3-1 Piecewise NCP Function for New Nonmonotone QP-Free Infeasible Method

Ailan Liu; Dingguo Pu

doi:10.20965/jrm.2014.p0566

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JRM Vol.26 No.5 pp. 566-572

doi: 10.20965/jrm.2014.p0566

(2014)

Paper:

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3-1 Piecewise NCP Function for New Nonmonotone QP-Free Infeasible Method

Ailan Liu^*,** and Dingguo Pu^*,***

^*Department of Mathematics, Tongji University, No.1239, Siping Road, Shanghai 200092, China

^**School of Mathematics and Physics, Shanghai University of Electric Power, No.2588, Changyang Road, Yangpu District, Shanghai, China

^***School of Mathematics and Statistics, Henan University of Science and Technology, No.263, Kaiyuan Road, Luoyang 471023, China

Received:

April 13, 2014

Accepted:

July 16, 2014

Published:

October 20, 2014

Keywords:

QP-free method, NCP function, nonmonotone, convergence

Abstract

Algorithm flow chart

We propose a nonmonotone QP-free infeasible method for inequality-constrained nonlinear optimization problems based on a 3-1 piecewise linear NCP function. This nonmonotone QP-free infeasible method is iterative and is based on nonsmooth reformulation of KKT first-order optimality conditions. It does not use a penalty function or a filter in nonmonotone line searches. This algorithm solves only two systems of linear equations with the same nonsingular coefficient matrix, and is implementable and globally convergent without a linear independence constraint qualification or a strict complementarity condition. Preliminary numerical results are presented.

Cite this article as:

A. Liu and D. Pu, “3-1 Piecewise NCP Function for New Nonmonotone QP-Free Infeasible Method,” J. Robot. Mechatron., Vol.26 No.5, pp. 566-572, 2014.

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References

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[2] C. H. Kim, H. Tsujino, and S. Sugano, “Rapid short-time path planning for phase space,” J. of Robotics and Mechatronics, Vol.23, No.2, pp. 271-280. 2011.
[3] P. Di, J. Huang, S. Nakagawa et al., “Optimized motion control of an intelligent cane robot for easing muscular fatigue in the elderly during walking,” J. of Robotics and Mechatronics, Vol.25, No.6, pp. 1070-1077. 2013.
[4] N. I. M. Gould and P. L. Toint, “Nonlinear programming without a penalty function and a filter,” Math. Program., Vol.122, pp. 155-196, 2010.
[5] X. W. Liu and Y. X. Yuan, “A sequential quadratic programming method a penalty function and a filter for equality constrained nonlinear programming,” SIAM J. on Optimization, Vol.21, No.2, pp. 545-571, 2011.
[6] D. G. Pu, A. L. Liu, Y. L. Shang et al., “QP-free infeasible method without a penalty function and a filter,” Operations Research Trans., Vol.17, No.1, pp. 106-116, 2013.
[7] Y. Zhou and D. G. Pu, “Piecewise linear NCP function for QP-free feasible method,” J. Appl. Math., Vol.21, No.3, pp. 289-301, 2006.
[8] H. D. Qi and L. Q. Qi, “A new QP-free, globally convergent, locally superlinearly convergent algorithm for inequality constrained optimization,” SIAM J Optim, Vol.11, pp. 113-132, 2000.
[9] L. Q. Qi and Y. F. Yang, “A globally and superlinearly convergent QP-free for nonlinear constrained optimization,” J. Optim. Theory Appl., Vol.113, pp. 297-323, 2002.
[10] Z. B. Zhu and J. B. Jin, “An improved feasible QP-free algorithm for inequality constrained optimization,” Acta. Math. Sinica, Vol.28, No.12, pp. 2475-2488, 2012.
[11] E. R. Panier, A. L. Tits, and J. N. Herskovits, “A QP-free, globally, locally superlinear convergent method for the inequality constrained optimization problems,” SIAM J. Control Optim., Vol.36, pp. 788-811, 1988.
[12] M. Ulbrich and S. Ulbrich, “Non-monotone trust region methods for nonlinear equality constrained optimization without a penalty function,” Math Program, Vol.95, pp. 103-135, 2003.
[13] W. Hock and K. Schittkowski, “Test examples for nonlinear programming codesLecture Notes in Economics and Mathematical Systems,” Vol.187, Springer-Verlag, Berlin Heidelberg New York, 1981.

This article is published under a Creative Commons Attribution-NoDerivatives 4.0 Internationa License.

[1] [1] M. Yokozuka and O. Matsumoto, “A reasonable path planning via path energy minimization,” J. of Robotics and Mechatronics, Vol.26, No.2, pp. 236-244, 2014.

[2] [2] C. H. Kim, H. Tsujino, and S. Sugano, “Rapid short-time path planning for phase space,” J. of Robotics and Mechatronics, Vol.23, No.2, pp. 271-280. 2011.

[3] [3] P. Di, J. Huang, S. Nakagawa et al., “Optimized motion control of an intelligent cane robot for easing muscular fatigue in the elderly during walking,” J. of Robotics and Mechatronics, Vol.25, No.6, pp. 1070-1077. 2013.

[4] [4] N. I. M. Gould and P. L. Toint, “Nonlinear programming without a penalty function and a filter,” Math. Program., Vol.122, pp. 155-196, 2010.

[5] [5] X. W. Liu and Y. X. Yuan, “A sequential quadratic programming method a penalty function and a filter for equality constrained nonlinear programming,” SIAM J. on Optimization, Vol.21, No.2, pp. 545-571, 2011.

[6] [6] D. G. Pu, A. L. Liu, Y. L. Shang et al., “QP-free infeasible method without a penalty function and a filter,” Operations Research Trans., Vol.17, No.1, pp. 106-116, 2013.

[7] [7] Y. Zhou and D. G. Pu, “Piecewise linear NCP function for QP-free feasible method,” J. Appl. Math., Vol.21, No.3, pp. 289-301, 2006.

[8] [8] H. D. Qi and L. Q. Qi, “A new QP-free, globally convergent, locally superlinearly convergent algorithm for inequality constrained optimization,” SIAM J Optim, Vol.11, pp. 113-132, 2000.

[9] [9] L. Q. Qi and Y. F. Yang, “A globally and superlinearly convergent QP-free for nonlinear constrained optimization,” J. Optim. Theory Appl., Vol.113, pp. 297-323, 2002.

[10] [10] Z. B. Zhu and J. B. Jin, “An improved feasible QP-free algorithm for inequality constrained optimization,” Acta. Math. Sinica, Vol.28, No.12, pp. 2475-2488, 2012.

[11] [11] E. R. Panier, A. L. Tits, and J. N. Herskovits, “A QP-free, globally, locally superlinear convergent method for the inequality constrained optimization problems,” SIAM J. Control Optim., Vol.36, pp. 788-811, 1988.

[12] [12] M. Ulbrich and S. Ulbrich, “Non-monotone trust region methods for nonlinear equality constrained optimization without a penalty function,” Math Program, Vol.95, pp. 103-135, 2003.

[13] [13] W. Hock and K. Schittkowski, “Test examples for nonlinear programming codesLecture Notes in Economics and Mathematical Systems,” Vol.187, Springer-Verlag, Berlin Heidelberg New York, 1981.

3-1 Piecewise NCP Function for New Nonmonotone QP-Free Infeasible Method

Ailan Liu*,** and Dingguo Pu*,***

Ailan Liu^*,** and Dingguo Pu^*,***