Hestenes–Stiefel-Type Conjugate Direction Algorithm for Interval-Valued Multiobjective Optimization Problems

Pandey, Rupesh Krishna; Upadhyay, Balendu Bhooshan; Poddar, Subham; Stancu-Minasian, Ioan

doi:10.3390/a18070381

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Hestenes–Stiefel-Type Conjugate Direction Algorithm for Interval-Valued Multiobjective Optimization Problems

by

Rupesh Krishna Pandey

^1,†

,

Balendu Bhooshan Upadhyay

^1,†

,

Subham Poddar

^1,†

and

Ioan Stancu-Minasian

^2,*,†

¹

Department of Mathematics, Indian Institute of Technology Patna, Patna 801106, Bihar, India

²

“Gheorghe Mihoc-Caius Iacob” Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy, 050711 Bucharest, Romania

^*

Author to whom correspondence should be addressed.

^†

These authors contributed equally to this work.

Algorithms 2025, 18(7), 381; https://doi.org/10.3390/a18070381

Submission received: 17 May 2025 / Revised: 19 June 2025 / Accepted: 20 June 2025 / Published: 23 June 2025

(This article belongs to the Special Issue Multi-Objective and Multi-Level Optimization: Algorithms and Applications (2nd Edition))

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Abstract

This article investigates a class of interval-valued multiobjective optimization problems (IVMOPs). We define the Hestenes–Stiefel (HS)-type direction for the objective function of IVMOPs and establish that it has a descent property at noncritical points. An Armijo-like line search is employed to determine an appropriate step size. We present an HS-type conjugate direction algorithm for IVMOPs and establish the convergence of the sequence generated by the algorithm. We deduce that the proposed algorithm exhibits a linear order of convergence under appropriate assumptions. Moreover, we investigate the worst-case complexity of the sequence generated by the proposed algorithm. Furthermore, we furnish several numerical examples, including a large-scale IVMOP, to demonstrate the effectiveness of our proposed algorithm and solve them by employing MATLAB. To the best of our knowledge, the HS-type conjugate direction method has not yet been explored for the class of IVMOPs.

Keywords: Hestenes–Stiefel method; interval-valued optimization; generalized Hukuhara derivative; conjugate direction method; multiobjective optimization

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MDPI and ACS Style

Pandey, R.K.; Upadhyay, B.B.; Poddar, S.; Stancu-Minasian, I. Hestenes–Stiefel-Type Conjugate Direction Algorithm for Interval-Valued Multiobjective Optimization Problems. Algorithms 2025, 18, 381. https://doi.org/10.3390/a18070381

AMA Style

Pandey RK, Upadhyay BB, Poddar S, Stancu-Minasian I. Hestenes–Stiefel-Type Conjugate Direction Algorithm for Interval-Valued Multiobjective Optimization Problems. Algorithms. 2025; 18(7):381. https://doi.org/10.3390/a18070381

Chicago/Turabian Style

Pandey, Rupesh Krishna, Balendu Bhooshan Upadhyay, Subham Poddar, and Ioan Stancu-Minasian. 2025. "Hestenes–Stiefel-Type Conjugate Direction Algorithm for Interval-Valued Multiobjective Optimization Problems" Algorithms 18, no. 7: 381. https://doi.org/10.3390/a18070381

APA Style

Pandey, R. K., Upadhyay, B. B., Poddar, S., & Stancu-Minasian, I. (2025). Hestenes–Stiefel-Type Conjugate Direction Algorithm for Interval-Valued Multiobjective Optimization Problems. Algorithms, 18(7), 381. https://doi.org/10.3390/a18070381

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

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Hestenes–Stiefel-Type Conjugate Direction Algorithm for Interval-Valued Multiobjective Optimization Problems

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