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On regularization in Banach spaces. (English) Zbl 0873.49028

The aim of the paper is to extend to Banach spaces previous results on regularization given in J. M. Lasry and P. L. Lions [Isr. J. Math. 55, 257-266 (1986; Zbl 0631.49018)]. The author proves that it is possible to associate to any bounded below lower semicontinuous (l.s.c.) proper function \(f\), defined in a Banach space, a family of \(C^1\) functions approximating \(f\) from below obtaining good properties for minimization. The method reduces, in case when \(X\) is a Hilbert space, to that one introduced by J. M. Lasry and P. L. Lions, which is based upon the Moreau-Yosida approximation.
The main results are contained in Theorem 1, where under suitable assumptions on the function \(f\), which is defined in a Banach space whose norm and dual norm are both locally uniformly rotund, regularization properties for the two-parameter family of approximates \(f_{t,s}\) and for \(df_{t,s}\) are established. Moreover, an approximation from below, i.e., \(f_{t,s}\leq f\), is proved and it is shown that the method preserves the infimum of \(f\) and the associated set of minimizers. Furthermore, convergence properties for \(f_{t,s}\) to \(f\) are also obtained.
Several corollaries show that some known results on approximation of uniformly continuous functions can be derived from Theorem 1. Also approximation by twice Gâteaux differentiable \(C^{1,1}\) functions in separable Banach spaces are studied. Finally, a convergence result for the derivatives of the approximates \(df_{t,s}\) to the Clarke subdifferential \(\partial f\) is obtained in case when \(f\) is defined in a Hilbert space.
Reviewer: C.Vinti (Perugia)

MSC:

49N60 Regularity of solutions in optimal control
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
46B25 Classical Banach spaces in the general theory

Citations:

Zbl 0631.49018
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References:

[1] Asplund, E. andRockafellar, R. T., Gradients of convex functions,Trans. Amer. Math. Soc. 139 (1969), 443–467. · Zbl 0181.41901 · doi:10.1090/S0002-9947-1969-0240621-X
[2] Attouch, H., Viscosity solutions of optimization problems. Epi-convergence and scaling,Sém. Anal. Convexe 22:8 (1992), 1–48. · Zbl 1267.49027
[3] Attouch, H. andAzé, D., Approximation and regularization of arbitrary functions in Hilbert spaces by the Lasry-Lions method,Ann. Inst. H. Poincaré Anal. Non Linéaire 10 (1993), 289–312. · Zbl 0780.41021
[4] Beer, G.,Topologies on Closed and Closed Convex Sets, Mathematics and its Applications268, Kluwer Academic Publishers Group, Dordrecht, 1993. · Zbl 0792.54008
[5] Benoist, J., Convergence de la dérivée de la régularisée Lasry-Lions,C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), 941–944. · Zbl 0772.49009
[6] Deville, R., Godefroy, G., andZizler, V.,Smoothness and Renormings in Banach Spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics64, Pitman, New York, 1993.
[7] Ekeland, I. andLasry, J. M., On the number of periodic trajectories for a Hamiltonian flow on a convex energy surface,Ann. of Math. 112 (1980), 283–319. · Zbl 0449.70014 · doi:10.2307/1971148
[8] Fabian, M., Whitfield, J. H. M. andZizler, V., Norms with locally Lipschitzian derivatives,Israel J. Math. 44 (1983), 262–276. · Zbl 0521.46009 · doi:10.1007/BF02760975
[9] Frontisi, J., Smooth partitions of unity in Banach spaces,Rocky Mountain J. Math. 25 (1995), 1295–1304. · Zbl 0853.46015 · doi:10.1216/rmjm/1181072147
[10] Lasry, J. M. andLions, P. L., A remark on regularization in Hilbert spaces,Israel J. Math. 55 (1986), 257–266. · Zbl 0631.49018 · doi:10.1007/BF02765025
[11] Lions, P. L.,Generalized Solutions of Hamilton-Jacobi Equations, Research Notes in Mathematics,69, Pitman (Advanced Publishing Program), Boston-London-Melbourne, 1982. · Zbl 0497.35001
[12] McLaughlin, D., Smooth partitions of unity and approximating norms in Banach spaces,Rocky Mountain J. Math. 25 (1995), 1137–1148. · Zbl 0869.46006 · doi:10.1216/rmjm/1181072210
[13] McLaughlin, D., Poliquin, R., Vanderwerff, J. andZizler, V., Second-order Gâteaux differentiable bump functions and approximations in Banach spaces,Canad. J. Math. 45 (1993), 612–625. · Zbl 0796.46005 · doi:10.4153/CJM-1993-032-9
[14] Nemirovskiî, A. S. andSemenov, S. M., On polynomial approximation of functions on Hilbert Space.Mat. Sb. 92 (134) (1973), 257–281, 344 (Russian). English transl.:Math. USSR-Sb. 21 (1973), 255–277.
[15] Poliquin, R., Vanderwerff, J. andZizler, V., Renormings and convex composite representations of functions,Preprint. · Zbl 0824.46009
[16] Rockafellar, R. T., Favorable classes of Lipschitz-continuous functions in subgradient optimization, inProgress in nondifferentiable optimization (Numinski, E. A., ed.), pp. 125–143, Internat. Inst. Appl. Systems Anal., Laxenburg, 1982. · Zbl 0511.26009
[17] Vanderwerff, J., Smooth approximations in Banach spaces,Proc. Amer. Math. Soc. 115 (1992), 113–120. · Zbl 0812.46005 · doi:10.1090/S0002-9939-1992-1081100-8
[18] Zâlinescu, C., On uniformly convex functions,J. Math. Anal. Appl. 95 (1983), 344–374. · Zbl 0519.49010 · doi:10.1016/0022-247X(83)90112-9
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