Fixed-Point Iteration Method for Solving the Convex Quadratic Programming with Mixed Constraints

Affiliation(s)

Department of Mathematics & Physics, Beijing Institute of Petrochemical Technology, Beijing, China.

Department of Mathematics & Physics, Beijing Institute of Petrochemical Technology, Beijing, China.

ABSTRACT

The present paper is devoted to a novel smoothing function method for convex quadratic programming problem with mixed constrains, which has important application in mechanics and engineering science. The problem is reformulated as a system of non-smooth equations, and then a smoothing function for the system of non-smooth equations is proposed. The condition of convergences of this iteration algorithm is given. Theory analysis and primary numerical results illustrate that this method is feasible and effective.

KEYWORDS

Fixed-Point Iteration; Convex Quadratic Programming Problem; Convergence; Smoothing Function

Fixed-Point Iteration; Convex Quadratic Programming Problem; Convergence; Smoothing Function

Cite this paper

R. Wang, H. Shi, K. Ruan and X. Gao, "Fixed-Point Iteration Method for Solving the Convex Quadratic Programming with Mixed Constraints,"*Applied Mathematics*, Vol. 5 No. 2, 2014, pp. 256-262. doi: 10.4236/am.2014.52027.

R. Wang, H. Shi, K. Ruan and X. Gao, "Fixed-Point Iteration Method for Solving the Convex Quadratic Programming with Mixed Constraints,"

References

[1] D. Goldfarb and A. Idinabi, “A Numerical Stable Dual Method for Solving Strictly Convex Quadratic Programs,” Mathematical Programming, Vol. 27, No. 2, 1983, pp. 1-33.

http://dx.doi.org/10.1007/BF02591962

[2] Y. Ye and E. Tse, “An Extension of Kamarker’s Algorithm to Convex Quadratic Programming,” Mathematical Programming, Vol. 47, No. 4, 1989, pp. 157-179.

http://dx.doi.org/10.1007/BF01587086

[3] J. L. Zhang and X. S. Zhang, “A Predictor-Corrector Method for Convex Quadratic Programming,” Journal of System Science and Mathematical Science, Vol. 23, No. 3, 2003, pp. 353-366.

[4] R. D. Monteiro, I. Adler and M. G. Resende, “A Polynomial-Time Primal-Dual Affine Scaling Algorithm for Linear and Convex Quadratic Programming and Its Powerseries Extension,” Mathematics of Operations Research, Vol. 15, No. 2, 1990, pp. 191-214. http://dx.doi.org/10.1287/moor.15.2.191

[5] M. W. Zhang and C. C. Huang, “A Primal-Dual Infeasible Interior Point Algorithm for Convex Quadratic Programming Problem with Box Constraints,” Journal of Engineering Mathematics, Vol. 18, No. 2, 2001, pp. 85-90.

[6] Y. X. Yuan and W. Y. Sun, “Optimization Theory and Method,” Science Press, 2002, pp. 111-117.

[7] R. P. Wang, “Fixed Iterative Method for Solving the Equality Constrained Convex Quadratic Programming Problem,” Journal of Beijing Institute of Petro-Chemical Technology, Vol. 16, No. 3, 2008, pp. 64-66.

[8] R. P. Wang, “Fixed Iterative Method for Solving the Inequality Constrained Quadratic Programming Problem,” Journal of Beijing Institute of Petro-Chemical Technology, Vol. 15, No. 1, 2007, pp. 1-4.

[9] X. S. Li, “An Effective Algorithm for Non-differentiable Problem,” Science in China (Series A), Vol. 23, No. 3, 1994, pp. 353366.

[10] G. Q. Chen and Y. Q. Chen, “An Entropy Function Method for Nonlinear Complementary Problems,” Acta Scientiarum Naturalium Universitatis NeiMongol, Vol. 31, No. 5, 2000, pp. 447-451.

[11] H. W. Tang and L. W. Zhang, “An Entropy Function Method for Nonlinear Programming Problems,” Chinese Science Bulletin, Vol. 39, No. 8, 1994, pp. 682-684.

[12] Q. Y. Li, Z. Z. Mo and L. Q. Qi, “Numerical Method for System of Nonlinear Equations,” Beijing Science Press, Beijing, 1999.

[1] D. Goldfarb and A. Idinabi, “A Numerical Stable Dual Method for Solving Strictly Convex Quadratic Programs,” Mathematical Programming, Vol. 27, No. 2, 1983, pp. 1-33.

http://dx.doi.org/10.1007/BF02591962

[2] Y. Ye and E. Tse, “An Extension of Kamarker’s Algorithm to Convex Quadratic Programming,” Mathematical Programming, Vol. 47, No. 4, 1989, pp. 157-179.

http://dx.doi.org/10.1007/BF01587086

[3] J. L. Zhang and X. S. Zhang, “A Predictor-Corrector Method for Convex Quadratic Programming,” Journal of System Science and Mathematical Science, Vol. 23, No. 3, 2003, pp. 353-366.

[4] R. D. Monteiro, I. Adler and M. G. Resende, “A Polynomial-Time Primal-Dual Affine Scaling Algorithm for Linear and Convex Quadratic Programming and Its Powerseries Extension,” Mathematics of Operations Research, Vol. 15, No. 2, 1990, pp. 191-214. http://dx.doi.org/10.1287/moor.15.2.191

[5] M. W. Zhang and C. C. Huang, “A Primal-Dual Infeasible Interior Point Algorithm for Convex Quadratic Programming Problem with Box Constraints,” Journal of Engineering Mathematics, Vol. 18, No. 2, 2001, pp. 85-90.

[6] Y. X. Yuan and W. Y. Sun, “Optimization Theory and Method,” Science Press, 2002, pp. 111-117.

[7] R. P. Wang, “Fixed Iterative Method for Solving the Equality Constrained Convex Quadratic Programming Problem,” Journal of Beijing Institute of Petro-Chemical Technology, Vol. 16, No. 3, 2008, pp. 64-66.

[8] R. P. Wang, “Fixed Iterative Method for Solving the Inequality Constrained Quadratic Programming Problem,” Journal of Beijing Institute of Petro-Chemical Technology, Vol. 15, No. 1, 2007, pp. 1-4.

[9] X. S. Li, “An Effective Algorithm for Non-differentiable Problem,” Science in China (Series A), Vol. 23, No. 3, 1994, pp. 353366.

[10] G. Q. Chen and Y. Q. Chen, “An Entropy Function Method for Nonlinear Complementary Problems,” Acta Scientiarum Naturalium Universitatis NeiMongol, Vol. 31, No. 5, 2000, pp. 447-451.

[11] H. W. Tang and L. W. Zhang, “An Entropy Function Method for Nonlinear Programming Problems,” Chinese Science Bulletin, Vol. 39, No. 8, 1994, pp. 682-684.

[12] Q. Y. Li, Z. Z. Mo and L. Q. Qi, “Numerical Method for System of Nonlinear Equations,” Beijing Science Press, Beijing, 1999.