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June 2019, 9(2): 133-145. doi: 10.3934/naco.2019010

## Second order modified objective function method for twice differentiable vector optimization problems over cone constraints

 a. Department of Applied Mathematics, Indian Institute of Technology (Indian School of Mines), Dhanbad-826004, Jharkhand, India b. Faculty of Mathematics and Computer Science, University of Lodz, Banacha 22, 90-238 Lodz, Poland

* Corresponding author: Shalini Jha, Visiting Scientist, Machine Intelligence Unit (MIU), ​Indian Statistical Institute (ISI) Kolkata, India - 700108

Received  January 2017 Revised  August 2018 Published  January 2019

In the paper, a vector optimization problem with twice differentiable functions and cone constraints is considered. The second order modified objective function method is used for solving such a multiobjective programming problem. In this method, for the considered twice differentiable multi-criteria optimization problem, its associated second order vector optimization problem with the modified objective function is constructed at the given arbitrary feasible solution. Then, the equivalence between the sets of (weakly) efficient solutions in the original twice differentiable vector optimization problem with cone constraints and its associated modified vector optimization problem is established. Further, the relationship between an (weakly) efficient solution in the original vector optimization problem and a saddle-point of the second order Lagrange function defined for the modified vector optimization problem is also analyzed.

Citation: Anurag Jayswala, Tadeusz Antczakb, Shalini Jha. Second order modified objective function method for twice differentiable vector optimization problems over cone constraints. Numerical Algebra, Control & Optimization, 2019, 9 (2) : 133-145. doi: 10.3934/naco.2019010
##### References:
 [1] B. Aghezzaf and M. Hachimi, Second-order optimality conditions in multiobjective optimization problems, J. Optim. Theory Appl., 102 (1999), 37-50. doi: 10.1023/A:1021834210437. [2] T. Antczak, A new approach to multiobjective programming with a modified objective function, J. Global Optim., 27 (2003), 485-495. doi: 10.1023/A:1026080604790. [3] T. Antczak, Saddle-point criteria and duality in multiobjective programming via an $\eta$-approximation method, Anziam J., 47 (2005), 155-172. doi: 10.1017/S1446181100009962. [4] T. Antczak, Saddle-point criteria via a second order $\eta$-approximation approach for nonlinear mathematical programming problem involving second order invex functions, Kybernetika, 47 (2011), 222-240. [5] C. R. Bector and B. K. Bector, Generalized-bonvex functions and second order duality for a nonlinear programming problem, Congr. Numer., 52 (1985), 37-52. [6] C. R. Bector and B. K. Bector, On various duality theorems for second order duality in nonlinear programming, Cahiers Centre Etudes Rech. Oper., 28 (1986), 283-292. [7] J. W. Chen, Y. J. Cho, J. K. Kim and J. Li, Multiobjective optimization problems with modified objective functions and cone constraints and applications, J. Global Optim., 49 (2011), 137-147. doi: 10.1007/s10898-010-9539-3. [8] M. K. Ghosh and A. J. Shaiju, Existence of value and saddle-point in infinite-dimensional differential games, J. Optim. Theory Appl., 121 (2004), 301-325. doi: 10.1023/B:JOTA.0000037407.15482.72. [9] T. R. Gulati, H. Saini and S. K. Gupta, Second-order multiobjective symmetric duality with cone constraints, European J. Oper. Res., 205 (2010), 247-252. doi: 10.1016/j.ejor.2009.12.024. [10] L. Li and J. Li, Equivalence and existence of weak Pareto optima for multiobjective optimization problems with cone constraints, Appl. Math. Lett., 21 (2008), 599-606. doi: 10.1016/j.aml.2007.07.012. [11] Z. F. Li and S. Y. Wang, Lagrange multipliers and saddle-points in multiobjective programming, J. Optim. Theory Appl., 83 (1994), 63-81. doi: 10.1007/BF02191762. [12] T. Li, Y. J. Wang, Z. Liang and P. M. Pardalos, Local saddle-point and a class of convexification methods for nonconvex optimization problems, J. Global Optim., 38 (2007), 405-419. doi: 10.1007/s10898-006-9090-4. [13] S. K. Suneja, M. B. Grover and M. Kapoor, Second order multiobjective symmetric duality in vector optimization over cones involving $\rho$-invexity, Amer. J. Oper. Res., 4 (2014), 1-9. doi: 10.1016/S0377-2217(01)00258-2. [14] S. K. Suneja, S. Sharma and M. Kapoor, Modified objective function method in nonsmooth vector optimization over cones, Optim. Lett., 8 (2014), 1361-1373. doi: 10.1007/s11590-013-0661-2. [15] S. K. Suneja, S. Sharma and Vani, Second order duality in vector optimization over cones, J. Appl. Math. Inform., 26 (2008), 251–261.

