A Different Approach to Cone-Convex Optimization

DOI: 10.4236/ajor.2013.36052   PDF   HTML   XML   3,541 Downloads   5,743 Views   Citations


In classical convex optimization theory, the Karush-Kuhn-Tucker (KKT) optimality conditions are necessary and sufficient for optimality if the objective as well as the constraint functions involved is convex. Recently, Lassere [1] considered a scalar programming problem and showed that if the convexity of the constraint functions is replaced by the convexity of the feasible set, this crucial feature of convex programming can still be preserved. In this paper, we generalize his results by making them applicable to vector optimization problems (VOP) over cones. We consider the minimization of a cone-convex function over a convex feasible set described by cone constraints that are not necessarily cone-convex. We show that if a Slater-type cone constraint qualification holds, then every weak minimizer of (VOP) is a KKT point and conversely every KKT point is a weak minimizer. Further a Mond-Weir type dual is formulated in the modified situation and various duality results are established.

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S. Suneja, S. Sharma, M. Grover and M. Kapoor, "A Different Approach to Cone-Convex Optimization," American Journal of Operations Research, Vol. 3 No. 6, 2013, pp. 536-541. doi: 10.4236/ajor.2013.36052.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] J. B. Lasserre, “On Representations of the Feasible Set in Convex Optimization,” Optimization Letters, Vol. 4, No. 1, 2010, pp. 1-5.
[2] C. R. Bector, S. Chandra and M. K. Bector, “Sufficient Optimality Conditions and Duality for a Quasiconvex Programming Problem,” Journal of Optimization Theory and Applications, Vol. 59, No. 2, 1988, pp. 209-221.
[3] O. L. Mangasian, “Non-Linear Programming,” McGraw Hill, New York, 1969.
[4] S. Nobakhtian, “Sufficiency in Nonsmooth Multiobjective Programming Involving Generalized (F, p)-Convexity,” Journal of Optimization Theory and Applications, Vol. 130, No. 2, 2006, pp. 359-365.
[5] J. Dutta and C. S. Lalitha, “Optimality Conditions in Convex Optimization Revisited,” Optimization Letters, Vol. 7, No. 2, 2013, pp. 221-229.
[6] F. H. Clarke, “Optimization and Non-smooth Analysis,” Wiley, New York, 1983.
[7] L. Coladas, Z. Li and S. Wang, “Optimality Conditions for Multiobjective and Nonsmooth Minimization in Abstract Spaces,” Bulletin of Australian Mathematical Society, Vol. 50, No. 2, 1994, pp. 205-218.
[8] S. Aggarwal, “Optimality and Duality in Mathematical Programming Involving Generalized Convex Functions,” Ph.D. Thesis, University of Delhi, Delhi, 1998.
[9] T. Weir, B. Mond and B. D. Craven, “Weak Minimization and Duality,” Numerical Functional Analysis and Optimization, Vol. 9, No. 1-2 ,1987, pp. 181-192.
[10] J. Jahn, “Vector Optimization,” Springer Verlag, New York, 2003.
[11] S. K. Suneja, S. Aggarwal and S. Davar, “Multiobjective Symmetric Duality Involving Cones,” European Journal of Operational Research, Vol. 141, No. 3, 2002, pp. 471479.

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