Abstract
We derive stability criteria for saddle points of a class of nonsmooth optimization problems in Hilbert spaces arising in PDE-constrained optimization, using metric regularity of infinite-dimensional set-valued mappings. A main ingredient is an explicit pointwise characterization of the regular coderivative of the subdifferential of convex integral functionals. This is applied to several stability properties for parameter identification problems for an elliptic partial differential equation with non-differentiable data fitting terms.
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Attouch, H., Bolte, J., Svaiter, B.F.: Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward-backward splitting, and regularized Gauss–Seidel methods. Math. Program. 137(1–2), 91–129 (2013). doi:10.1007/s10107-011-0484-9
Aubin, J., Frankowska, H.: Set-Valued Analysis. Modern Birkhäuser Classics. Birkhäuser, Boston (2009). doi:10.1007/978-0-8176-4848-0. Reprint of the 1990 edition
Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. Springer, New York (2011). doi:10.1007/978-1-4419-9467-7
Bolte, J., Daniilidis, A., Ley, O., Mazet, L.: Characterizations of Lojasiewicz inequalities: subgradient flows, talweg, convexity. Trans. Am. Math. Soc. 362(6), 3319–3363 (2010). doi:10.1090/S0002-9947-09-05048-X
Bonnans, J.F., Shapiro, A.: Perturbation analysis of optimization problems. Springer series in operations research. Springer, New York. doi:10.1007/978-1-4612-1394-9 (2000)
Casas, E., Tröltzsch, F.: Second order optimality conditions and their role in PDE control. Jahresbericht der Deutschen Mathematiker-Vereinigung 117(1), 3–44 (2015). doi:10.1365/s13291-014-0109-3
Clarke, F.H.: Optimization and Nonsmooth Analysis, 2 edn. Classics Appl. Math, vol. 5. SIAM, Philadelphia (1990). doi:10.1137/1.9781611971309
Clason, C.: L∞ fitting for inverse problems with uniform noise. Inverse Prob. 28(104), 007 (2012). doi:10.1088/0266-5611/28/10/104007
Clason, C., Jin, B.: A semismooth Newton method for nonlinear parameter identification problems with impulsive noise. SIAM J. Imag. Sci. 5, 505–536 (2012). doi:10.1137/110826187
Clason, C., Kunisch, K.: Multi-bang control of elliptic systems. Annales de l’Institut Henri Poincaré (C) Analyse Non Linéaire. doi:10.1016/j.anihpc.2013.08.005. Article in press (2013)
Clason, C., Rund, A., Kunisch, K., Barnard, R.C.: A convex penalty for switching control of partial differential equations. Syst. Control Lett. 89, 66–73 (2016). doi:10.1016/j.sysconle.2015.12.013
Dong, Y., Hintermüller, M., Rincon-Camacho, M.: Automated regularization parameter selection in multi-scale total variation models for image restoration. J. Math. Imaging Vision 40(1), 82–104 (2011). doi:10.1007/s10851-010-0248-9
Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Mappings, 2nd edn. Springer, New York. doi:10.1007/978-1-4939-1037-3 (2014)
Drusvyatskiy, D., Lewis, A.S.: Tilt stability, uniform quadratic growth, and strong metric regularity of the subdifferential. SIAM J. Optim. 23(1), 256–267 (2013). doi:10.1137/120876551
Eberhard, A., Wenczel, R.: A study of tilt-stable optimality and sufficient conditions. Nonlinear Anal. Theory Methods Appl. 75(3), 1260–1281 (2012). doi:10.1016/j.na.2011.08.014. Variational Analysis and Its Applications
Ekeland, I., Temam, R.: Convex analysis and variational problems. SIAM. doi:10.1137/1.9781611971088 (1999)
