Duality results on mathematical programs with vanishing constraints involving generalized invex functions

Bhuwan Chandra Joshi

doi:10.2478/candc-2023-0042

.blurhash-client-img { display: none !important; }

Duality results on mathematical programs with vanishing constraints involving generalized invex functions

Control and Cybernetics

Volume 52 (2023): Issue 4 (December 2023)

By: Bhuwan Chandra Joshi

Open Access

|Nov 2024

Abstract

In the here presented research, we investigate Wolfe and Mond-Weir duality models applied to a specific category of generalized convex functions known as p-invex functions. We establish various dualities between the primal MPVC and its Wolfe type dual, as well as between the primal MPVC (mathematical program with vanishing constraints) and its Mond-Weir type dual under p- invexity assumptions. To illustrate these theorems, we will include some examples.

References

ACHTZIGER,W. and KANZOW, C. (2008) Mathematical programs with vanishing constraints: Optimality conditions and constraints qualifications. Mathematical Programming, 114 (1), 69–99.
Search in Google Scholar Back to article
ANTCZAK, T. (2001) On (p, r)-invexity-type nonlinear programming problems. Journal of Mathematical Analysis and Applications, 264 (2), 382–397.
Search in Google Scholar Back to article
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.
Search in Google Scholar Back to article
ANTCZAK, T. and SINGH, V. (2009) First-order optimality conditions and duality results for multi-objective optimization problems. Annals of Operations Research, 172 (1), 277–289.
Search in Google Scholar Back to article
ANTCZAK, T. and SINGH, V. (2013) Optimality and duality for minimax fractional programming with support functions under B-(p, r)- Type I assumptions. Mathematical and Computer Modelling, 57 (5-6), 1083 – 1100.
Search in Google Scholar Back to article
BOT, R. I. and GRAD, S.-M. (2010) Wolfe duality and Mond–Weir duality via perturbations. Nonlinear Analysis: Theory, Methods and Applications, 73 (2), 374–384.
Search in Google Scholar Back to article
CHINCHULUUN, A., YUAN, D. and PARDALOS, P. M. (2007) Optimality conditions and duality for nondifferentiable multiobjective fractional programming with generalized convexity. Annals of Operations Research, 154 (1), 133–147.
Search in Google Scholar Back to article
DUSSAULT, J.-P., MOUNIR H. and TANGI M. (2019) Mathematical programs with vanishing constraints: constraint qualifications, their applications, and a new regularization method. Optimization, 68 (2-3), 509–538.
Search in Google Scholar Back to article
HOHEISEL T. and KANZOW C. (2007) First and second-order optimality conditions for mathematical programs with vanishing constraints. Applications of Mathematics, 52, 495–514
Search in Google Scholar Back to article
HOHEISEL T. and KANZOW C. (2008) Stationary conditions for mathematical programs with vanishing constraints using weak constraint qualifications. Journal of Mathematical Analysis and Applications, 337, 292–310.
Search in Google Scholar Back to article
HOHEISEL T. and KANZOW C. (2009) On the Abadie and Guignard constraint qualification for mathematical progams with vanishing constraints. Optimization, 58, 431–448.
Search in Google Scholar Back to article
HU, Q. J., CHEN, Y., ZHU, Z. B. and 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. https://doi.org/10.1155/2014/715015
Search in Google Scholar Back to article
HU, Q. J., WANG, J. G., CHEN, Y. and ZHU, Z. B. (2017) On an l1 exact penalty result for mathematical programs with vanishing constraints. Optimization Letters, 11 (3), 641–653.
Search in Google Scholar Back to article
HU, Q., WANG, J. and CHEN, Y. (2020) New dualities for mathematical programs with vanishing constraints. Annals of Operations Research, 287 (1), 233–255.
Search in Google Scholar Back to article