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##### References:
 [1] B. Aghezzaf and M. Hachimi, Second-order optimality conditions in multiobjective optimization problems, J. Optim. Theory Appl., 102 (1999), 37-50. doi: 10.1023/A:1021834210437. [2] T. Antczak, A new approach to multiobjective programming with a modified objective function, J. Global Optim., 27 (2003), 485-495. doi: 10.1023/A:1026080604790. [3] T. Antczak, Saddle-point criteria and duality in multiobjective programming via an $\eta$-approximation method, Anziam J., 47 (2005), 155-172. doi: 10.1017/S1446181100009962. [4] T. Antczak, Saddle-point criteria via a second order $\eta$-approximation approach for nonlinear mathematical programming problem involving second order invex functions, Kybernetika, 47 (2011), 222-240. [5] C. R. Bector and B. K. Bector, Generalized-bonvex functions and second order duality for a nonlinear programming problem, Congr. Numer., 52 (1985), 37-52. [6] C. R. Bector and B. K. Bector, On various duality theorems for second order duality in nonlinear programming, Cahiers Centre Etudes Rech. Oper., 28 (1986), 283-292. [7] J. W. Chen, Y. J. Cho, J. K. Kim and J. Li, Multiobjective optimization problems with modified objective functions and cone constraints and applications, J. Global Optim., 49 (2011), 137-147. doi: 10.1007/s10898-010-9539-3. [8] M. K. Ghosh and A. J. Shaiju, Existence of value and saddle-point in infinite-dimensional differential games, J. Optim. Theory Appl., 121 (2004), 301-325. doi: 10.1023/B:JOTA.0000037407.15482.72. [9] T. R. Gulati, H. Saini and S. K. Gupta, Second-order multiobjective symmetric duality with cone constraints, European J. Oper. Res., 205 (2010), 247-252. doi: 10.1016/j.ejor.2009.12.024. [10] L. Li and J. Li, Equivalence and existence of weak Pareto optima for multiobjective optimization problems with cone constraints, Appl. Math. Lett., 21 (2008), 599-606. doi: 10.1016/j.aml.2007.07.012. [11] Z. F. Li and S. Y. Wang, Lagrange multipliers and saddle-points in multiobjective programming, J. Optim. Theory Appl., 83 (1994), 63-81. doi: 10.1007/BF02191762. [12] T. Li, Y. J. Wang, Z. Liang and P. M. Pardalos, Local saddle-point and a class of convexification methods for nonconvex optimization problems, J. Global Optim., 38 (2007), 405-419. doi: 10.1007/s10898-006-9090-4. [13] S. K. Suneja, M. B. Grover and M. Kapoor, Second order multiobjective symmetric duality in vector optimization over cones involving $\rho$-invexity, Amer. J. Oper. Res., 4 (2014), 1-9. doi: 10.1016/S0377-2217(01)00258-2. [14] S. K. Suneja, S. Sharma and M. Kapoor, Modified objective function method in nonsmooth vector optimization over cones, Optim. Lett., 8 (2014), 1361-1373. doi: 10.1007/s11590-013-0661-2. [15] S. K. Suneja, S. Sharma and Vani, Second order duality in vector optimization over cones, J. Appl. Math. Inform., 26 (2008), 251–261.
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