# New dualities for mathematical programs with vanishing constraints

- 21 Downloads

## Abstract

Recently, Mishra et al. (Ann Oper Res 243(1):249–272, 2016) formulate and study the Wolfe and the Mond–Weir type dual models for the mathematical programs with vanishing constraints. They establish the weak, strong, converse, restricted converse and strict converse duality results between the primal mathematical programs with vanishing constraints and the corresponding dual model under some assumptions. However, their models contain the calculation of the index sets, this makes it difficult to solve them from algorithm point of view. In this paper, we propose the new Wolfe and Mond–Weir type dual models for the mathematical programs with vanishing constraints, which do not involve the calculation of the index set. We show that the weak, strong, converse and restricted converse duality results hold between the primal mathematical programs with vanishing constraints and the corresponding new dual models under the same assumptions as the ones of Mishra et al.

## Keywords

Mathematical programs with vanishing constraints Wolfe dual Mond–Weir dual## Mathematics Subject Classification

49J52 90C33## Notes

## References

- Achtziger, W., Hoheisel, T., & Kanzow, C. (2012). On a relaxation method for mathematical programs with vanishing constraints.
*GAMM-Mitteilungen*,*35*(2), 110–130.Google Scholar - Achtziger, W., Hoheisel, T., & Kanzow, C. (2013). A smoothing-regularization approach to mathematical programs with vanishing constraints.
*Computation Optimization and Applications*,*55*(3), 733–767.Google Scholar - Achtziger, W., & Kanzow, C. (2008). Mathematical programs with vanishing constraints: Optimality conditions and constraints qualifications.
*Mathematical Programming*,*114*(1), 69–99.Google Scholar - Antczak, T. (2010). G-saddle point criteria and G-Wolfe duality in differentiate mathematical programming.
*Journal of Information and Optimization Sciences*,*31*(1), 63–85.Google Scholar - Askar, S. S., & Tiwari, A. (2009). First-order optimality conditions and duality results for multi-objective optimization problems.
*Annals of Operations Research*,*172*(1), 277–289.Google Scholar - Benko, M., & Gfrerer, H. (2017). An SQP method for mathematical programs with vanishing constraints with strong convergence properties.
*Computation Optimization and Applications*,*11*(3), 641–653.Google Scholar - Bot, R. I., & Heinrich, A. (2014). Regression tasks in machine learning via Fenchel duality.
*Annals of Operations Research*,*222*(1), 197–211.Google Scholar - Chinchuluun, A., Yuan, D., & Pardalos, P. M. (2007). Optimality conditions and duality for nondifferentiable multiobjective fractional programming with generalized convexity.
*Annals of Operations Research*,*154*(1), 133–147.Google Scholar - Dorsch, D., Shikhman, V., & Stein, O. (2012). Mathematical programs with vanishing constraints: Critical point theory.
*Journal of Global Optimization*,*52*(3), 591–605.Google Scholar - Gulati, T. R., & Mehndiratta, G. (2010). Nondifferentiable multiobjective Mond-Weir type second-order symmetric duality over cones.
*Optimization Letters*,*4*(2), 293–309.Google Scholar - Hoheisel, T., & Kanzow, C. (2007). First- and second-order optimality conditions for mathematical programs with vanishing constraints.
*Applied Mathematics*,*52*(6), 495–514.Google Scholar - Hoheisel, T., & Kanzow, C. (2008). Stationary conditions for mathematical programs with vanishing constraints using weak constraint qualification.
*Journal of Mathematical Analysis and Applications*,*337*(1), 292–310.Google Scholar - Hoheisel, T., & Kanzow, C. (2009). On the Abadie and Guignard constraint qualification for mathematical programs with vanishing constraints.
*Optimization*,*58*(4), 431–448.Google Scholar - Hoheisel, T., Kanzow, C., & Outrata, J. V. (2010). Exact penalty results for mathematical programs with vanishing constraints.
*Nonlinear Analysis*,*72*(5), 2514–2526.Google Scholar - Hoheisel, T., Kanzow, C., & Schwartz, A. (2012). Convergence of a local regularization approach for mathematical programs with complementarity or vanishing constraints.
*Optimization Methods and Software*,*27*(3), 483–512.Google Scholar - Hu, Q. J., Chen, Y., Zhu, Z. B., & Zhang, B. S. (2014). Notes on some convergence properties for a smoothing-regularization approach to mathematical programs with vanishing constraints.
