Recently, the author in 2024 studied the $ V $-Moreau envelope $ e^V_{\lambda, S}f $ of nonconvex functions $ f $ with index $ \lambda > 0 $ and associated with closed sets $ S $ in smooth Banach spaces. In this paper, we developed a corresponding local theory for the local $ V $-Moreau envelope of $ f $ on a neighborhood of a point $ \bar x $ in the domain of $ f $. A local concept of generalized projection is introduced in connection with the local $ V $-Moreau envelope. We investigated the Lipschitz continuity, Fréchet differentiability, and $ V $-proximal subdifferentiability of the local $ V $-Moreau envelope of $ f $. Moreover, we established the local Hölder continuity of the local generalized projection for the class of $ V $-primal lower nice functions introduced and studied in 2020 by Bounkhel et al. Our results extended several known theorems from Hilbert spaces to a more general framework of smooth Banach spaces.
Citation: Messaoud Bounkhel. Local $ V $-Moreau envelope of nonconvex functions[J]. AIMS Mathematics, 2026, 11(7): 21012-21035. doi: 10.3934/math.2026853
Recently, the author in 2024 studied the $ V $-Moreau envelope $ e^V_{\lambda, S}f $ of nonconvex functions $ f $ with index $ \lambda > 0 $ and associated with closed sets $ S $ in smooth Banach spaces. In this paper, we developed a corresponding local theory for the local $ V $-Moreau envelope of $ f $ on a neighborhood of a point $ \bar x $ in the domain of $ f $. A local concept of generalized projection is introduced in connection with the local $ V $-Moreau envelope. We investigated the Lipschitz continuity, Fréchet differentiability, and $ V $-proximal subdifferentiability of the local $ V $-Moreau envelope of $ f $. Moreover, we established the local Hölder continuity of the local generalized projection for the class of $ V $-primal lower nice functions introduced and studied in 2020 by Bounkhel et al. Our results extended several known theorems from Hilbert spaces to a more general framework of smooth Banach spaces.
| [1] | Y. Alber, I. Ryazantseva, Nonlinear Ill-posed problems of monotone type, Dordrecht: Springer, 2006. http://doi.org/10.1007/1-4020-4396-1 |
| [2] | Y. I. Alber, A. I. Notik, On some estimates for projection operators in Banach spaces, Communications on Applied Nonlinear Analysis, 2 (1993), 47–55. |
| [3] | Y. I. Alber, Generalised projection operators in Banach spaces: properties and applications, Functional Differential Equations, 1 (1994), 1–21. |
| [4] | Y. I. Alber, A. N. Iusem, M. V. Solodov, Minimization of nonsmooth convex functionals in Banach spaces, J. Convex Anal., 4 (1997), 235–255. |
| [5] | H. H. Bauschke, P. L. Combettes, Convex analysis and monotone operator theory in Hilbert spaces, 2 Eds., Cham: Springer, 2017. https://doi.org/10.1007/978-3-319-48311-5 |
| [6] | F. Bernard, L. Thibault, N. Zlateva, Characterizations of prox-regular sets in uniformly convex Banach space, J. Convex Anal., 13 (2006), 525–559. |
| [7] | M. Bounkhel, Regularity concepts in nonsmooth analysis: theory and applications, New York: Springer, 2012. http://doi.org/10.1007/978-1-4614-1019-5 |
| [8] |
M. Bounkhel, Calculus rules for V-proximal subdifferentials in smooth Banach spaces, J. Funct. Space., 2016 (2016), 1917387. https://doi.org/10.1155/2016/1917387 doi: 10.1155/2016/1917387
|
| [9] |
M. Bounkhel, Generalized Projections on closed nonconvex sets in uniformly convex and uniformly smooth Banach spaces, J. Funct. Space., 2015 (2015), 478437. https://doi.org/10.1155/2015/478437 doi: 10.1155/2015/478437
|
| [10] |
M. Bounkhel, Generalized $(f, \lambda)$-projection operator on closed nonconvex sets and its applications in reflexive smooth Banach spaces, AIMS Mathematics, 8 (2023), 29555–29568. http://doi.org/10.3934/math.20231513 doi: 10.3934/math.20231513
|
| [11] |
M. Bounkhel, $V$-Moreau envelope of nonconvex functions on smooth Banach spaces, AIMS Mathematics, 9 (2024), 28589–28610. https://doi.org/10.3934/math.20241387 doi: 10.3934/math.20241387
|
| [12] |