Emich, K., Henrion, R.: A simple formula for the second-order subdifferential of maximum functions. Vietnam J. Math. 42(4), 467–478 (2014). doi:10.1007/s10013-013-0052-0
Griepentrog, J.A., Recke, L.: Linear elliptic boundary value problems with non-smooth data: normal solvability on Sobolev-Campanato spaces. Math. Nachr. 225, 39–74 (2001). doi:10.1002/1522-2616(200105)225:1<39::AID-MANA39>3.3.CO;2-X
Henrion, R., Jourani, A., Outrata, J.: On the calmness of a class of multifunctions. SIAM J. Optim. 13(2), 603–618 (2002). doi:10.1137/S1052623401395553
Henrion, R., Kruger, A.Y., Outrata, J.V.: Some remarks on stability of generalized equations. J. Optim. Theory Appl. 159(3), 681–697 (2013). doi:10.1007/s10957-012-0147-x
Henrion, R., Mordukhovich, B.S., Nam, N.M.: Second-order analysis of polyhedral systems in finite and infinite dimensions with applications to robust stability of variational inequalities. SIAM J. Optim. 20(5), 2199–2227 (2010). doi:10.1137/090766413
Hiriart-Urruty, J.B., Lemaréchal, C.: Fundamentals of convex analysis springer. doi:10.1007/978-3-642-56468-0 (2001)
Ioffe, A.D.: Metric regularity. Theory and applications – a survey. arXiv:1505.07920 (2015)
Kaltenbacher, B., Kirchner, A., Vexler, B.: Adaptive discretizations for the choice of a Tikhonov regularization parameter in nonlinear inverse problems. Inverse Prob. 27(125), 008 (2011). doi:10.1088/0266-5611/27/12/125008
Klatte, D., Kummer, B.: Nonsmooth Equations in Optimization: regularity, Calculus, Methods and Applications, Nonconvex Optimization and Its Applications, vol. 60. Springer US. doi:10.1007/b130810 (2002)
Klatte, D., Kummer, B.: Optimization methods and stability of inclusions in Banach spaces. Math. Program. 117(1-2), 305–330 (2009). doi:10.1007/s10107-007-0174-9
Kröner, A., Vexler, B.: A priori error estimates for elliptic optimal control problems with a bilinear state equation. J. Comput. Appl. Math 230(2), 781–802 (2009). doi:10.1016/j.cam.2009.01.023
Kunze, M., Monteiro Marques, M.D.P.: An introduction to Moreau’s sweeping process. In: Brogliato, B. (ed.) Impacts in Mechanical Systems, Lecture Notes in Physics, vol. 551, pp. 1–60. Springer, Berlin (2000). doi:10.1007/3-540-45501-9_1
Kurdyka, K.: On gradients of functions definable in o-minimal structures. Annales de l’Institut Fourier 48(3), 769–783 (1998). http://eudml.org/doc/75302
Levy, A.B., Poliquin, R.A., Rockafellar, R.T.: Stability of locally optimal solutions. SIAM J. Optim. 10(2), 580–604 (2000). doi:10.1137/S1052623498348274
Lewis, A.S., Zhang, S.: Partial smoothness, tilt stability, and generalized hessians. SIAM J. Optim. 23(1), 74–94 (2013). doi:10.1137/110852103
de Los Reyes, J.C., Schönlieb, C.B., Valkonen, T.: The structure of optimal parameters for image restoration problems. J. Math. Anal. Appl. 434(1), 464–500 (2016). doi:10.1016/j.jmaa.2015.09.023. Accepted
Mordukhovich, B.S.: Sensitivity analysis in nonsmooth optimization. In: Field, D. A., Komkov, V. (eds.) Proceedings of the SIAM Regional Conference on Industrial Design Theory, Ohio, April 25–26, 1990, pp 32–46. SIAM, Philadelphia (1992)
Mordukhovich, B.S.: Complete characterization of openness, metric regularity, and Lipschitzian properties of multifunctions. Trans. Am. Math. Soc. 340(1), 1–35 (1993). doi:10.1090/S0002-9947-1993-1156300-4
Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation I: Basic Theory, Grundlehren der mathematischen Wissenschaften, vol. 330 Springer-Verlag. doi:10.1007/3-540-31247-1 (2006)
Mordukhovich, B.S., Nghia, T.T.A.: Second-order variational analysis and characterizations of tilt-stable optimal solutions in infinite-dimensional spaces. Nonlinear Anal. Theory Methods Appl. 86, 159–180 (2013). doi:10.1016/j.na.2013.03.014
Mordukhovich, B.S., Nghia, T.T.A.: Full Lipschitzian and Hölderian stability in optimization with applications to mathematical programming and optimal control. SIAM J. Optim. 24(3), 1344–1381 (2014). doi:10.1137/130906878
Mordukhovich, B.S., Outrata, J.V.: On second-order subdifferentials and their applications. SIAM J. Optim. 12(1), 139–169 (2001). doi:10.1137/S1052623400377153
Mordukhovich, B.S., Outrata, J.V., Ramírez Cabrera, H.: Graphical derivatives and stability analysis for parameterized equilibria with conic constraints. Set-Valued and Variational Analysis, 1–18 (2015). doi:10.1007/s11228-015-0328-5
Mordukhovich, B.S., Rockafellar, R.T.: Second-order subdifferential calculus with applications to tilt stability in optimization. SIAM J. Optim. 22(3), 953–986 (2012). doi:10.1137/110852528
Lojasiewicz, S.: Une propriété topologique des sous-ensembles analytiques réels. In: Les équations aux dérivées partielles (paris, 1962), pp. 87–89. éditions du centre national de la recherche scientifique, paris (1963)
Lojasiewicz, S.: Sur les ensembles semi-analytiques. Actes Congrés Intern Math. 2, 237–241 (1970). http://www.mathunion.org/ICM/ICM1970.2/Main/icm1970.2.0237.0242.ocr.pdf
Poliquin, R.A., Rockafellar, R.T.: Tilt stability of a local minimum. SIAM J. Optim. 8(2), 287–299 (1998). doi:10.1137/S1052623496309296
Rockafellar, R.T.: Integral Functionals, Normal Integrands and Measurable Selections. In: Nonlinear Operators and the Calculus of Variations (Summer School, Univ. Libre Bruxelles, Brussels, 1975), Lecture Notes in Math., vol. 543, pp. 157–207. Springer, Berlin (1976). doi:10.1007/BFb0079944
Rockafellar, R.T., Wets, R.J.B.: Variational Analysis, Grundlehren der mathematischen Wissenschaften, vol. 317 Springer-Verlag. doi:10.1007/978-3-642-02431-3 (1998)
Tuy, H.: D.C. Optimization: Theory, Methods and Algorithms. In: Horst, R., Pardolos, P.M. (eds.) Handbook of Global Optimization, pp. 149–216. Kluwer Academic Publishers (1995). doi:10.1007/978-1-4615-2025-2_4
Valkonen, T.: Diff-convex combinations of Euclidean distances: a search for optima. No. 99 in Jyväskylä Studies in Computing. University of Jyväskylä. http://tuomov.iki.fi/mathematics/thesis.pdf. Ph. D. Thesis (2008)
Valkonen, T.: Refined optimality conditions for differences of convex functions. J. Glob. Optim. 48(2), 311–321 (2010). doi:10.1007/s10898-009-9495-y
Valkonen, T.: Extension of primal-dual interior point methods to diff-convex problems on symmetric cones. Optimization, 62(3), 345–377 (2013). doi:10.1080/02331934.2011.585465
Valkonen, T.: A primal-dual hybrid gradient method for nonlinear operators with applications to MRI. Inverse Prob. 30(5), 055012 (2014). doi:10.1088/0266-5611/30/5/055012
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Clason, C., Valkonen, T. Stability of Saddle Points Via Explicit Coderivatives of Pointwise Subdifferentials. Set-Valued Var. Anal 25, 69–112 (2017). https://doi.org/10.1007/s11228-016-0366-7
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DOI: https://doi.org/10.1007/s11228-016-0366-7