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Second-order optimality conditions for cone constrained multi-objective optimization
School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China |
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.
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. |
| [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. |
| [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. |
| [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. |
| [5] |
J. F. Bonnans and A. Shapiro,
Perturbation Analysis of Optimization Problems, Springer, New York, 2000.
doi: 10.1007/978-1-4612-1394-9. |
| [6] |
J. F. Bonnas and C. H. RamÃÂrez,
Perturbation anylsis of second-order cone programming problems, Mathematical Programming, 104 (2005), 205-227.
doi: 10.1007/s10107-005-0613-4. |
| [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. |
| [8] |
J. G. Lin,
Maximal vectors and multiobjective optimization, Journal of Optimization Theory and Applications, 18 (1976), 41-64.
doi: 10.1007/BF00933793. |
| [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. |
| [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. |
| [11] |
O. L. Mangasarian,
Nonlinear Programming, McGraw Hill, New York, 1969. |
| [12] |
C. Singh,
Optimality conditions in multiobjective differentiable programming, Journal of Optimization Theory and Applications, 53 (1987), 115-123.
doi: 10.1007/BF00938820. |
| [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. |
| [14] |
R. E. Wendell and D. N. Lee,
Efficiency in multiple objective optimization problems, Mathematical Programming, 12 (1977), 406-414.
doi: 10.1007/BF01593807. |
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. |
| [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. |
| [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. |
| [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. |
| [5] |
J. F. Bonnans and A. Shapiro,
Perturbation Analysis of Optimization Problems, Springer, New York, 2000.
doi: 10.1007/978-1-4612-1394-9. |
| [6] |
J. F. Bonnas and C. H. RamÃÂrez,
Perturbation anylsis of second-order cone programming problems, Mathematical Programming, 104 (2005), 205-227.
doi: 10.1007/s10107-005-0613-4. |
| [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. |
| [8] |
J. G. Lin,
Maximal vectors and multiobjective optimization, Journal of Optimization Theory and Applications, 18 (1976), 41-64.
doi: 10.1007/BF00933793. |
| [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. |
| [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. |
| [11] |
O. L. Mangasarian,
Nonlinear Programming, McGraw Hill, New York, 1969. |
| [12] |
C. Singh,
Optimality conditions in multiobjective differentiable programming, Journal of Optimization Theory and Applications, 53 (1987), 115-123.
doi: 10.1007/BF00938820. |
| [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. |
| [14] |
R. E. Wendell and D. N. Lee,
Efficiency in multiple objective optimization problems, Mathematical Programming, 12 (1977), 406-414.
doi: 10.1007/BF01593807. |
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