*Abstract and Applied Analysis*,*2014*(1), 1–7.Google Scholar - Hu, Q. J., Wang, J. G., Chen, Y., & Zhu, Z. B. (2017). On an \(l_1\) exact penalty result for mathematical programs with vanishing constraints.
*Optimization Letters*,*11*(3), 641–653.Google Scholar - Izmailov, A. F., & Pogosyan, A. L. (2009). Optimality conditions and Newton-type methods for mathematical programs with vanishing constraints.
*Computation Mathematics and Mathematics Physics*,*49*(7), 1128–1140.Google Scholar - Izmailov, A. F., & Solodov, M. V. (2009). Mathematical programs with vanishing constraints: Optimality conditions, sensitivity and a relaxation method.
*Journal of Optimization Theory and Applications*,*142*(3), 501–532.Google Scholar - Jabr, R. A. (2012). Solution to economic dispatching with disjoint feasible regions via semidefinite programming.
*IEEE Transactions on Power Systems*,*27*(1), 572–573.Google Scholar - Jefferson, T. R., & Scott, C. H. (2001). Quality tolerancing and conjugate duality.
*Annals of Operations Research*,*105*(1–4), 185–200.Google Scholar - Kirches, C., Potschka, A., Bock, H. G., & Sager, S. (2013). A parametric active set method for quadratic programs with vanishing constraints.
*Pacific Jounal of Optimization*,*9*(2), 275–299.Google Scholar - Lai, H. C., & Huang, T. Y. (2012). Nondifferentiable minimax fractional programming in complex spaces with parametric duality.
*Journal of Global Optimization*,*53*(2), 243–254.Google Scholar - Lee, J. C., & Lai, H. C. (2005). Parameter-free dual models for fractional programming with generalized invexity.
*Annals of Operations Research*,*133*(1–4), 47–61.Google Scholar - Michael, N. J., Kirches, C., & Sager, S. (2013). On perspective functions and vanishing constraints in mixedinteger nonlinear optimal control. In M. Jünger & G. Reinelt (Eds.),
*Facets of combinatorial optimization*(pp. 387–417). Berlin: Springer.Google Scholar - Mishra, S. K., & Jaiswal, M. (2015). Optimality conditions and duality for semi-infinite mathematical programming problem with equilibrium constraints.
*Journal Numerical Functional Analysis and Optimization*,*36*(4), 460–480.Google Scholar - Mishra, S. K., Jaiswal, M., & An, L. T. H. (2012). Duality for nonsmooth semi-infinite programming problems.
*Optimization Letters*,*6*(2), 261–271.Google Scholar - Mishra, S. K., & Shukla, K. (2010). Nonsmooth minimax programming problems with V-r-invex functions.
*Optimization*,*59*(1), 95–103.Google Scholar - Mishra, S. K., Singh, V., & Laha, V. (2016). On duality for mathematical programs with vanishing constraints.
*Annals of Operations Research*,*243*(1), 249–272.Google Scholar - Mishra, S. K., Singh, V., Laha, V., & Mohapatra, R. N. (2015). On constraint qualifications for multiobjective optimization problems with vanishing constraints. In H. Xu, S. Wang & S. Y. Wu (Eds.),
*Optimization methods, theory and applications*(pp. 95–135). Berlin: Springer.Google Scholar - Mond, B., & Weir, T. (1981). Generalized concavity and duality. In S. Schaible & W. T. Ziemba (Eds.),
*Generalized concavity in optimization and economics*(pp. 263–279). New York: Academic Press.Google Scholar - Pandey, Y., & Mishra, S. K. (2016). Duality for nonsmooth optimization problems with equilibrium constraints using convexificators.
*Journal of Optimization Theory and Applications*,*171*(2), 694–707.Google Scholar - Pandey, Y., & Mishra, S. K. (2017). Duality of mathematical programming problems with equilibrium constraints.
*Pacific Journal of Optimization*,*13*(1), 105–122.Google Scholar - Pandey, Y., & Mishra, S. K. (2018). Optimality conditions and duality for semi-infinite mathematical programming problems with equilibrium constraints, using convexificators.
*Annals of Operations Research*,*269*(2), 549–564.Google Scholar - Peterson, E. L. (2001). The fundamental relations between geometric programming duality, parametric programming duality, and ordinary Lagrangian duality.
*Annals of Operations Research*,*105*(1–4), 109–153.Google Scholar - Rockafellar, R. T. (1999). Duality and optimality in multistagestochastic programming.
*Annals of Operations Research*,*85*(1), 1–19.Google Scholar - Wolfe, P. (1961). A duality theorem for nonlinear programming.
*Quarterly of Applied Mathematics*,*19*, 239–244.Google Scholar