Geodesic
Maximization of a~function with Lipschitz continuous gradient
Fundamentalʹnaâ i prikladnaâ matematika, Tome 18 (2013) no. 5, pp. 17-25.

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In the present paper, we consider (nonconvex in the general case) functions that have Lipschitz continuous gradient. We prove that the level sets of such functions are proximally smooth and obtain an estimate for the constant of proximal smoothness. We prove that the problem of maximization of such function on a strongly convex set has a unique solution if the radius of strong convexity of the set is sufficiently small. The projection algorithm (similar to the gradient projection algorithm for minimization of a convex function on a convex set) for solving the problem of maximization of such a function is proposed. The algorithm converges with the rate of geometric progression. 
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     title = {Maximization of a~function with {Lipschitz} continuous gradient},
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M. V. Balashov. Maximization of a~function with Lipschitz continuous gradient. Fundamentalʹnaâ i prikladnaâ matematika, Tome 18 (2013) no. 5, pp. 17-25. http://geodesic.mathdoc.fr/item/FPM_2013_18_5_a1/

[1] Balashov M. V., Ivanov G. E., “Slabo vypuklye i proksimalno gladkie mnozhestva v banakhovykh prostranstvakh”, Izv. RAN. Ser. mat., 73:3 (2009), 23–66 | DOI | MR | Zbl

[2] Ivanov G. E., Slabo vypuklye mnozhestva i funktsii, Fizmatlit, M., 2006

[3] Polovinkin E. S., Balashov M. V., Elementy vypuklogo i silno vypuklogo analiza, Fizmatlit, M., 2007 | Zbl

[4] Balashov M. V., Golubev M. O., “About the Lipschitz property of the metric projection in the Hilbert space”, J. Math. Anal. Appl., 394 (2012), 545–551 | DOI | MR | Zbl

[5] Bernard F., Thibault L., Zlateva N., “Characterization of proximal regular sets in super reflexive Banach spaces”, J. Convex Anal., 13:3–4 (2006), 525–559 | MR | Zbl

[6] Canino A., “On $p$-convex sets and geodesics”, J. Differ. Equ., 75 (1988), 118–157 | DOI | MR | Zbl

[7] Clarke F. H., Stern R. J., Wolenski P. R., “Proximal smoothness and lower-$C^{2}$ property”, J. Convex Anal., 2:1–2 (1995), 117–144 | MR | Zbl

[8] Hiriart-Urruty J.-B., Ledyaev Yu. S., “A note on the characterization of the global maxima of a (tangentially) convex function over a convex set”, J. Convex Anal., 3:1 (1996), 55–61 | MR | Zbl

[9] Poliquin R. A., Rockafellar R. T., Thibault L., “Local differentiability of distance functions”, Trans. Am. Math. Soc., 352 (2000), 5231–5249 | DOI | MR | Zbl