2023 Volume 13 Issue 5
Article Contents
Article navigation > Journal of Applied Analysis & Computation > 2023 > 13(5): 2613-2629
Next Article Previous Article

Tadeusz Antczak. OPTIMALITY RESULTS FOR NONDIFFERENTIABLE VECTOR OPTIMIZATION PROBLEMS WITH VANISHING CONSTRAINTS[J]. Journal of Applied Analysis & Computation, 2023, 13(5): 2613-2629. doi: 10.11948/20220465
Citation: Tadeusz Antczak. OPTIMALITY RESULTS FOR NONDIFFERENTIABLE VECTOR OPTIMIZATION PROBLEMS WITH VANISHING CONSTRAINTS[J]. Journal of Applied Analysis & Computation, 2023, 13(5): 2613-2629. doi: 10.11948/20220465
shu

OPTIMALITY RESULTS FOR NONDIFFERENTIABLE VECTOR OPTIMIZATION PROBLEMS WITH VANISHING CONSTRAINTS

  • 1.

    Department of Mathematics and Computer Science, University of Lodz, Banacha 22, 90-238 Lodz, Poland

    Abstract

  • At present, some real extremum problems related to the activity of modern man, for example, in industry, economy, optimal control, engineering, mechanics, are modeled by optimization problems with vanishing constraints. In this paper, a class of nondifferentiable vector optimization problems with vanishing constraints is considered in which every component of the involved functions is locally Lipschitz. This kind of extremum problems is generally difficult to deal with, because of a special structure of constraints. The Karush-Kuhn-Tucker necessary optimality conditions are established for foregoing nonsmooth multicriteria optimization problems under the VC-Cottle constraint qualification. Sufficient optimality conditions are also proved for the considered nondifferentiable vector optimization problem with vanishing constraints under convexity hypotheses.

    MSC: 90C29, 90C30, 90C46, 90C25, 49J52
    • FullText(HTML)
  • 加载中
    • References
  • [1] W. Achtziger and C. Kanzow, Mathematical programs with vanishing constraints: optimality conditions and constraint qualifications, Math. Program., 2008, 114, 69–99. doi: 10.1007/s10107-006-0083-3

    CrossRef Google Scholar

    [2] W. Achtziger, T. Hoheisel and C. A. Kanzow, Smoothing-regularization approach to mathematical programs with vanishing constraints, Comput. Optim. Appl., 2013, 55, 733–767. doi: 10.1007/s10589-013-9539-6

    CrossRef Google Scholar

    [3] I. Ahmad, K. Kummari and S. Al-Homidan, Sufficiency and duality for interval-valued optimization problems with vanishing constraints using weak constraint qualifications, Int. J. Anal. Appl., 2020, 18, 784–798.

    Google Scholar

    [4] T. Antczak, Optimality conditions and Mond-Weir duality for a class of differentiable semi-infinite multiobjective programming problems with vanishing constraints, 4OR, 2022, 20, 417–442. doi: 10.1007/s10288-021-00482-1

    CrossRef Google Scholar

    [5] J. S. Ardakani, R. S. H. Farahmand, N. Kanzi and P.R. Ardabili, Necessary stationary conditions for multiobjective optimization, Iran. J. Sci. Technol. Trans. A Sci., 2019, 43, 2913–2919. doi: 10.1007/s40995-019-00768-4

    CrossRef Google Scholar

    [6] D. Barilla, G. Caristi and N. Kanzi, Stationarity condition for nonsmooth MPVCs with constraint set, in: Y. D. Sergeyev and D. E. Kvasov (eds.), NUMTA 2019, LNCS 11974, 314–321, Springer Nature Switzerland AG, 2020.

    Google Scholar

    [7] F H. Clarke, Optimization and Nonsmooth Analysis, A Wiley-Interscience Publication, John Wiley & Sons, Inc., 1983.

    Google Scholar

    [8] A. Dhara and A. Mehra, Metric regularity and optimality conditions in nonsmooth optimization. in: Mishra, S K. (ed.), Topics in Nonconvex Optimization: Theory and Applications. Springer Optimization and its Applications Vol. 50, 101–114, Springer New York Dordrecht Heidelberg London, 2011.

    Google Scholar

    [9] D. Dorsch, V. Shikhman and O. Stein, Mathematical programs with vanishing constraints: Critical point theory, J. Global Optim., 2012, 52, 591–605. doi: 10.1007/s10898-011-9805-z

    CrossRef Google Scholar

    [10] J. P. Dussault, M. Haddou and T. Migot, Mathematical programs with vanishing constraints: constraint qualifications, their applications and a new regularization method, Optimization, 2018, 68, 509–538.

