# American Institute of Mathematical Sciences

• Previous Article
Performance optimization of parallel-distributed processing with checkpointing for cloud environment
• JIMO Home
• This Issue
• Next Article
Times until service completion and abandonment in an M/M/$m$ preemptive-resume LCFS queue with impatient customers
doi: 10.3934/jimo.2017089

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

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

* Corresponding authorr: Liwei Zhang

Received  February 2017 Revised  June 2017 Published  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.

Citation: Liwei Zhang, Jihong Zhang, Yule Zhang. Second-order optimality conditions for cone constrained multi-objective optimization. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2017089
##### 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. [6] J. F. Bonnas and C. H. RamÃƒÂ­rez, 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. [6] J. F. Bonnas and C. H. RamÃƒÂ­rez, 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.
 [1] Xiaoni Chi, Zhongping Wan, Zijun Hao. Second order sufficient conditions for a class of bilevel programs with lower level second-order cone programming problem. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2015.11.1111 [2] Yi Zhang, Yong Jiang, Liwei Zhang, Jiangzhong Zhang. A perturbation approach for an inverse linear second-order cone programming. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2013.9.171 [3] Ye Tian, Shu-Cherng Fang, Zhibin Deng, Wenxun Xing. Computable representation of the cone of nonnegative quadratic forms over a general second-order cone and its application to completely positive programming. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2013.9.703 [4] Yanhong Yuan, Hongwei Zhang, Liwei Zhang. A smoothing Newton method for generalized Nash equilibrium problems with second-order cone constraints. Numerical Algebra, Control & Optimization, doi: 10.3934/naco.2012.2.1 [5] Xi-De Zhu, Li-Ping Pang, Gui-Hua Lin. Two approaches for solving mathematical programs with second-order cone complementarity constraints. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2015.11.951 [6] Ye Tian, Qingwei Jin, Zhibin Deng. Quadratic optimization over a polyhedral cone. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2016.12.269 [7] Shiyun Wang, Yong-Jin Liu, Yong Jiang. A majorized penalty approach to inverse linear second order cone programming problems. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2014.10.965 [8] Qilin Wang, Xiao-Bing Li, Guolin Yu. Second-order weak composed epiderivatives and applications to optimality conditions. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2013.9.455 [9] Xiaoling Guo, Zhibin Deng, Shu-Cherng Fang, Wenxun Xing. Quadratic optimization over one first-order cone. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2014.10.945 [10] Yafeng Li, Guo Sun, Yiju Wang. A smoothing Broyden-like method for polyhedral cone constrained eigenvalue problem. Numerical Algebra, Control & Optimization, doi: 10.3934/naco.2011.1.529 [11] Qiong Meng, X. H. Tang. Solutions of a second-order Hamiltonian system with periodic boundary conditions. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2010.9.1053 [12] B. Bonnard, J.-B. Caillau, E. Trélat. Second order optimality conditions with applications. Conference Publications, doi: 10.3934/proc.2007.2007.145 [13] José F. Cariñena, Javier de Lucas Araujo. Superposition rules and second-order Riccati equations. Journal of Geometric Mechanics, doi: 10.3934/jgm.2011.3.1 [14] Eugenii Shustin, Emilia Fridman, Leonid Fridman. Oscillations in a second-order discontinuous system with delay. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2003.9.339 [15] Piotr Pokora, Tomasz Szemberg. Minkowski bases on algebraic surfaces with rational polyhedral pseudo-effective cone. Electronic Research Announcements, doi: 10.3934/era.2014.21.126 [16] Fengming Ma, Yiju Wang, Hongge Zhao. A potential reduction method for the generalized linear complementarity problem over a polyhedral cone. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2010.6.259 [17] Leonardo Colombo. Second-order constrained variational problems on Lie algebroids: Applications to Optimal Control. Journal of Geometric Mechanics, doi: 10.3934/jgm.2017001 [18] Rui Li, Yingjing Shi. Finite-time optimal consensus control for second-order multi-agent systems. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2014.10.929 [19] He Zhang, Xue Yang, Yong Li. Lyapunov-type inequalities and solvability of second-order ODEs across multi-resonance. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2017061 [20] Qilin Wang, Shengji Li, Kok Lay Teo. Continuity of second-order adjacent derivatives for weak perturbation maps in vector optimization. Numerical Algebra, Control & Optimization, doi: 10.3934/naco.2011.1.417

2016 Impact Factor: 0.994

Article outline

[Back to Top]