M. Bounkhel, M. Bachar, Generalized prox-regularity in reflexive Banach spaces with smooth dual norm, J. Math. Anal. Appl., 475 (2019), 699–729. https://doi.org/10.1016/j.jmaa.2019.02.064 doi: 10.1016/j.jmaa.2019.02.064
|
| [13] |
M. Bounkhel, M. Bachar, Primal lower nice functions in reflexive smooth Bnach spaces, Mathematics, 8 (2020), 2066. https://doi.org/10.3390/math8112066 doi: 10.3390/math8112066
|
| [14] |
M. Bounkhel, R. Al-Yusof, Proximal analysis in reflexive smooth Banach spaces, Nonlinear Anal. Theor., 73 (2010), 1921–1939. https://doi.org/10.1016/j.na.2010.04.077 doi: 10.1016/j.na.2010.04.077
|
| [15] |
M. Bounkhel, M. Bachar, $V$-Prox-regular functions in smooth Banach spaces, J. Funct. Space., 2020 (2020), 9465492. https://doi.org/10.1155/2020/9465492 doi: 10.1155/2020/9465492
|
| [16] |
M. Bounkhel, M. Bachar, $V$-Proximal trustworthy Banach spaces, J. Funct. Space., 2020 (2020), 4274160. https://doi.org/10.1155/2020/4274160 doi: 10.1155/2020/4274160
|
| [17] | F. H. Clarke, R. J. Stern, P. R. Wolenski, Proximal smoothness and the lower-$C^2$ property, J. Convex Anal., 2 (1995), 117–144. |
| [18] |
P. L. Combettes, V. R. Wajs, Signal recovery by proximal forward-backward splitting, Multiscale Model. Sim., 4 (2005), 1168–1200. https://doi.org/10.1137/050626090 doi: 10.1137/050626090
|
| [19] | H. Federer, Curvature measures, Trans. Amer. Math. Soc., 93 (1959), 418–491. https://doi.org/10.1090/S0002-9947-1959-0110078-1 |
| [20] | W. L. Hare, R. A. Poliquin, Prox-regularity and stability of the proximal mapping, J. Convex Anal., 14 (2007), 589–606. |
| [21] | W. L. Hare, C. Planiden, Parametrically prox-regular functions, J. Convex Anal., 21 (2014), 901–923. |
| [22] |
J. L. Li, The generalized projection operator on reflexive Banach spaces and its applications, J. Math. Anal. Appl., 306 (2005), 55–71. https://doi.org/10.1016/j.jmaa.2004.11.007 doi: 10.1016/j.jmaa.2004.11.007
|
| [23] |
J. L. Li, On the existence of solutions of variational inequalities in Banach spaces, J. Math. Anal. Appl., 295 (2004), 115–126. https://doi.org/10.1016/j.jmaa.2004.03.010 doi: 10.1016/j.jmaa.2004.03.010
|
| [24] | J. J. Moreau, Fonctions convexes duales et points proximaux dans un espace Hilbertien, Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences, 255 (1962), 2897–2899. |
| [25] | J. J. Moreau, Proximité et dualité dans un espace Hilbertien, Bulletin de la Société Mathématique de France 93 (1965), 273–299. https://doi.org/10.24033/bsmf.1625 |
| [26] |
R. A. Poliquin, R. T. Rockafellar, Prox-regular functions in variational analysis, Trans. Amer. Math. Soc., 348 (1996), 1805–1838. https://doi.org/10.1090/s0002-9947-96-01544-9 doi: 10.1090/s0002-9947-96-01544-9
|
| [27] |
R. A. Poliquin, R. T. Rockafellar, L. Thibault, Local differentiability of distance functions, Trans. Amer. Math. Soc., 352 (2000), 5231–5249. https://doi.org/10.1090/s0002-9947-00-02550-2 doi: 10.1090/s0002-9947-00-02550-2
|
| [28] | R. T. Rockafellar, R. J. B. Wets, Variational analysis, 1 Eds., Heidelberg: Springer, 1998. https://doi.org/10.1007/978-3-642-02431-3 |
| [29] |
A. Shapiro, Existence and differentiability of metric projections in Hilbert spaces, SIAM J. Optimiz., 4 (1994), 130–141. https://doi.org/10.1137/0804006 doi: 10.1137/0804006
|
| [30] | W. Takahashi, Nonlinear functional analysis, Yokohama: Yokohama Publishers, 2000. |
| [31] |
K. Q. Wu, N. J. Huang, The generalized $f$-projection operator with an application, B. Aust. Math. Soc., 73 (2006), 307–317. http://doi.org/10.1017/S0004972700038892 doi: 10.1017/S0004972700038892
|
| [32] |
K. Q. Wu, N. J. Huang, Properties of the generalized $f$-projection operator and its application in Banach spaces, Comput. Math. Appl., 54 (2007), 399–406. http://doi.org/10.1016/j.camwa.2007.01.029 doi: 10.1016/j.camwa.2007.01.029
|
Messaoud Bounkhel. Local $ V $-Moreau envelope of nonconvex functions[J]. AIMS Mathematics, 2026, 11(7): 21012-21035. doi: 10.3934/math.2026853
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