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A Theoretical Framework for Optimality Conditions of Nonlinear Type-2 Interval-Valued Unconstrained and Constrained Optimization Problems Using Type-2 Interval Order Relations

Author

Listed:
  • Md Sadikur Rahman

    (Department of Mathematics, The University of Burdwan, Golapbag, Bardhaman 713104, West Bengal, India)

  • Ali Akbar Shaikh

    (Department of Mathematics, The University of Burdwan, Golapbag, Bardhaman 713104, West Bengal, India)

  • Irfan Ali

    (Department of Statistic and Operations Research, AMU Campus, Aligarh Muslim University, Shiv Puri, Aligarh 202001, Uttar Pradesh, India)

  • Asoke Kumar Bhunia

    (Department of Mathematics, The University of Burdwan, Golapbag, Bardhaman 713104, West Bengal, India)

  • Armin Fügenschuh

    (Institute for Mathematics, Faculty of Mathematics, Computer Science, Physics, Electrical Engineering and Information Technology, Brandenburg University of Technology Cottbus-Senftenberg, Platz der Deutschen Einheit 1, 03046 Cottbus, Germany)

Abstract

In the traditional nonlinear optimization theory, the Karush-Kuhn-Tucker (KKT) optimality conditions for constrained optimization problems with inequality constraints play an essential role. The situation becomes challenging when the theory of traditional optimization is discussed under uncertainty. Several researchers have discussed the interval approach to tackle nonlinear optimization uncertainty and derived the optimality conditions. However, there are several realistic situations in which the interval approach is not suitable. This study aims to introduce the Type-2 interval approach to overcome the limitation of the classical interval approach. This study introduces Type-2 interval order relation and Type-2 interval-valued function concepts to derive generalized KKT optimality conditions for constrained optimization problems under uncertain environments. Then, the optimality conditions are discussed for the unconstrained Type-2 interval-valued optimization problem and after that, using these conditions, generalized KKT conditions are derived. Finally, the proposed approach is demonstrated by numerical examples.

Suggested Citation

  • Md Sadikur Rahman & Ali Akbar Shaikh & Irfan Ali & Asoke Kumar Bhunia & Armin Fügenschuh, 2021. "A Theoretical Framework for Optimality Conditions of Nonlinear Type-2 Interval-Valued Unconstrained and Constrained Optimization Problems Using Type-2 Interval Order Relations," Mathematics, MDPI, vol. 9(8), pages 1-22, April.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:8:p:908-:d:539031
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    References listed on IDEAS

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    3. Panigrahi, Motilal & Panda, Geetanjali & Nanda, Sudarsan, 2008. "Convex fuzzy mapping with differentiability and its application in fuzzy optimization," European Journal of Operational Research, Elsevier, vol. 185(1), pages 47-62, February.
    4. Luciano Stefanini & Barnabas Bede, 2008. "Generalized Hukuhara Differentiability of Interval-valued Functions and Interval Differential Equations," Working Papers 0803, University of Urbino Carlo Bo, Department of Economics, Society & Politics - Scientific Committee - L. Stefanini & G. Travaglini, revised 2008.
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