A modified constraint shifting homotopy method for solving general nonlinear multiobjective programming
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Authors
Zhichuan Zhu
- School of Statistics, Jilin University of Finance and Economics, Changchun, Jilin 130117, China.
- School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin 130024, China.
Yonghong Yao
- School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin 130024, China.
Abstract
In this paper, for solving the general nonconvex multiobjective
programming with both inequality and equality constraints, a
modified constraint shifting homotopy is constructed, and the
existence and global convergence of the smooth homotopy pathway is
proven for any initial point in the almost Euclidean space under
some mild conditions. The advantage of the newly proposed method
requires that the initial point can be chosen much more
conveniently, which needs to be only in the shifted feasible set not
necessarily in the original feasible set. Meanwhile, the normal cone
condition for proving the global convergence, which is much weaker than
the existing interior method, need only be satisfied at the boundary
of the shifted feasible set but not the original constraint set.
Share and Cite
ISRP Style
Zhichuan Zhu, Yonghong Yao, A modified constraint shifting homotopy method for solving general nonlinear multiobjective programming, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 9, 4687--4694
AMA Style
Zhu Zhichuan, Yao Yonghong, A modified constraint shifting homotopy method for solving general nonlinear multiobjective programming. J. Nonlinear Sci. Appl. (2017); 10(9):4687--4694
Chicago/Turabian Style
Zhu, Zhichuan, Yao, Yonghong. "A modified constraint shifting homotopy method for solving general nonlinear multiobjective programming." Journal of Nonlinear Sciences and Applications, 10, no. 9 (2017): 4687--4694
Keywords
- Homotopy method
- nonconvex programming
- multiobjective programming
- global convergence.
MSC
References
-
[1]
E. L. Allgower, K. Georg , Numerical Continuation Methods: An Introduction, Springer, Berlin (1990)
-
[2]
T. Antczak, A. Pitea , Parametric approach to multitime multiobjective fractional variational problems under \((F, \rho)\)- convexity, Optim. Control Appl. Meth., 37 (2016), 831–847.
-
[3]
S. N. Chow, J. Mallet-Paret, J. A. Yorke, Finding zeros of maps: homotopy methods that are constructive with probability one, Math. Comp., 32 (1978), 887–899.
-
[4]
G. Isac, M. M. Kosteva, M. M. Wiecek, Multiple-objective appproximation of feasible but unsolvable linear complementarity problems, J. Optim. Theory Appl., 86 (1995), 389–405.
-
[5]
A. Jayswal, S. Choudhury, An exact \(l_1\) exponential penalty function method for multiobjective optimization problems with exponential-type invexity, J. Oper. Res. Soc. China, 2 (2014), 75–91.
-
[6]
R. B. Kellogg, T. Y. Li, J. Yorke, A constructive proof the Brouwer fixed-point theorem and computational results, SIAM J. Numer. Anal., 13 (1976), 473–483.
-
[7]
H. Konno, T. Kuno, Linear multiplicative programming, Math. Programming, 56 (1992), 51–64.
-
[8]
T. C. Koopmans, Activity Analysis of Production and Allocation, Wiley, New York (1951)
-
[9]
M. M. Kosterva, M. M. Wiecek , Linear complementarity problems and multiple objective programming, Math. Programming, 60 (1993), 349–359.
-
[10]
Q. H. Liu, Z. H. Lin, Homotopy interior point method of solving weak efficient solutions for multiobjective programming problems, Acta Math. Appl. Sin. Engl. Ser., 23 (2000), 188–195.
-
[11]
Z.-H. Lin, B. Yu, G.-C. Feng, A combined homotopy interior point method for convex nonlinear programming, Appl. Math. Comput., 84 (1997), 193–211.
-
[12]
Z. H. Lin, D. L. Zhu, Z. P. Sheng, Finding a minimal efficient solution of a convex multiobjective program, J. Optim. Theory Appl., 118 (2003), 587–600.
-
[13]
V. Pareto , Course d’Economic Politique , Librairie Droz, Switzerland (1964)
-
[14]
V. M. Perez, J. E. Renaud, L. T. Watson, Homotopy curve tracking in approximate interior point optimization, Optim. Eng., 10 (2009), 91–108.
-
[15]
A. Pitea, T. Antczak , Proper efficiency and duality for a new class of nonconvex multitime multiobjective variational problems, J. Inequal. Appl., 2014 (2014), 20 pages.
-
[16]
A. Pitea, M. Postolache , Duality theorems for a new class of multitime multiobjective variational problems, J. Global Optim., 54 (2012), 47–58.
-
[17]
A. Pitea, M. Postolache, Minimization of vectors of curvilinear functionals on the second order jet bundle Necessary conditions, Optim. Lett., 6 (2012), 459–470.
-
[18]
A. Pitea, M. Postolache , Minimization of vectors of curvilinear functionals on the second order jet bundle Sufficient efficiency conditions, Optim. Lett., 6 (2012), 1657–1669.
-
[19]
L. Qi, Z.-X. Wei, On the constant positive linear dependence condition and its application to SQP methods, SIAM J. Optim., 10 (2000), 963–981.
-
[20]
S. Smale, A convergent process of price adjustment and global newton methods, J. Math. Econom., 3 (1976), 107–120.
-
[21]
W. Song, G. M. Yao, Homotopy Method for a General Multiobjective Programming Problem, J. Optim. Theory Appl., 138 (2008), 139–153.
-
[22]
J. Von Neumann, O. Morgenstern, Theory of Games and Economic Behavior, Princeton University Press, New Jersey (1944)
-
[23]
Y. H. Yang, X. R. Lu, Q. H. Liu , Infeasible interior-point homotopy method for non-convex programming under normal cone condition , J. Jilin Univ. Sci., 45 (2007), 365–368.
-
[24]
L. Yang, B. Yu, Q. Xu, A constraint shifting homotopy method for general non-linear programming, Optimization, 63 (2014), 585–600.
-
[25]
B. Yu, Y. Shang, Boundary moving combined homotopy method for nonconvex nonlinear programming, J. Math. Res. Expos., 26 (2006), 831–834.
-
[26]
X. Zhao, S. G. Zhang, Q. H. Liu , Homotopy Interior-Point Method for a General Multiobjective Programming Problem, J. Appl. Math., 2012 (2012), 12 pages.
-
[27]
Z.-C. Zhu, H.-J. Xiong, A constraint shifting homotopy method for finding a minimal efficient solution of nonconvex multiobjective programming, Optim. Methods Softw., 30 (2015), 634–642.
-
[28]
, , , (),
-
[29]
Z.-C. Zhu, B. Yu, A modified homotopy method for solving the principal-agent bilevel programming problem , Comput. Appl. Math., 2016 (2016), 26 pages.
-
[30]
Z.-C. Zhu, B. Yu, Globally convergent homotopy algorithm for solving the KKT systems to the principal agent bilevel programming, Optim. Methods Softw., 32 (2017), 69–85.
-
[31]
Z.-C. Zhu, B. Yu, Y.-F. Shang, A modified homotopy method for solving nonconvex fixed points problems, Fixed Point Theory, 14 (2013), 531–544.
-
[32]
Z.-C. Zhu, B. Yu, L. Yang, Globally convergent homotopy method for designing piecewise linear deterministic contractual function, J. Ind. Manag. Optim., 10 (2014), 717–741.
-
[33]
Z. Zhu, Z. Zhou, Y.-C. Liou, Y.-H. Yao, Y. Xing , A globally convergent method for computing fixed point of self-mapping on general nonconvex set, J. Nonlinear Convex Anal., 18 (2017), 1067–1078.