2969/jmsj/01940508.

[12]

T. Kato, On the Trotter-Lie product formula, Proc. Japan Acad., 50 (1974), 694-698.  doi: 10.3792/pja/1195518790.

[13]

Y. Kobayashi, Difference approximation of Cauchy problems for quasi-dissipative operators and generation of nonlinear semigroups, J. Math. Soc. Japan, 27 (1975), 640-665.  doi: 10.2969/jmsj/02740640.

[14]

K. KobayasiY. Kobayashi and S. Oharu, Nonlinear evolution operators in Banach spaces, Osaka J. Math., 21 (1984), 281-310. 

[15]

G. Lumer and R. S. Phillips, Dissipative operators in a Banach space, Pacific J. Math., 11 (1961), 679-698.  doi: 10.2140/pjm.1961.11.679.

[16]

B. Martinet, Régularisation d'inéquations variationnelles par approximations successives, Rev. Française Informat. Recherche Opérationnelle, 4 (1970), 154–158.

[17]

I. Miyadera and K. Kobayasi, On the asymptotic behavior of almost-orbits of nonlinear contractions in Banach spaces, Nonlinear Anal., 6 (1982), 349-365.  doi: 10.1016/0362-546X(82)90021-9.

[18]

O. Nevanlinna and S. Reich, Strong convergence of contraction semigroups and of iterative methods for accretive operators in Banach spaces, Israel J. Math., 32 (1979), 44-58.  doi: 10.1007/BF02761184.

[19]

G. B. Passty, Preservation of the asymptotic behavior of a nonlinear contraction semigroup by backward differencing, Houston J. Math., 7 (1981), 103-110. 

[20]

J. Peypouquet, Convex Optimization in Normed Spaces. Theory, Methods and Examples, Springer, Cham, 2015. doi: 10.1007/978-3-319-13710-0.

[21]

J. Peypouquet and S. Sorin, Evolution equations for maximal monotone operators: Asymptotic analysis in continuous and discrete time, J. Convex Anal., 17 (2010), 1113-1163. 

[22]

S. Rasmussen, Nonlinear semigroups, evolution equations and product integral representations, Various Publication Series, Vol. 20, Aarhus Universitet, (1971/72).

[23]

R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), 877-898.  doi: 10.1137/0314056.

[24]

H. F. Trotter, On the product of semi-groups of operators, Proc. Amer. Math. Soc., 10 (1959), 545-551.  doi: 10.1090/S0002-9939-1959-0108732-6.

[25]

P. Tseng, A modified forward-backward splitting method for maximal monotone mappings, SIAM J. Control Optim., 38 (2000), 431-446.  doi: 10.1137/S0363012998338806.

[26]

T. Sugimoto and M. Koizumi, On the asymptotic behaviour of a nonlinear contraction semigroup and the resolvente iteration, Proc. Japan Acad. Ser. A. Math. Sci., 59 (1983), no. 6,238–240. doi: 10.3792/pjaa.59.238.

[27]

G. Vigeral, Evolution equations in discrete and continuous time for nonexpansive opreators in Banach spaces, ESAIM, Control Optim. Calc. Var., 16 (2010), 809-832.  doi: 10.1051/cocv/2009026.

[28]

H. K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal., 16 (1991), 1127-1138.  doi: 10.1016/0362-546X(91)90200-K.

[29]

K. Yosida, On the differentiability and the representation of one-parameter semi-group of linear operators, J. Math. Soc. Japan, 1 (1948), 15-21.  doi: 10.2969/jmsj/00110015.

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