JABR, R. A. (2012) Solution to economic dispatching with disjoint feasible regions via semidefinite programming. IEEE Transactions on Power Systems, 27 (1), 572–573.
Search in Google Scholar Back to article
JOSHI, B. C. (2021a) Optimality and duality for nonsmooth semi-infinite mathematical program with equilibrium constraints involving generalized invexity of order α > 0. RAIRO - Operations Research, 55, 2221–2240.
Search in Google Scholar Back to article
JOSHI, B. C. (2021b) On generalized approximate convex functions and variational inequalities. RAIRO-Operations Research, 55, S2999–S3008.
Search in Google Scholar Back to article
JOSHI, B. C., MISHRA, S. K. and KUMAR, P. (2020) On Semi-infinite Mathematical Programming Problems with Equilibrium Constraints Using Generalized Convexity. Journal of the Operations Research Society of China, 8 (4), 619–636.
Search in Google Scholar Back to article
KAZEMI, S. and KANZI, N. (2018) Constraint qualifications and stationary conditions for mathematical programming with non-differentiable vanishing constraints. Journal of Optimization Theory and Applications, 179 (3), 800–819.
Search in Google Scholar Back to article
MISHRA, S. K. and SHUKLA, K. (2010) Nonsmooth minimax programming problems with V -r-invex functions. Optimization, 59 (1), 95–103.
Search in Google Scholar Back to article
MISHRA, S. K., SINGH, V. and LAHA, V. (2016) On duality for mathematical programs with vanishing constraints. Annals of Operations Research, 243 (1), 249–272.
Search in Google Scholar Back to article
MICHAEL, N. J., KIRCHES, C. and SAGER, S. (2013) On perspective functions and vanishing constraints in mixedinteger nonlinear optimal control. In: M. Jünger and G. Reinelt, eds., Facets of Combinatorial Optimization, 387–417. Berlin: Springer.
Search in Google Scholar Back to article
MOND, B. and WEIR, T. (1981) Generalized concavity and duality. In: S. Schaible and W. T. Ziemba, eds., Generalized Concavity in Optimization and Economics, 263–279. New York: Academic Press.
Search in Google Scholar Back to article
SAGLAM, S. D. and MAHMUDOV, E. (2022) On Duality in Convex Optimization of Second-Order Differential Inclusions with Periodic Boundary Conditions. Hacettepe Journal of Mathematics and Statistics, 51 (6), 1588–1599.
Search in Google Scholar Back to article
TUNG, L. T. (2020) Karush–Kuhn–Tucker optimality conditions and duality for multiobjective semi-infinite programming with vanishing constraints. Annals of Operations Research, 311, 1307–1334.
Search in Google Scholar Back to article
TUNG, L. T. and TAM, D. H. (2021) Optimality Conditions and Duality for Multi-objective Semi-infinite Programming on Hadamard Manifolds. Bulletin of the Iranian Mathematical Society, 48, 2191–2219.
Search in Google Scholar Back to article
WOLFE, P. (1961) A duality theorem for nonlinear programming. Quarterly of Applied Mathematics, 19, 239–244.
Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.2478/candc-2023-0042 | Journal eISSN: 2720-4278 | Journal ISSN: 0324-8569

Journal RSS Feed

Language: English

Page range: 351 - 379

Submitted on: Nov 1, 2022

Accepted on: Jun 1, 2024

Published on: Nov 9, 2024

Published by: Systems Research Institute Polish Academy of Sciences

In partnership with: Paradigm Publishing Services

Publication frequency: 4 issues per year

Keywords:

vanishing constraints,

dual models,

generalized convexity,

constraint qualifications

Related subjects:

Computer sciences,

Computer sciences, other,

Engineering,

Electrical engineering,

Fundamentals of electrical engineering,

Mechanical engineering,

Fundamentals of mechanical engineering,

Mathematics,

General mathematics

© 2024 Bhuwan Chandra Joshi, published by Systems Research Institute Polish Academy of Sciences
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Volume 52 (2023): Issue 4 (December 2023)