A Variable Metric Algorithm with Broyden Rank One Modifications for Nonlinear Equality Constraints Optimization

Affiliation(s)

School of Electronic Engineering and Automation, Guilin University of Electronic Technology, Guilin, China.

School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin, China.

School of Electronic Engineering and Automation, Guilin University of Electronic Technology, Guilin, China.

School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin, China.

ABSTRACT

In this paper, a variable metric algorithm is proposed with Broyden rank one modifications for the equality constrained optimization. This method is viewed expansion in constrained optimization as the quasi-Newton method to unconstrained optimization. The theoretical analysis shows that local convergence can be induced under some suitable conditions. In the end, it is established an equivalent condition of superlinear convergence.

Cite this paper

C. Hu and Z. Zhu, "A Variable Metric Algorithm with Broyden Rank One Modifications for Nonlinear Equality Constraints Optimization,"*Open Journal of Optimization*, Vol. 2 No. 1, 2013, pp. 33-37. doi: 10.4236/ojop.2013.21005.

C. Hu and Z. Zhu, "A Variable Metric Algorithm with Broyden Rank One Modifications for Nonlinear Equality Constraints Optimization,"

References

[1] M. J. D. Powell, “A Fast Algorithm for Nonlinearly Constrained Optimization Calculations,” In: G. A. Watson, Ed., Numerical Analysis, Springer-Verlag, Berlin, 1978, pp. 144-157. doi:10.1007/BFb0067703

[2] S.-P. Han, “Superlinearly Convergent Variable Metric Algorithm for General Nolinear Programming Problem,” Mathematical Programming, Vol. 11, No. 1, 1976, pp. 263-282. doi:10.1007/BF01580395

[3] D. Q. Mayne and E. Polak, “A Superlinearly Convergent Algorithm for Constrained Optimization Problems,” Mathematical Programming Studies, Vol. 16, 1982, pp. 45- 61. doi:10.1007/BFb0120947

[4] J.-B. Jian, C.-M. Tang, Q.-J. Hu and H.-Y. Zheng, “A Feasible Descent SQP Algorithm for General Constrained Optimization without Strict Complementarity,” Journal of Computational and Applied Mathematics, Vol. 180, No. 2, 2005, pp. 391-412. doi:10.1016/j.cam.2004.11.008

[5] Z. Jin and Y. Q. Wang, “A Simple Feasible SQP Method for Inequality Constrained Optimization with Global and Superlinear Convergence,” Journal of Computational and Applied Mathematics, Vol. 233, No. 1, 2010, pp. 3060- 3073. doi:10.1016/j.cam.2009.11.061

[6] Y.-F. Yang, D.-H. Li and L. Q. Qi, “A Feasible Sequential Linear Equation Method for Inequality Constrained Optimization,” SIAM Journal on Optimization, Vol. 13, No. 4, 2003, pp. 1222-1244. doi:10.1137/S1052623401383881

[7] Z. B. Zhu, “An Interior Point Type QP-Free Algorithm with Superlinear Convergence for Inequality Constrained Optimization,” Applied Mathematical Modelling, Vol. 31, No. 6, 2007, pp. 1201-1212. doi:10.1016/j.apm.2006.04.019

[8] J. B. Jian, D. L. Han and Q. J. Xu, “A New Sequential Systems of Linear Equations Algorithm of Feasible Descent for Inequality Constrained Optimization,” Acta Mathematica Sinica, English Series, Vol. 26, No. 12, 2010, pp. 2399-2420. doi:10.1007/s10114-010-7432-0

[9] Y. Yuan and W. Sun, “Theory and Methods of Optimization,” Science Press, Beijing, 1997.

[1] M. J. D. Powell, “A Fast Algorithm for Nonlinearly Constrained Optimization Calculations,” In: G. A. Watson, Ed., Numerical Analysis, Springer-Verlag, Berlin, 1978, pp. 144-157. doi:10.1007/BFb0067703

[2] S.-P. Han, “Superlinearly Convergent Variable Metric Algorithm for General Nolinear Programming Problem,” Mathematical Programming, Vol. 11, No. 1, 1976, pp. 263-282. doi:10.1007/BF01580395

[3] D. Q. Mayne and E. Polak, “A Superlinearly Convergent Algorithm for Constrained Optimization Problems,” Mathematical Programming Studies, Vol. 16, 1982, pp. 45- 61. doi:10.1007/BFb0120947

[4] J.-B. Jian, C.-M. Tang, Q.-J. Hu and H.-Y. Zheng, “A Feasible Descent SQP Algorithm for General Constrained Optimization without Strict Complementarity,” Journal of Computational and Applied Mathematics, Vol. 180, No. 2, 2005, pp. 391-412. doi:10.1016/j.cam.2004.11.008

[5] Z. Jin and Y. Q. Wang, “A Simple Feasible SQP Method for Inequality Constrained Optimization with Global and Superlinear Convergence,” Journal of Computational and Applied Mathematics, Vol. 233, No. 1, 2010, pp. 3060- 3073. doi:10.1016/j.cam.2009.11.061

[6] Y.-F. Yang, D.-H. Li and L. Q. Qi, “A Feasible Sequential Linear Equation Method for Inequality Constrained Optimization,” SIAM Journal on Optimization, Vol. 13, No. 4, 2003, pp. 1222-1244. doi:10.1137/S1052623401383881

[7] Z. B. Zhu, “An Interior Point Type QP-Free Algorithm with Superlinear Convergence for Inequality Constrained Optimization,” Applied Mathematical Modelling, Vol. 31, No. 6, 2007, pp. 1201-1212. doi:10.1016/j.apm.2006.04.019

[8] J. B. Jian, D. L. Han and Q. J. Xu, “A New Sequential Systems of Linear Equations Algorithm of Feasible Descent for Inequality Constrained Optimization,” Acta Mathematica Sinica, English Series, Vol. 26, No. 12, 2010, pp. 2399-2420. doi:10.1007/s10114-010-7432-0

[9] Y. Yuan and W. Sun, “Theory and Methods of Optimization,” Science Press, Beijing, 1997.