Abstract
The paper is devoted to the question of existence of continuous implicit functions for non-smooth equations. The equation \(f(x,\sigma )=0\) is studied with the unknown x and the parameter \(\sigma \). We assume that the mapping which defines the equation is locally Lipschitzian in the unknown variable, the parameter belongs to a topological space, and both the unknown variable and the value of the mapping belong to finite-dimensional spaces. Sufficient conditions for the existence of a continuous implicit function in a neighbourhood of a given value of the parameter and conditions for the existence of a continuous implicit function on a given subset of the parameter space are obtained in terms of the Clarke generalized Jacobian. The key tool of this development is the smoothing of the initial equation and application of recent results on solvability of smooth nonlinear equations and a priori solution estimates. Applications to control and optimization are discussed.
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Arutyunov, A.V., Izmailov, A.F., Zhukovskiy, S.E.: Continuous selections of solutions for locally Lipschitzian equations. J. Optim. Theory Appl. 185, 679–699 (2020)
Arutyunov, A.V., Zhukovskiy, S.E.: Global and semilocal theorems on implicit and inverse functions in Banach spaces. Sb. Math. 213, 1–41 (2022)
Arutyunov, A.V., Zhukovskiy, S.E.: On implicit function theorem for locally Lipschitz equations. Math. Program. (2022). https://doi.org/10.1007/s10107-021-01750-y
Bartle, R.G., Graves, L.M.: Mappings between function spaces. Trans. AMS 72, 400–413 (1952)
Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)
Clarke, F.: On the inverse function theorem. Pac. J. Math. 64, 97–102 (1976)
Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)
Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Mappings, 2nd edn. Springer, New York (2014)
Engelking, R.: General Topology. Herdelman Verlag, Berlin (1989)
Gfrerer, H., Mordukhovich, B.: Robinson regularity of parametric constraint systems via variational analysis. SIAM J. Optim. 27, 438–465 (2017)
Hartman, P.: Ordinary Differential Equations. Wiley, New York (1964)
Ioffe, A.D.: Implicit functions: a metric theory. Set-Valued Var. Anal. 25, 679–699 (2017)
Izmailov, A.F.: Strongly regular nonsmooth generalized equations. Math. Program. Ser. A 147, 581–590 (2014)
Izmailov, A.F., Kurennoy, A.S.: On regularity conditions for complementarity problems. Comput. Optim. Appl. 57, 667–684 (2014)
Izmailov, A.F., Solodov, M.V.: Newton-Type Methods for Optimization and Variational Problems. Springer, Cham (2014)
Jaramillo, L.S., Lajara, S., Madiedo, O.: Inversion of nonsmooth maps between Banach spaces. Set-Valued Var. Anal. 27, 921–947 (2019)
Levitin, E.S., Milyutin, A.A., Osmolovskii, N.P.: Conditions of high order for a local minimum in problems with constraints. Russ. Math. Surv. 33, 97–168 (1978)
Magaril-Il’aev, G.G.: The implicit function theorem for Lipschitz maps. Russ. Math. Surv. 33, 209–210 (1978)
Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, Vol. I: Basic Theory. Springer, New York (2006)
Mordukhovich, B.S.: Variational Analysis and Applications. Springer, Cham (2018)
Mordukhovich, B.S., Wang, B.: Restrictive metric regularity and generalized differential calculus in Banach spaces. Int. J. Math. Math. Sci. 2004(683907), 279–285 (2004)
Pourciau, B.H.: Analysis and optimization of Lipschitz continuous mappings. J. Optim. Theory Appl. 22, 311–351 (1977)
Pourciau, B.H.: Hadamard’s theorem for locally Lipschitzian maps. J. Math. Anal. Appl. 85, 279–285 (1982)
Robinson, S.M.: Strongly regular generalized equations. Math. Oper. Res. 5, 43–62 (1980)
Rockafellar, R.T.: Lagrange multipliers and subderivatives of optimal value functions in nonlinear programming. Math. Program Stud. 17, 28–66 (1982)
Rockafellar, R.T.: Directional differentiability of the optimal value function in a nonlinear programming problem. Math. Program Stud. 21, 213–226 (1984)
Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998). (Corrected 3rd printing (2009))
Tsar’kov, I.G.: On global existence of an implicit function. Russ. Acad. Sci. Sb. Math. 79, 287–313 (1994)
Vinter, R.: Optimal Control. Birkhauser, Basel (2010)
Warga, J.: Derivate containers, inverse functions, and controllability. In: Russell, D.L. (ed.) Calculus of Variations and Control Theory. Academic Press, New York (1976)
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Communicated by Alexey F. Izmailov.
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The research is supported by the Grant of Russian Science Foundation (Project No 22-21-00863, https://rscf.ru/en/project/22-21-00863/).
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Arutyunov, A.V., Zhukovskiy, S.E. Smoothing Procedure for Lipschitzian Equations and Continuity of Solutions. J Optim Theory Appl 199, 112–142 (2023). https://doi.org/10.1007/s10957-023-02244-x
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DOI: https://doi.org/10.1007/s10957-023-02244-x