A unified approach to zero duality gap for convex optimization problems
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Abstract
In this paper we establish necessary and sufficient condition for zero duality gap of the optimization problem involving the general perturbation mapping via characteringsetunder the convex setting. An application to the class of composite optimization problems will also be given to show that our general results can be applied to various classes of optimization problems.
Article Details
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
Keywords
Characterizing set, composite optimization problem, perturbation function, zero duality gap
References
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Huang, X. X., & Yang, X. Q. (2003). A unified augmented Lagrangian approach to duality and exact penalization. Mathematics of Operations Research, 28(3), 533-552. https://doi.org/10.1287/moor.28.3.533.16395.
Huang, X. X., & Yang, X. Q. (2005). Further study on augmented Lagrangian duality theory. Journal of Global Optimization, 31(2), 193-210. https://doi.org/10.1007/s10898-004-5695-7.
Jeyakumar, V., & Li, G. Y. (2009). Stable zero duality gaps in convex programming: complete dual characterisations with applications to semidefinite programs. Journal of mathematical analysis and applications, 360(1), 156-167. https://doi.org/10.1016/j.jmaa.2009.06.043.
Jeyakumar, V., & Li, G. Y. (2009). New dual constraint qualifications characterizing zero duality gaps of convex programs and semidefinite programs. Nonlinear Analysis: Theory, Methods & Applications, 71(12), e2239-e2249. https://doi.org/10.1016/j.na.2009.05.009.
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Li, D. (1995). Zero duality gap for a class of nonconvex optimization problems. Journal of Optimization Theory and Applications, 85, 309-324. https://doi.org/10.1007/BF02192229.
Li, D. (1999). Zero duality gap in integer programming: P-norm surrogate constraint method. Operations Research Letters, 25(2), 89-96. https://doi.org/10.1016/S0167-6377(99)00039-5.
Long, F., & Zeng, B. (2021). The zero duality gap property for an optimal control problem governed by a multivalued hemivariational inequality. Applied Mathematics & Optimization, 84, 2629-2643. https://doi.org/10.1007/s00245-020-09721-z.
Nguyen, D., Dang, H. L., Tran, H. M., & Yao, J. C. (2020). Approximate Farkas lemmas for vector systems with applications to convex vector optimization problems. Journal of Nonlinear and Convex Analysis, 21(5), 1225-1246.
Nguyen, D. & Tran, H. M. (2012). Qualification conditions and Farkas-type results for systems involving composite functions. Vietnam Journal of Math, 40(4), 407-437.
Pham, D. K., Tran, H. M., & Tran, T. T. T. (2019). Necessary and sufficient conditions for qualitative properties of infinite dimensional linear programming problems. Numerical Functional Analysis and Optimization, 40(8), 924-943. https://doi.org/10.1080/01630563.2019.1566244.
Rubinov, A. M., Huang, X. X., & Yang, X. Q. (2002). The zero duality gap property and lower semicontinuity of the perturbation function. Mathematics of Operations Research, 27(4), 775-791. https://doi.org/10.1287/moor.27.4.775.295.
Rudin, W. (1991). Functional Analysis (2nd Edition). New York: McGraw-Hill.
Yang, X. Q., & Huang, X. X. (2001). A nonlinear Lagrangian approach to constrained optimization problems. SIAM Journal on Optimization, 11(4), 1119-1144. https://doi.org/10.1137/S1052623400371806.
Boţ, R. I. (2009). Conjugate duality in convex optimization (Vol. 637). Springer Science & Business Media.
Boţ, R. I., Hodrea, I. B., & Wanka, G. (2005, August). Composed convex programming: duality and Farkas-type results. In Proceeding of the International Conference In Memoriam Gyula Farkas, 23-26.
Feizollahi, M. J., Ahmed, S., & Sun, A. (2017). Exact augmented Lagrangian duality for mixed integer linear programming. Mathematical Programming, 161, 365-387. https://doi.org/10.1007/s10107-016-1012-8.
Huang, X. X., & Yang, X. Q. (2003). A unified augmented Lagrangian approach to duality and exact penalization. Mathematics of Operations Research, 28(3), 533-552. https://doi.org/10.1287/moor.28.3.533.16395.
Huang, X. X., & Yang, X. Q. (2005). Further study on augmented Lagrangian duality theory. Journal of Global Optimization, 31(2), 193-210. https://doi.org/10.1007/s10898-004-5695-7.
Jeyakumar, V., & Li, G. Y. (2009). Stable zero duality gaps in convex programming: complete dual characterisations with applications to semidefinite programs. Journal of mathematical analysis and applications, 360(1), 156-167. https://doi.org/10.1016/j.jmaa.2009.06.043.
Jeyakumar, V., & Li, G. Y. (2009). New dual constraint qualifications characterizing zero duality gaps of convex programs and semidefinite programs. Nonlinear Analysis: Theory, Methods & Applications, 71(12), e2239-e2249. https://doi.org/10.1016/j.na.2009.05.009.
Jeyakumar, V., & Wolkowicz, H. (1990). Zero duality gaps in infinite-dimensional programming. Journal of Optimization Theory and Applications, 67, 87-108. https://doi.org/10.1007/BF00939737.
Li, D. (1995). Zero duality gap for a class of nonconvex optimization problems. Journal of Optimization Theory and Applications, 85, 309-324. https://doi.org/10.1007/BF02192229.
Li, D. (1999). Zero duality gap in integer programming: P-norm surrogate constraint method. Operations Research Letters, 25(2), 89-96. https://doi.org/10.1016/S0167-6377(99)00039-5.
Long, F., & Zeng, B. (2021). The zero duality gap property for an optimal control problem governed by a multivalued hemivariational inequality. Applied Mathematics & Optimization, 84, 2629-2643. https://doi.org/10.1007/s00245-020-09721-z.
Nguyen, D., Dang, H. L., Tran, H. M., & Yao, J. C. (2020). Approximate Farkas lemmas for vector systems with applications to convex vector optimization problems. Journal of Nonlinear and Convex Analysis, 21(5), 1225-1246.
Nguyen, D. & Tran, H. M. (2012). Qualification conditions and Farkas-type results for systems involving composite functions. Vietnam Journal of Math, 40(4), 407-437.
Pham, D. K., Tran, H. M., & Tran, T. T. T. (2019). Necessary and sufficient conditions for qualitative properties of infinite dimensional linear programming problems. Numerical Functional Analysis and Optimization, 40(8), 924-943. https://doi.org/10.1080/01630563.2019.1566244.
Rubinov, A. M., Huang, X. X., & Yang, X. Q. (2002). The zero duality gap property and lower semicontinuity of the perturbation function. Mathematics of Operations Research, 27(4), 775-791. https://doi.org/10.1287/moor.27.4.775.295.
Rudin, W. (1991). Functional Analysis (2nd Edition). New York: McGraw-Hill.
Yang, X. Q., & Huang, X. X. (2001). A nonlinear Lagrangian approach to constrained optimization problems. SIAM Journal on Optimization, 11(4), 1119-1144. https://doi.org/10.1137/S1052623400371806.
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