Geodesic
Continuous second order minimization method with variable metric projection operator
Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 21 (2019) no. 1, pp. 34-47.

Voir la notice de l'article provenant de la source Math-Net.Ru

The paper examines a new continuous projection second order method of minimization of continuously Frechet differentiable convex functions on the convex closed simple set in separable, normed Hilbert space with variable metric. This method accelerates common continuous projection minimization method by means of quasi-Newton matrices. In the method, apart from variable metric operator, vector of search direction for motion to minimum, constructed in auxiliary extrapolated point, is used. By other word, complex continuous extragradient variable metric method is investigated. Short review of allied methods is presented and their connections with given method are indicated. Also some auxiliary inequalities are presented which are used for theoretical reasoning of the method. With their help, under given supplemental conditions, including requirements on operator of metric and on method parameters, convergence of the method for convex smooth functions is proved. Under conditions completely identical to those in convergence theorem, without additional requirements to the function, estimates of the method's convergence rate are obtained for convex smooth functions. It is pointed out, that one must execute computational implementation of the method by means of numerical methods for ODEs solution and by taking into account the conditions of proved theorems.
Keywords: convex function, continuous minimization method, projection in variable metric, rate of convergence.
Mots-clés : convergence 
@article{SVMO_2019_21_1_a2,
     author = {V. G. Malinov},
     title = {Continuous second order minimization method with variable metric projection operator},
     journal = {\v{Z}urnal Srednevol\v{z}skogo matemati\v{c}eskogo ob\^{s}estva},
     pages = {34--47},
     publisher = {mathdoc},
     volume = {21},
     number = {1},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SVMO_2019_21_1_a2/}
}
TY  - JOUR
AU  - V. G. Malinov
TI  - Continuous second order minimization method with variable metric projection operator
JO  - Žurnal Srednevolžskogo matematičeskogo obŝestva
PY  - 2019
SP  - 34
EP  - 47
VL  - 21
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SVMO_2019_21_1_a2/
LA  - ru
ID  - SVMO_2019_21_1_a2
ER  - 
%0 Journal Article
%A V. G. Malinov
%T Continuous second order minimization method with variable metric projection operator
%J Žurnal Srednevolžskogo matematičeskogo obŝestva
%D 2019
%P 34-47
%V 21
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SVMO_2019_21_1_a2/
%G ru
%F SVMO_2019_21_1_a2
V. G. Malinov. Continuous second order minimization method with variable metric projection operator. Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 21 (2019) no. 1, pp. 34-47. http://geodesic.mathdoc.fr/item/SVMO_2019_21_1_a2/

[1] A. S. Antipin, “Continuous and iterative proceses with projection ana projection-type operators”, Voprosy kibernetiki. Vychislitelnie voprosy analiza bolshih sistem, AN SSSR Publ., Moscow, 1989, 5–43 (In Russ.)

[2] A. S. Antipin, “Minimization of convex functions on convex sets by differential equations”, Differential equations, 30:9 (1994), 1365–1375 (In Russ.) | MR | Zbl

[3] A. S. Antipin, F. P. Vasil'ev, “On continuous method of minimization in variable metric spaces”, Russian Mathematics, 39:12 (1995), 1–6 (In Russ.) | Zbl

[4] T. V. Amochkina, “Continuous second order variable metric gradient projection method”, Journal of Computational Mathematics and Mathematical Physics, 37:10 (1997), 1134–1142 (In Russ.) | MR

[5] V. G. Malinov, “Continuous projection variable metric second order minimization method”, Functionalniy analiz, 39 (2006), 53–64, UlPU Publ., Ulyanovsk (In Russ.)

[6] V. G. Malinov, “On the version of continuous projection second order variable metric method”, Zhurnal Srednevolzhskogo matematicheskogo obshchestva, 16:1 (2014), 121–134 (In Russ.) | Zbl

[7] V. G. Malinov, “Continuous second order minimization method with variable metric projection operator”, VIII Moscow International Conference on Operations Research(ORM2016): Moscow, October 17–22, 2016: Proceedings (Moscow, October 17-22, 2016), v. II, FRC CSC RAS Publ., Moscow, 48–50 (In Russ)

[8] V. G. Malinov, “PGTM with variable metric projection operator”, Zhurnal Srednevolzhskogo matematicheskogo obshchestva, 14:4 (2012), 44–56 (In Russ.)

[9] F. P. Vasil'ev, Numerical methods solution of Extremal problems, Nauka Publ., Moscow, 1988, 552 pp. (In Russ.)

[10] A. N. Kolmogorov, S. V. Fomin, Elements of function theory and functional analysis, Nauka Publ., Moscow, 1976, 544 pp. (In Russ.) | MR

[11] A. S. Antipin, Methods of nonlinear programming, based on direct and dual modifications Lagrange function, VNII systems research Publ., Moscow, 1979, 73 pp. (In Russ.)