¹Krasovskii Institute of Mathematics and Mechanics UB RAS, Yekaterinburg, Russian Federation

The paper focuses on the Fréchet subdifferential of the pointwise supremum of the family of functions taken over arbitrary index sets. The all functions in the corresponding family are defined on a Fréchet smooth space; this class of Banach spaces includes all reflexive spaces and all separable Asplund spaces. The new upper estimates, including non-convex ones, establish for the Frechet subdifferentials of the suprema of continuous functions and lower semicontinuous functions. In these estimates, an additional requirement is imposed on every ε-active index that corresponds continuous function: the ε-closeness of the considered point of the graph of the supremum to the graph of this continuous function. The key two-sided inequalities with respect to the graph of continuous function, corresponding to ε-active index, are based on the two-sided unidirectional mean value inequality. The method of proving upper estimates combines the approaches of works of J. S. Treiman, Y. S. Ledyaev, B. S. Mordukhovich, T. Nghia, and P. Pérez–Aros

https://doi.org/10.26516/1997-7670.2025.52.21

supremum of continuous functions, Fréchet smooth space, Fréchet subdifferential

1. Borwein J.M., Zhu Q.J. Techniques of variational analysis. New York, SpringerVerlag, 2005, 362 p.
2. Borwein J.M., Zhu Q.J. A survey of subdifferential calculus with applications. Nonlinear Analysis, 1999, vol. 38, no. 6, pp. 687–774. https://doi.org/10.1016/S0362-546X(98)00142-4
3. Correa R., Hantoute A., López M.A. Fundamentals of convex analysis and optimization. Cham, Springer, 2023, 444 p.
4. Ioffe A.D., Tikhomirov V.M. Theory of Extremal Problems. Amsterdam, NorthHolland, 1979, 460 p.
5. Khlopin D. On two-sided unidirectional mean value inequality in a Fréchet smooth space. Ural Mathematical Journal, 2023, vol. 9, no. 2, pp. 132–140. http://doi.org/10.15826/umj.2023.2.011
6. Ledyaev Y.S., Treiman J.S. Sub-and supergradients of envelopes, semicontinuous closures, and limits of sequences of functions. Russ. Math. Surv., 2012, vol. 67, pp. 345–373. https://doi.org/10.1070/rm2012v067n02abeh004789
7. Mordukhovich B.S. Variational analysis and generalized differentiation, I: Basic theory. Springer-Verlag: Berlin, 2006. xxii+579 p.
8. Mordukhovich B.S. Variational analysis and applications. Cham, Springer, 2018, 622 p.
9. Mordukhovich B.S., Nghia T. Subdifferentials of nonconvex supremum functions and their applications to semi-infinite and infinite programs with lipschitzian data. SIAM J. Control Optim, 2013, vol. 23, pp. 406–431. https://doi.org/10.1137/110857738
10. Penot J.P. Calculus without derivatives. New York, Springer, 2013, 524 p.
11. Pérez-Aros P. Subdifferential formulae for the supremum of an arbitrary family of functions, SIAM Journal on Optimization, 2019, vol. 29, no. 2, pp. 1714–1743. https://doi.org/10.1137/17M1163141
12. Rockafellar R.T., Wets R.J.B. Variational analysis. Berlin, Springer-Verlag, 2009, 735 p