×

zbMATH — the first resource for mathematics

Modified projection method for solving a system of monotone equations with convex constraints. (English) Zbl 1225.90128
Summary: In this paper, we propose a modified projection method for solving a system of monotone equations with convex constraints. At each iteration of the method, we first solve a system of linear equations approximately, and then perform a projection of the initial point onto the intersection set of the feasible set and two half spaces containing the current iterate to obtain the next one. The iterate sequence generated by the proposed algorithm possesses an expansive property with regard to the initial point. Under mild condition, we show that the proposed algorithm is globally convergent. Preliminary numerical experiments are also reported.

MSC:
90C30 Nonlinear programming
15A06 Linear equations (linear algebraic aspects)
Software:
STRSCNE; levmar
PDF BibTeX Cite
Full Text: DOI
References:
[1] Bellavia, S., Macconi, M., Morini, B.: An affine scaling trust-region approach to bound-constrained nonlinear systems. Appl. Numer. Math. 44, 257–280 (2003) · Zbl 1018.65067
[2] Calamai, P.H., More, J.J.: Projected gradient methods for linearly constrained problems. Math. Program. 39, 93–116 (1987) · Zbl 0634.90064
[3] Gafni, E.M., Bertsekas, D.P.: Two-metric projection methods for constrained optimization. SIAM J. Control Optim. 22, 936–964 (1984) · Zbl 0555.90086
[4] Gabriel, S.A., Pang, J.S.: A trust region method for constrained nonsmooth equations. In: Hager, W.W., Hearn, D.W., Pardalos, P.M. (eds.) Large Scale Optimization: State of the Art, pp. 155–181. Kluwer, Dordrecht (1984) · Zbl 0813.65091
[5] Kanzow, C., Yamashita, N., Fukushima, M.: Levenberg-Marquardt methods with strong local convergence properties for solving nonlinear equations with convex constraints. J. Comput. Appl. Math. 172, 375–397 (2004) · Zbl 1064.65037
[6] Maranas, C.D., Floudas, C.A.: Finding all solutions of nonlinearly constrained systems of equations. J. Global Optim. 7, 143–182 (1995) · Zbl 0841.90115
[7] Solodov, M.V., Svaiter, B.F.: A new projection method for variational inequality problems. SIAM J. Control. Optim. 37, 765–776 (1999) · Zbl 0959.49007
[8] Tong, X.J., Qi, L.: On the convergence of a trust-region method for solving constrained nonlinear equations with degenerate solution. J. Optim. Theory. Appl. 123, 187–211 (2004) · Zbl 1069.65055
[9] Wang, C.W., Wang, Y.J., Xu, C.L.: A projection method for a system of nonlinear monotone equations with convex constraints. Math. Method Oper. Res. 66, 33–46 (2007) · Zbl 1126.90067
[10] Wang, Y.J., Wang, C.W., Xiu, N.H.: A family of supermemory gradient projection methods for constrained optimization. Optim. 51, 889–905 (2002) · Zbl 1145.90465
[11] Wang, Y.J., Xiu, N.H., Zhang, J.Z.: Modified extragradient method for variational inequalities and verification of solution existence. J. Optim. Theory. Appl. 119, 167–183 (2003) · Zbl 1045.49017
[12] Xiu, N.H., Zhang, J.Z.: Some recent advances in projection-type methods for variational inequalities. J. Comput. Appl. Math. 152, 559–585 (2003) · Zbl 1018.65083
[13] Zarantonello, E.H.: Projections on Convex Sets in Hilbert Space and Spectral Theory. Academic Press, New York (1971) · Zbl 0281.47043
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.