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July  2018, 14(3): 1041-1054. doi: 10.3934/jimo.2017089

Second-order optimality conditions for cone constrained multi-objective optimization

Liwei Zhang , , Jihong Zhang and  Yule Zhang

School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China

* Corresponding authorr: Liwei Zhang

Received  February 2017 Revised  June 2017 Published  July 2018 Early access  September 2017

Fund Project: Supported by the National Natural Science Foundation of China under project grant No. 11571059,11731013 and No. 91330206.

The aim of this paper is to develop second-order necessary and second-order sufficient optimality conditions for cone constrained multi-objective optimization. First of all, we derive, for an abstract constrained multi-objective optimization problem, two basic necessary optimality theorems for weak efficient solutions and a second-order sufficient optimality theorem for efficient solutions. Secondly, basing on the optimality results for the abstract problem, we demonstrate, for cone constrained multi-objective optimization problems, the first-order and second-order necessary optimality conditions under Robinson constraint qualification as well as the second-order sufficient optimality conditions under upper second-order regularity for the conic constraint. Finally, using the optimality conditions for cone constrained multi-objective optimization obtained, we establish optimality conditions for polyhedral cone, second-order cone and semi-definite cone constrained multi-objective optimization problems.

Keywords: Cone constrained multi-objective optimization, second-order optimality conditions, polyhedral cone, second-order cone, semi-definite cone.
Mathematics Subject Classification: Primary: 90C29, 90C46.
Citation: Liwei Zhang, Jihong Zhang, Yule Zhang. Second-order optimality conditions for cone constrained multi-objective optimization. Journal of Industrial & Management Optimization, 2018, 14 (3) : 1041-1054. doi: 10.3934/jimo.2017089
References:
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B. Aghezzaf and M. Hachimi, Second-order optimality conditions in multiobjective optimization problems, Journal of Optimization Theory and Applications, 102 (1999), 37-50.  doi: 10.1023/A:1021834210437.  Google Scholar

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H. Kawasaki, Second-order necessary conditions of the Kuhn-Tucker type under new constraint qualification, Journal of Optimization Theory and Applications, 57 (1988), 253-264.  doi: 10.1007/BF00938539.  Google Scholar

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J. G. Lin, Maximal vectors and multiobjective optimization, Journal of Optimization Theory and Applications, 18 (1976), 41-64.  doi: 10.1007/BF00933793.  Google Scholar

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T. Maeda, Constraint qualification in multiobjective optimization problems: Differentiable case, Journal of Optimization Theory and Applications, 80 (1994), 483-500.  doi: 10.1007/BF02207776.  Google Scholar

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A. A. K. Majumdar, Optimality conditions in differentiable multiobjective programming, Journal of Optimization Theory and Applications, 92 (1997), 419-427.  doi: 10.1023/A:1022667432420.  Google Scholar

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O. L. Mangasarian, Nonlinear Programming, McGraw Hill, New York, 1969.  Google Scholar

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C. Singh, Optimality conditions in multiobjective differentiable programming, Journal of Optimization Theory and Applications, 53 (1987), 115-123.  doi: 10.1007/BF00938820.  Google Scholar

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S. Wang, Second-order necessary and sufficient conditions in multiobjective programming, Numerical Functional Analysis and Optimization, 12 (1991), 237-252.  doi: 10.1080/01630569108816425.  Google Scholar

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R. E. Wendell and D. N. Lee, Efficiency in multiple objective optimization problems, Mathematical Programming, 12 (1977), 406-414.  doi: 10.1007/BF01593807.  Google Scholar

show all references

References:
[1]

B. Aghezzaf and M. Hachimi, Second-order optimality conditions in multiobjective optimization problems, Journal of Optimization Theory and Applications, 102 (1999), 37-50.  doi: 10.1023/A:1021834210437.  Google Scholar

[2]

H. P. Benson, Existence of efficient solutions for vector maximization problems, Journal of Optimization Theory and Applications, 26 (1978), 569-580.  doi: 10.1007/BF00933152.  Google Scholar

[3]

G. Bigi, On sufficient second order optimality conditions in multiobjective optimization, Math. Meth. Oper. Res., 63 (2006), 77-85.  doi: 10.1007/s00186-005-0013-9.  Google Scholar

[4]

G. Bigi and M. Castellani, Second-order optimality conditions for differentiable multiobjective problems, BAIRO Operations Research, 34 (2000), 411-426.  doi: 10.1051/ro:2000122.  Google Scholar

[5]

J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer, New York, 2000. doi: 10.1007/978-1-4612-1394-9.  Google Scholar

[6]

J. F. Bonnas and C. H. Ramírez, Perturbation anylsis of second-order cone programming problems, Mathematical Programming, 104 (2005), 205-227.  doi: 10.1007/s10107-005-0613-4.  Google Scholar

[7]

H. Kawasaki, Second-order necessary conditions of the Kuhn-Tucker type under new constraint qualification, Journal of Optimization Theory and Applications, 57 (1988), 253-264.  doi: 10.1007/BF00938539.  Google Scholar

[8]

J. G. Lin, Maximal vectors and multiobjective optimization, Journal of Optimization Theory and Applications, 18 (1976), 41-64.  doi: 10.1007/BF00933793.  Google Scholar

[9]

T. Maeda, Constraint qualification in multiobjective optimization problems: Differentiable case, Journal of Optimization Theory and Applications, 80 (1994), 483-500.  doi: 10.1007/BF02207776.  Google Scholar

[10]

A. A. K. Majumdar, Optimality conditions in differentiable multiobjective programming, Journal of Optimization Theory and Applications, 92 (1997), 419-427.  doi: 10.1023/A:1022667432420.  Google Scholar

[11]

O. L. Mangasarian, Nonlinear Programming, McGraw Hill, New York, 1969.  Google Scholar

[12]

C. Singh, Optimality conditions in multiobjective differentiable programming, Journal of Optimization Theory and Applications, 53 (1987), 115-123.  doi: 10.1007/BF00938820.  Google Scholar

[13]

S. Wang, Second-order necessary and sufficient conditions in multiobjective programming, Numerical Functional Analysis and Optimization, 12 (1991), 237-252.  doi: 10.1080/01630569108816425.  Google Scholar

[14]

R. E. Wendell and D. N. Lee, Efficiency in multiple objective optimization problems, Mathematical Programming, 12 (1977), 406-414.  doi: 10.1007/BF01593807.  Google Scholar

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