    Google Scholar

    [11] S. M. Guu, Y. Singh, and S.K. Mishra, On strong KKT type sufficient optimality conditions for multiobjective semi-infinite programming problems with vanishing constraints, J. Inequal. Appl., 2017, 2017, 282. doi: 10.1186/s13660-017-1558-x

    CrossRef Google Scholar

    [12] T. Hoheisel and C. Kanzow, First- and second-order optimality conditions for mathematical programs with vanishing constraints, Appl. Math., 2007, 52, 495–514. doi: 10.1007/s10492-007-0029-y

    CrossRef Google Scholar

    [13] T. Hoheisel and C. Kanzow, Stationary conditions for mathematical programs with vanishing constraints using weak constraint qualifications, J. Math. Anal. Appl., 2008, 337, 292–310. doi: 10.1016/j.jmaa.2007.03.087

    CrossRef Google Scholar

    [14] T. Hoheisel and C. Kanzow, On the Abadie and Guignard constraint qualifications for mathematical programmes with vanishing constraints, Optimization, 2009, 58, 431–448. doi: 10.1080/02331930701763405

    CrossRef Google Scholar

    [15] T. Hoheisel, C. Kanzow and A. Schwartz, Mathematical programs with vanishing constraints: a new regularization approach with strong convergence properties, Optimization, 2012, 61, 619–636. doi: 10.1080/02331934.2011.608164

    CrossRef Google Scholar

    [16] Q. Hu, J. Wang, and Y. Chen, New dualities for mathematical programs with vanishing constraints, Ann. Oper. Res., 2020, 287, 233–255. doi: 10.1007/s10479-019-03409-6

    CrossRef Google Scholar

    [17] A. F. Izmailov and M. F. Solodov, Mathematical programs with vanishing constraints: optimality conditions. sensitivity, and relaxation method, J. Optim. Theory Appl., 2009, 142, 501–532. doi: 10.1007/s10957-009-9517-4

    CrossRef Google Scholar

    [18] R. A. Jabr, Solution to economic dispatching with disjoint feasible regions via semidefinite programming, IEEE Trans. Power Syst., 2012, 27, 572–573. doi: 10.1109/TPWRS.2011.2166009

    CrossRef Google Scholar

    [19] M. N. Jung, Ch. Kirches and S. Sager, On perspective functions and vanishing constraints in mixed-integer nonlinear optimal control, in: M. Jünger and G. Reinelt (eds.), Facets of Combinatorial Optimization, Springer, Heidelberg, 2013, 387–417.

    Google Scholar

    [20] S. Kazemi and N. Kanzi, Constraint qualifications and stationary conditions for mathematical programming with non-differentiable vanishing constraints, J. Optim. Theory Appl., 2018, 179, 800–819. doi: 10.1007/s10957-018-1373-7

    CrossRef Google Scholar

    [21] 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., 2019, 472, 1042–1077. doi: 10.1016/j.jmaa.2018.11.063

    CrossRef Google Scholar

    [22] C. Kirches, A. Potschka, H. G. Bock, and S. Sager, A parametric active set method for quadratic programs with vanishing constraints, Pacific J. Optim., 2013, 9, 275–299.

    Google Scholar

    [23] N. J. Michael, C. Kirches and S. Sager, On perspective functions and vanishing constraints in mixedinteger nonlinear optimal control, in: M. Jünger and G. Reinelt (eds.), Facets of combinatorial optimization. Berlin, Springer, 2013, 387–417.

    Google Scholar

    [24] S. K. Mishra, V. Singh, V. Laha and R. N. Mohapatra, On constraint qualifications for multiobjective optimization problems with vanishing constraints, in: H., Xu, S. Wang and S. -Y. Wu (eds.), Optimization Methods, Theory and Applications, 95–135, Berlin, Heidelberg Springer, 2015.

    Google Scholar

    [25] S. K. Mishra, V. Singh and V. Laha, On duality for mathematical programs with vanishing constraints, Ann. Oper. Res., 2016, 243, 249–272. doi: 10.1007/s10479-015-1814-8

    CrossRef Google Scholar

    [26] R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970.

    Google Scholar

    [27] T. V. Su and D. D. Hang, Optimality conditions and duality theorems for nonsmooth semi-infinite interval-valued mathematical programs with vanishing constraints, Comput. Appl. Math., 2022, 41, 422. doi: 10.1007/s40314-022-02139-z

    CrossRef Google Scholar

    [28] L. T. Tung, Karush-Kuhn-Tucker optimality conditions and duality for multiobjective semi-infinite programming with vanishing constraints, Ann. Oper. Res., 2022, 311, 1307–1334. doi: 10.1007/s10479-020-03742-1

    CrossRef Google Scholar

    [29] H. Wang and H. Wang, Duality theorems for nondifferentiable semi-infinite interval-valued optimization problems with vanishing constraints, J. Inequal. Appl., 2021, 2021, 182. doi: 10.1186/s13660-021-02717-5

    CrossRef Google Scholar

    • Supplements
    • Cited by
Export Citation
PDF
XML

Article Metrics

Article views(1617) PDF downloads(432) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Published by Wilmington Scientific Publisher