Geodesic
Non-convex minimization of a quadratic function on a sphere
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 18 (2015) no. 2, pp. 163-176 
E. A. Kotel'nikov. Non-convex minimization of a quadratic function on a sphere. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 18 (2015) no. 2, pp. 163-176. http://geodesic.mathdoc.fr/item/SJVM_2015_18_2_a4/
@article{SJVM_2015_18_2_a4,
     author = {E. A. Kotel'nikov},
     title = {Non-convex minimization of a~quadratic function on a~sphere},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
     pages = {163--176},
     year = {2015},
     volume = {18},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJVM_2015_18_2_a4/}
}
TY  - JOUR
AU  - E. A. Kotel'nikov
TI  - Non-convex minimization of a quadratic function on a sphere
JO  - Sibirskij žurnal vyčislitelʹnoj matematiki
PY  - 2015
SP  - 163
EP  - 176
VL  - 18
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/SJVM_2015_18_2_a4/
LA  - ru
ID  - SJVM_2015_18_2_a4
ER  - 
%0 Journal Article
%A E. A. Kotel'nikov
%T Non-convex minimization of a quadratic function on a sphere
%J Sibirskij žurnal vyčislitelʹnoj matematiki
%D 2015
%P 163-176
%V 18
%N 2
%U http://geodesic.mathdoc.fr/item/SJVM_2015_18_2_a4/
%G ru
%F SJVM_2015_18_2_a4

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

The minimization of convex functions on a sphere reduces to a sequence of problems minimizing its convex majorants on a sphere. To build majorants, the representation of the target function as a difference of convex quadratic functions and the solutions of the problem at the previous step is used. Representation of the target function in the form of a difference of convex quadratic functions is based on a modified procedure of decomposition of the Cholesky symmetric alternating-sign matrices.

[1] Dennis Dzh., Shnabel' R., Chislennye metody bezuslovnoj optimizatsii i resheniya nelinejnykh uravnenij, Mir, M., 1988 | MR

[2] Nechaeva M. S., KHamisov O. V., “Metod vetvej i granits dlya zadachi minimizatsii kvadratichnoj funktsii pri vypuklykh kvadratichnykh ogranicheniyakh”, Diskretnyj analiz i issledovanie operatsij. Seriya 2, 7:2 (2000), 74–88 | MR | Zbl

[3] Gay D. M., “Computing optimal locally constrained steps”, SIAM J. Sci. Comput., 2:2 (1981), 186–197 | DOI | MR | Zbl

[4] Ye Y., “On affine scaling algorithms for nonconvex quadratic programming”, Math. Programming, 56:3 (1992), 285–300 | MR | Zbl

[5] Hager W. W., “Minimizing a quadratic over a sphere”, SIAM J. Optim., 12:1 (2001), 188–208 | MR | Zbl

[6] Polyak B. T., Vvedenie v optimizatsiyu, Nauka, M., 1983 | MR

[7] Kotel'nikov E. A., Ob odnom sposobe ischerpyvaniya dlya simmetrichnykh matrits, Preprint No 1083, RAN. Sib. otd-nie. VTs, Novosibirsk, 1997 | MR

[8] Gill F., Myurrej U., Rajt M., Prakticheskaya optimizatsiya, Mir, M., 1985 | MR

[9] Kotel'nikov E. A., “Minimizatsiya kvadratichnoj funktsii na share”, Sib. zhurn. vychisl. matematiki (Novosibirsk), 17:4 (2014), 329–338