Abstract
The theory of subdifferentials provides adequate methods and tools to put descent methods for nonsmooth optimization problems into practice. However, in applications it is often difficult to decide on a suitable subdifferential concept to construct a descent method. Therefore, we introduce subdifferentials in terms of their properties to indicate a selection of subdifferentials worth considering. This initials the first part of the construction of a continuous outer subdifferential (COS). Typically, methods based on e.g. the Clarke subdifferential are non-convergent without assumptions like semismoothness on the objective function. In cases in which only supersets of the Clarke subdifferential are known, semismoothness cannot be proved or is even violated. Therefore, in the second part of the construction, a previously selected subdifferential will be expanded to a continuous mapping, if necessary. This is also practicable for upper bounds of the subdifferential of current interest. Finally, based on COS we present a methodology for solving nonsmooth optimization problems. From a theoretical point of view, convergence is established through the construction of COS.
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References
Aubin, J.-P., Frankowska, H.: Set-Valued Analysis, 1st edn. Birkhäuser, Cambridge (1990)
Bagirov, A.M.: Continuous subdifferential approximations and their applications. J. Math. Sci. 115(5), 2567–2609 (2003)
Bagirov, A.M., Karmitsa, N., Mäkelä, M.M.: Introduction to Nonsmooth Optimization, 1st edn. Springer, Berlin (2014)
Bajaj, A., Hare, W., Lucet, Y.: Visualization of the ε-subdifferential of piecewise linear–quadratic functions. Comput. Optim. Appl. 67(2), 421–442 (2017)
Burke, J.V., Lewis, A.S., Overton, M.L.: A robust gradient sampling algorithm for nonsmooth, nonconvex optimization. SIAM J. Optim. 15(3), 751–779 (2005)
Clarke, F.H.: Optimization and Nonsmooth Analysis, 1st edn. Wiley, New York (1983)
Demyanov, V.F., Malozemov, V.N.: Einführung in Minimax-Probleme, 1st edn. Akademische Verlagsgesellschaft Geest & Portig K.-G. (1975)
Demyanov, V.F., Rubinov, A.M.: On Quasidifferentiable mappings. Math. Oper. Stat. Ser. Optim. 14, 3–21 (1983). https://doi.org/10.1080/02331938308842828
Demyanov, V.F., Rubinov, A.M.: Constructive Nonsmooth Analysis, 1st edn. Lang (1995)
Dubeau, F., Gauvin, J.: Differential properties of the marginal function in mathematical programming. Math. Program. Stud. 19, 101–119 (1982). https://doi.org/10.1007/BFb0120984
Goldstein, A.A.: Optimization of Lipschitz continuous functions. Math. Program. 13(1), 14–22 (1977)
Hiriart-Urruty, J.-B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms I, 2nd edn. Springer, Berlin (1996)
Hiriart-Urruty, J.-B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms II, 1st edn. Springer, Berlin (1993)
Hiriart-Urruty, J.-B., Lemaréchal, C.: Fundaments of Convex Analysis, 2nd edn. Springer, Berlin (2004)
Ioffe, A.D.: Calculus of Dini subdifferentials of functions and contingent coderivatives of set-valued maps. Nonlinear Anal. Theory Methods Appl. 8(5), 517–539 (1984)
Ioffe, A.D.: On the theory of subdifferential. North-Holland Math. Stud. 129, 183–200 (1986)
Ioffe, A.D., Outrata, J.V.: On metric and calmness qualification conditions in subdifferential calculus. Set-Valued Anal. 16(2), 199–227 (2008)
Kiwiel, K.C.: Methods of Descent for Nondifferentiable Optimization, 1st edn. Springer, Berlin (1985)
Mäkelä, M. M., Neittaanmäki, P.: Nonsmooth Optimization, 1st edn. World Scientific, Singapore (1992)
Mifflin, R.: Semismooth and semiconvex functions in constrained optimization. SIAM J. Control Optim. 15(6), 959–972 (1976)
Mordukhovich, B.S., Shao, Y.: On nonconvex subdifferentials calculus in Banach spaces. J. Convex Anal. 2(1), 211–227 (1995)
Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation I, 1st edn. Springer, Berlin (2006)
Penot, J.-P.: Calcul Sous-Differential et Optimisation. J. Funct. Anal. 27, 248–276 (1978)
Rockafellar, R.T.: Convex Analysis, 1st edn. Princeton University Press, Princeton (1970)
Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14(5), 877–898 (1976)
Rockafellar, R.T., Wets, R.J-B.: Variational Analysis, 3rd edn. Springer, Berlin (2009)
Wolfe, P.: A method of conjugate subgradients for minimizing nondifferentiable functions. Math. Program. Stud. 3, 145–173 (1975)
Acknowledgements
The author wishes to thank the reviewers and the associate editor for their very grateful suggestions, which greatly have improve the presentation of the paper. Also, the author likes to thank W. Achtziger for his time to discuss several results.
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Knossalla, M. Continuous Outer Subdifferentials in Nonsmooth Optimization. Set-Valued Var. Anal 27, 665–692 (2019). https://doi.org/10.1007/s11228-018-0481-8
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DOI: https://doi.org/10.1007/s11228-018-0481-8