Abstract
The paper presents a method for solving optimal control problem with free right end and linear differential equations constraints. The proposed iterative process of extragradient-type is formulated in the functional subspace of piecewise continuous controls of L 2. The convergence of the method is proved.
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Vasiliev, F.P.: Optimization Methods. Factorial Press, Moscow (2002, in Russian)
Karush, W.: Minima of functions of several variables with inequalities as side conditions. Master’s thesis, Department of Mathematics, University of Chicago (1939)
Kuhn, H.W., Tucker, A.W.: Nonlinear programming. In: Neyman, J. (ed.) Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability. pp. 481–492. University of California Press, Berkeley (1951)
Intriligator M.: Mathematical Optimization and Economic Theory. Prentice-Hall, New York (1971)
Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Ekonomika i Matematicheskie Metody XII(6), 747–756 (1976, in Russian)
Antipin, A.S.: Method of saddle point finding for augmented Lagrangian. Ekonomika i Matematicheskie Metody XIII(3), 560–565 (1977, in Russian)
Antipin, A.S.: Equilibrium programming: methods of gradient type. Avtomatika i telemekhanika (8), 1337–1347 (1997, in Russian)
Antipin, A.S., Khoroshilova, E.V.: Extragradient methods for optimal control problems with linear restrictions. Izvestia IGU Ser. Math. 3, 2–20. http://isu.ru/izvestia/ (2010, in Russian)
Antipin A.S.: Differential gradient prediction-type methods for computing fixed points of extremal mappings. Differ. Equ. 31(11), 1786–1795 (1995)
Antipin, A.S.: Iterative methods of prediction-type for computing fixed points of extremal mapping. Izvestiya Vysshikh Uchebnykh Zavedeniy. Matematika. 11(402), 17–27 (1995, in Russian)
Ceng L.C., Teboulle M., Yao J.C.: Weak convergence of an iterative method for pseudomonotone variational inequalities and fixed-point problems. J. Optim. Theory Appl. 146(1), 19–31 (2010)
Li L., Song W.: A modified extragradient method for inverse-monotone operators in Banach spaces. J. Glob Optim. 44(4), 609–629 (2009)
Qin X., Cho S.Y., Kang S.M.: An extragradient-type method for generalized equilibrium problems involving strictly pseudocontractive mappings. J. Glob. Optim. 49(4), 679–693 (2011)
Censor Y., Gibali A., Reich S.: The subgradient extragradient method for solving variational inequalities in Hilbert space. J. Glob. Optim. 148(2), 318–335 (2011)
Ceng L.-Ch., Hadjisavvas N., Wong N.-C.: Strong convergence theorem by a hybrid extragradient-like approximation method for variational inequalities and fixed point problems. J. Glob. Optim. 46(4), 635–646 (2010)
Petrusel A., Yao J.-C.: An extragradient iterative scheme by viscosity approximation methods for fixed point problems and variational inequality problems. Cent. Eur. J. Math. 7(2), 335–347 (2009)
Jaiboon C., Kumam P., Humphries U.W.: An extragradient method for relaxed cocoercive variational inequality and equilibrium problems. Anal. Theory Appl. 25(4), 381–400 (2009)
Saejung, S., Wongchan, K.: A note on Ceng–Wang–Yao’s result [Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities, Math. Methods Oper. Res. (2008) 67: 375–390]. Math. Methods Oper. Res. 73(2), 153–157 (2011)
Yao Y., Noor M.A., Noor K.I., Liou Y.-C., Yaqoob H.: Modified extragradient methods for a system of variational inequalities in Banach spaces. Acta Applicandae Mathematicae 110(3), 1211–1224 (2010)
Vasilieva, O.V.: Optimal control in terms of smooth and bounded functions. Izvestia IGU Ser. Math. 2(1), 118–131. http://isu.ru/izvestia/ (2009)
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Khoroshilova, E.V. Extragradient-type method for optimal control problem with linear constraints and convex objective function. Optim Lett 7, 1193–1214 (2013). https://doi.org/10.1007/s11590-012-0496-2
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DOI: https://doi.org/10.1007/s11590-012-0496-2