An Implicit Smooth Conjugate Projection Gradient Algorithm for Optimization with Nonlinear Complementarity Constraints

Affiliation(s)

^{1}
Huarui College, Xinyang Normal University, Xinyang, China.

^{2}
School of Mathematics & Computational Science, Guilin University of Electronic Technology, Guilin, China.

^{3}
College of Science, Anhui University of Science and Technology, Huainan, China.

ABSTRACT

This paper discusses a special class of mathematical programs with equilibrium constraints. At first, by using a generalized complementarity function, the discussed problem is transformed into a family of general nonlinear optimization problems containing additional variable*μ*. Furthermore, combining the idea of penalty function, an auxiliary problem with inequality constraints is presented. And then, by providing explicit searching direction, we establish a new conjugate projection gradient method for optimization with nonlinear complementarity constraints. Under some suitable conditions, the proposed method is proved to possess global and superlinear convergence rate.

This paper discusses a special class of mathematical programs with equilibrium constraints. At first, by using a generalized complementarity function, the discussed problem is transformed into a family of general nonlinear optimization problems containing additional variable

KEYWORDS

Mathematical Programs with Equilibrium Constraints, Conjugate Projection Gradient, Global Convergence, Superlinear Convergence

Mathematical Programs with Equilibrium Constraints, Conjugate Projection Gradient, Global Convergence, Superlinear Convergence

Cite this paper

Zhang, C. , Sun, L. , Zhu, Z. and Fang, M. (2015) An Implicit Smooth Conjugate Projection Gradient Algorithm for Optimization with Nonlinear Complementarity Constraints.*Applied Mathematics*, **6**, 1712-1726. doi: 10.4236/am.2015.610152.

Zhang, C. , Sun, L. , Zhu, Z. and Fang, M. (2015) An Implicit Smooth Conjugate Projection Gradient Algorithm for Optimization with Nonlinear Complementarity Constraints.

References

[1] Kocvara, M. and Outrata, J. (1994) On Optimization Systems Govern by Implicit Complementarity Problems. Numerical Functional Analysis and Optimization, 15, 869-887.

http://dx.doi.org/10.1080/01630569408816597

[2] Fletcher, R., Leyffer, S., Ralph, D. and Scholtes, S. (2006) Local Convergence of SQP Methods for Mathematical Programs with Equilibrium Constraints. SIAM: SIAM Journal on Optimization, 17, 259-286.

http://dx.doi.org/10.1137/S1052623402407382

[3] Luo, Z.Q., Pang, J.S. and Ralph, D. (1996) Mathmetical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge.

http://dx.doi.org/10.1017/CBO9780511983658

[4] Jiang, H. (2000) Smooth SQP Methods for Mathematical Programs with Nonlinear Complementarity Constaints. SIAM Journal of Optimization, 10, 779-808.

http://dx.doi.org/10.1137/S1052623497332329

[5] Kocvara, M. and Outrata, J. (1995) A Nonsmmoth Approach to Optimization Problems with Equilibrium Constraints. In: Ierns, M.C. and Pang, J.D., Eds., Proceedings of the International Conference on Complementatity Problems, SIAM Publications, Baltimore, 148-164.

[6] Outrata, J. (1990) On the Numberical Solution of a Class of Stachelberg Problems. Zeitschrift for Operations Research, 4, 255-278.

[7] Zhang, C., Zhu, Z.B., Chen, F.H. and Fang, M.L. (2010) Sequential System of Linear Equations Algorithm for Optimization with Complementary Constraints. Mathematics Modelling and Applied Computing, 1, 71-80.

[8] Zhang, C., Zhu, Z.B. and Fang, M.L. (2010) A Superlinearly Covergent SSLE Algorithm for Optimization Problems with Linear Complementarity Constraints. Journal of Mathematical Science: Advance and Application, 6, 149-164.

[9] Huang, Z.H., Lin, G.H. and Xiu, N.H. (2014) Several Developments of Variational Inequalities and Complementarity Problems, Bilevel Programming and MPEC. Operations Research Transactions, 18, 113-133.

