Abstract
This paper is devoted to the study of a newly introduced tool, projectional coderivatives, and the corresponding calculus rules in finite dimensional spaces. We show that when the restricted set has some nice properties, more specifically, it is a smooth manifold, the projectional coderivative can be refined as a fixed-point expression. We will also improve the generalized Mordukhovich criterion to give a complete characterization of the relative Lipschitz-like property under such a setting. Chain rules and sum rules are obtained to facilitate the application of the tool to a wider range of parametric problems.
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References
Aubin, J.-P.: Lipschitz behavior of solutions to convex minimization problems. Math. Oper. Res. 9(1), 87–111 (1984)
Mordukhovich, B.S.: Variational Analysis and Applications. Springer, Cham (2018)
Mordukhovich, B.S.: Sensitivity analysis in nonsmooth optimization. In: Theoretical Aspects of Industrial Design, vol. 58, pp. 32–46 (1992)
Mordukhovich, B.S.: Generalized differential calculus for nonsmooth and set-valued mappings. J. Math. Anal. Appl. 183(1), 250–288 (1994)
Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (2009)
Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation I. Springer, Berlin (2006)
Mordukhovich, B.S., Outrata, J.V.: Coderivative analysis of quasi-variational inequalities with applications to stability and optimization. SIAM J. Optim. 18(2), 389–412 (2007)
Levy, A.B., Mordukhovich, B.S.: Coderivatives in parametric optimization. Math. Program. 99(2), 311–327 (2004)
Huyen, D.T.K., Yen, N.D.: Coderivatives and the solution map of a linear constraint system. SIAM J. Optim. 26(2), 986–1007 (2016)
Lee, G.M., Tam, N.N., Yen, N.D.: Quadratic Programming and Affine Variational Inequalities: A Qualitative Study. Springer, New York (2005)
Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, Berlin (2013)
Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Mappings. Springer, Heidelberg (2009)
Ioffe, A.D.: Variational Analysis of Regular Mappings: Theory and Applications. Springer, Cham (2017)
Klatte, D., Kummer, B.: Nonsmooth Equations in Optimization: Regularity, Calculus, Methods and Applications. Kluwer Academic, New York (2002)
Gfrerer, H.: On directional metric regularity, subregularity and optimality conditions for nonsmooth mathematical programs. Set-Valued Var. Anal. 21(2), 151–176 (2013)
Ginchev, I., Mordukhovich, B.S.: On directionally dependent subdifferentials. C. R. Acad. Bulgare Sci. 64(4), 497–508 (2011)
Clarke, F.H.: Optimization and Nonsmooth Analysis. SIAM, Philadelphia (1990)
Gfrerer, H., Outrata, J.V.: On Lipschitzian properties of implicit multifunctions. SIAM J. Optim. 26(4), 2160–2189 (2016)
Van Ngai, H., Théra, M.: Directional metric regularity of multifunctions. Math. Oper. Res. 40(4), 969–991 (2015)
Ioffe, A.D.: On regularity concepts in variational analysis. J. Fixed Point Theory Appl. 8(2), 339–363 (2010)
Arutyunov, A.V., Izmailov, A.F.: Directional stability theorem and directional metric regularity. Math. Oper. Res. 31(3), 526–543 (2006)
Mordukhovich, B.S., Wang, B.: Restrictive metric regularity and generalized differential calculus in Banach spaces. Int. J. Math. Math. Sci. 2004(50), 2653–2680 (2004)
Ioffe, A.D.: Regularity on a fixed set. SIAM J. Optim. 21(4), 1345–1370 (2011)
Benko, M., Gfrerer, H., Outrata, J.V.: Stability analysis for parameterized variational systems with implicit constraints. Set-Valued Var. Anal. 28(1), 167–193 (2020)
Meng, K.W., Li, M.H., Yao, W.F., Yang, X.Q.: Lipschitz-like property relative to a set and the generalized Mordukhovich criterion. Math. Program. 189(1), 455–489 (2021)
Yao, W., Yang, X.: Relative Lipschitz-like property of parametric systems via projectional coderivatives. SIAM J. Optim. 33(3), 2021–2040 (2023)
Daniilidis, A., Pang, J.C.: Continuity and differentiability of set-valued maps revisited in the light of tame geometry. J. Lond. Math. Soc. 83(3), 637–658 (2011)
Lee, J.M.: Introduction to Smooth Manifolds. Springer, New York (2013)
Lang, S.: Introduction to Linear Algebra, 5th edn. Wellesley – Cambridge Press, Wellesley (2016)
Acknowledgements
The authors are grateful to the anonymous reviewers for their careful reading and valuable suggestions.
Funding
Kaiwen Meng was supported in part by the National Natural Science Foundation of China (Ref No.: 11671329, 12001445). Minghua Li was supported by the National Natural Science Foundation of China (Ref No.: 12271072) and the Natural Science Foundation of Chongqing Municipal Science and Technology Commission (Ref No.: CSTB2022NSCQ-MSX0409, CSTB2022NSCQ-MSX0406). Yang Xiaoqi was partly supported by a project from the Research Grants Council of Hong Kong (Ref No.: 15209921).
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Yao, W., Meng, K., Li, M. et al. Projectional Coderivatives and Calculus Rules. Set-Valued Var. Anal 31, 36 (2023). https://doi.org/10.1007/s11228-023-00698-9
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DOI: https://doi.org/10.1007/s11228-023-00698-9
Keywords
- Projectional coderivative
- Calculus rules
- Generalized Mordukhovich criterion
- Relative Lipschitz-like property