Unified duality for mathematical programming problems with vanishing constraints

Document Type : Research Paper


1 Department of Mathematics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia

2 Center for Intelligent Secure Systems, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia

3 Department of Mathematics, School of Science, GITAM-Hyderabad Campus Hyderabad-502329, India

4 Center for Smart Mobility and Logistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia


In this article, we formulate a new mixed-type dual problem for a mathematical program with vanishing constraints. The presented dual problem does not involve the index set, however, the dual models contain the calculations of index sets, which makes it difficult to solve these models from an algorithm point of view. The weak, strong and strict converse duality theorems are discussed in order to establish the relationships between the mathematical program with vanishing constraints and its mixed type dual under generalized convexity. To validate the results, a non-trivial example is discussed. Our dual model unifies the dual models discussed in [Q. Hu, J. Wang, Y. Chen, New dualities for mathematical programs with vanishing constraints, Annals of Operations Research, 287 (2020) 233-255].


[1] W. Achtziger and C. Kanzow, Mathematical programs with vanishing constraints: Optimality conditions and constraint qualifications, Math. Program. 114 (2008), 69–99.
[2] I. Ahmad, K. Kummari and S. Al-Homidan, Sufficiency and duality for interval-valued optimization problems with vanishing constraints using weak constraint qualification, Int. J. Anal. Appl. 18 (2020), no. 5, 784–798.
[3] T. Antczak, Optimality conditions and Mond–Weir duality for a class of differentiable semi-infinite multiobjective programming problems with vanishing constraints, 4OR 20 (2022), no. 3, 417–442.
[4] A.A. Ardali, M. Zeinali and G.H. Shirdel, Some Pareto optimality results for nonsmooth multiobjective optimization problems with equilibrium constraints, Int. J. Nonlinear Anal. Appl. 13 (2022), no. 2, 2185–2196.
[5] M.S. Bazaraa, H.D. Sherali and C. M. Shetty, Nonlinear Programming. Theory and Algorithms, 2nd edition, John Wiley & Sons, Hoboken, 1993.
[6] M.P. Bendsøe and O. Sigmund, Topology Optimization-Theory, Methods and Applications, 2nd ed., Springer, Heidelberg, Germany, 2003.
[7] R. Dubey and V.N. Mishra, Second-order nondifferentiable multiobjective mixed type fractional programming problems, Int. J. Nonlinear Anal. Appl. 11 (2020), no. 1, 439–451.
[8] J.-P. Dussault, M. Haddou and T. Migot, Mathematical programs with vanishing constraints: constraint qualifications, their applications, and a new regularization method, Optim. 68 (2019), no. 2-3, 509–538.
[9] T. Hoheisel and C. Kanzow, On the Abadie and Guignard constraint qualifications for Mathematical Programmes with Vanishing Constraints, Optim. 58 (2009), no. 4, 431–448.
[10] Q. Hu, J. Wang and Y. Chen, New dualities for mathematical programs with vanishing constraints, Ann. Oper. Res., 287 (2020), 233–255.
[11] R.A. Jabr, Solution to economic dispatching with disjoint feasible regions via semidefinite programming, IEEE T. Power Syst. 27 (2012), no. 1, 572–573.
[12] A. Jayswal and V. Singh, The Characterization of efficiency and saddle point criteria for multiobjective optimization problem with vanishing constraints, Acta Math. Sci. 39 (2019), no. 2, 382–394.
[13] S. Kazemi and N. Kanzi, Constraint qualifications and stationary conditions for mathematical programming with non-differentiable vanishing constraints, J. Optim. Theory Appl. 179 (2018), no. 3, 800–819.
[14] A. Khare and T. Nath, Enhanced Fritz-John stationarity, new constraint qualifications and local error bound for mathematical programs with vanishing constraints, J. Math. Anal. Appl. 472 (2019), no. , 1042–1077.
[15] C. Kirches, A. Potschka, H. G. Bock and S. Sager, A parametric active set method for quadratic programs with vanishing constraints, Pacific J. Optim. 9 (2013), no. 2, 275–299.
[16] N. J. Michael, C. Kirches and S. Sager, On perspective functions and vanishing constraints in mixed-integer nonlinear optimal control, In: M. Junger & G. Reinelt (Eds.), Facets of combinatorial optimization (pp. 387-417), Springer, Berlin, 2013.
[17] S.K. Mishra, V. Singh and V. Laha, On duality for mathematical programs with vanishing constraints. Ann. Oper. Res. 243 (2016), no. 1, 249–272.
[18] L.T. Tung, Karush–Kuhn–Tucker optimality conditions and duality for multiobjective semi-infinite programming with vanishing constraints, Annal. Oper. Res. 311 (2022), 1307–1334.
Volume 13, Issue 2
July 2022
Pages 3191-3201
  • Receive Date: 06 January 2022
  • Revise Date: 21 July 2022
  • Accept Date: 07 August 2022