[10] Zhang, C., Sun, L.M., Zhu, Z.B. and Fang, M.L. (2015) Levenberg-Marquardt Method for Mathematical Programs with Linearly Complementarity Constraints. American Journal of Computational Mathematics, 5, 239-242.

http://dx.doi.org/10.4236/ajcm.2015.53020

[11] Outrata, J., Kocvara, M. and Zowe, J. (1998) Nonsmmoth Approach to Aptimization Problems with Equilibrium Constraints. Kluwer Academic Publishers, The Netherland.

http://dx.doi.org/10.1007/978-1-4757-2825-5

[12] Gao, Z.Y., He, G.P. and Wu, F. (2004) Sequential Systems of Linear Equations Algorithm for Nonlinear Optimization Problems—General Constrained Problems. Applied Mathematics and Computation, 147, 211-226.

http://dx.doi.org/10.1016/S0096-3003(02)00662-8

[13] Jian, J.B., Qin, Y. and Liang, Y.M. (2007) A Generalized Strongly Sub-Feasible Algorithm for Mathematical Problems with Nonliear Complementarity Constraints. Numerical Mathematics: A Journal of Chinese Universities, 29, 15-27.

[14] Zhu, Z.B. and Zhang, K.C. (2004) A New Conjugate Projuction Gradient Method and Superlinear Convergence. Acta Mathematicae Applicatae Sinica, 27, 149-161.

[15] Yuan, Y.X. and Sun, W.Y. (1997) Optimization Theory and Method. Science Press, Beijing.

[1] Kocvara, M. and Outrata, J. (1994) On Optimization Systems Govern by Implicit Complementarity Problems. Numerical Functional Analysis and Optimization, 15, 869-887.

http://dx.doi.org/10.1080/01630569408816597

[2] Fletcher, R., Leyffer, S., Ralph, D. and Scholtes, S. (2006) Local Convergence of SQP Methods for Mathematical Programs with Equilibrium Constraints. SIAM: SIAM Journal on Optimization, 17, 259-286.

http://dx.doi.org/10.1137/S1052623402407382

[3] Luo, Z.Q., Pang, J.S. and Ralph, D. (1996) Mathmetical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge.

http://dx.doi.org/10.1017/CBO9780511983658

[4] Jiang, H. (2000) Smooth SQP Methods for Mathematical Programs with Nonlinear Complementarity Constaints. SIAM Journal of Optimization, 10, 779-808.

http://dx.doi.org/10.1137/S1052623497332329

[5] Kocvara, M. and Outrata, J. (1995) A Nonsmmoth Approach to Optimization Problems with Equilibrium Constraints. In: Ierns, M.C. and Pang, J.D., Eds., Proceedings of the International Conference on Complementatity Problems, SIAM Publications, Baltimore, 148-164.

[6] Outrata, J. (1990) On the Numberical Solution of a Class of Stachelberg Problems. Zeitschrift for Operations Research, 4, 255-278.

[7] Zhang, C., Zhu, Z.B., Chen, F.H. and Fang, M.L. (2010) Sequential System of Linear Equations Algorithm for Optimization with Complementary Constraints. Mathematics Modelling and Applied Computing, 1, 71-80.

[8] Zhang, C., Zhu, Z.B. and Fang, M.L. (2010) A Superlinearly Covergent SSLE Algorithm for Optimization Problems with Linear Complementarity Constraints. Journal of Mathematical Science: Advance and Application, 6, 149-164.

[9] Huang, Z.H., Lin, G.H. and Xiu, N.H. (2014) Several Developments of Variational Inequalities and Complementarity Problems, Bilevel Programming and MPEC. Operations Research Transactions, 18, 113-133.

[10] Zhang, C., Sun, L.M., Zhu, Z.B. and Fang, M.L. (2015) Levenberg-Marquardt Method for Mathematical Programs with Linearly Complementarity Constraints. American Journal of Computational Mathematics, 5, 239-242.

http://dx.doi.org/10.4236/ajcm.2015.53020

[11] Outrata, J., Kocvara, M. and Zowe, J. (1998) Nonsmmoth Approach to Aptimization Problems with Equilibrium Constraints. Kluwer Academic Publishers, The Netherland.

http://dx.doi.org/10.1007/978-1-4757-2825-5

[12] Gao, Z.Y., He, G.P. and Wu, F. (2004) Sequential Systems of Linear Equations Algorithm for Nonlinear Optimization Problems—General Constrained Problems. Applied Mathematics and Computation, 147, 211-226.

http://dx.doi.org/10.1016/S0096-3003(02)00662-8

[13] Jian, J.B., Qin, Y. and Liang, Y.M. (2007) A Generalized Strongly Sub-Feasible Algorithm for Mathematical Problems with Nonliear Complementarity Constraints. Numerical Mathematics: A Journal of Chinese Universities, 29, 15-27.

[14] Zhu, Z.B. and Zhang, K.C. (2004) A New Conjugate Projuction Gradient Method and Superlinear Convergence. Acta Mathematicae Applicatae Sinica, 27, 149-161.

[15] Yuan, Y.X. and Sun, W.Y. (1997) Optimization Theory and Method. Science Press, Beijing.