Abstract
In this paper, a new projection method for solving a system of nonlinear equations with convex constraints is presented. Compared with the existing projection method for solving the problem, the projection region in this new algorithm is modified which makes an optimal stepsize available at each iteration and hence guarantees that the next iterate is more closer to the solution set. Under mild conditions, we show that the method is globally convergent, and if an error bound assumption holds in addition, it is shown to be superlinearly convergent. Preliminary numerical experiments also show that this method is more efficient and promising than the existing projection method.
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Calamai P.H., Moré J.J.: Projected gradient methods for linearly constrained problems. Math. Program. 39(1), 93–116 (1987)
Dirkse S.P., Ferris M.C.: MCPLIB: a collection of nonlinear mixed complementarity problems. Optim. Methods Software 5, 319–345 (1995)
El-Hawary M.E.: Optimal Power Flow: Solution Techniques, Requirement and Challenges. IEEE Service Center, Piscataway, NJ (1996)
Kanzow C., Yamashita N., Fukushima M.: Levenberg-Marquardt methods for constrained nonlinear equations with strong local convergence properties. J. Comput. Appl. Math. 172, 375–397 (2004)
Maranas C.D., Floudas C.A.: Finding all solutions of nonlinearly constrained systems of equations. J. Glob. Optim. 7(2), 143–182 (1995)
Meintjes K., Morgan A.P.: A methodology for solving chemical equilibrium systems. Appl. Math. Comput. 22, 333–361 (1987)
Solodov M.V., Svaiter B.F.: A new projection method for variational inequality problems. SIAM J. Control Optim. 37(3), 765–776 (1999)
Solodov M.V., Svaiter B.F.: A truly globally convergent Newton-type method for the monotone nonlinear complementarity problem. SIAM J. Optim. 10, 605–625 (2000)
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)
Wang Y.J., Xiu N.H., Zhang J.Z.: Unified framework of extragradient-type methods for pseudomonotone variational inequalities. J. Optim. Theory Appl. 111, 641–656 (2001)
Wang C.W., Wang Y.J., Xu C.L.: A projection method for a system of nonlinear equtions with convex constraints. Math. Methods Oper. Res. 66, 33–46 (2007)
Wood A.J., Wollenberg B.F.: Power Generations, Operations, and Control. Wiley, New York (1996)
Xiu N.H., Zhang J.Z.: Some recent advances in projection-type methods for variational inequalities. J. Comput. Appl. Math. 152, 559–585 (2003)
Xiu N.H., Wang C.Y., Zhang J.Z.: Convergence properties of projection and contraction methods for variational inequality problems. Appl. Math. Optim. 43, 147–168 (2001)
Yamashita N., Fukushima M.: On the rate of convergence of the Levenberg-Marquardt method. Computing (Suppl.) 15, 237–249 (2001)
Zarantonello E.H.: Projections on convex sets in Hilbert space and spectral theory, contributions to nonlinear functional analysis. In: Zarantonello, E.H. (eds) Contributions to Nonlinear Functional Analysis, Academic Press, New York, NY (1971)
Zhou G.L., Toh K.C.: Superlinear convergence of a Newton-type algorithm for monotone equations. J. Optim. Theory Appl. 125, 205–221 (2005)
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This work was done when Yiju Wang visited Chongqing Normal University.
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Wang, C., Wang, Y. A superlinearly convergent projection method for constrained systems of nonlinear equations. J Glob Optim 44, 283–296 (2009). https://doi.org/10.1007/s10898-008-9324-8
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DOI: https://doi.org/10.1007/s10898-008-9324-8