Optimality conditions for nonsmooth interval-valued multiobjective semi-infinite programming problem subject to switching constraints via tangential subdifferentials
Abstract
This paper explores optimality conditions for a nonsmooth interval-valued mul
tiobjective semi-infinite programming problem with switching constraints. Specif
ically, we use an appropriate constraint qualification to establish necessary M
stationary conditions utilizing tangential subdifferentials. Furthermore, sufficient
optimality conditions are derived based on generalized convexity. Results are well
illustrated by example.
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References
References
[1] N. A. Gadhi, and M. El Idrissi, Necessary optimality conditions for a multiobjective
semi-infinite interval-valued programming problem, Optimization Letters, vol. 16,
no. 2, pp. 653–666, 2022.
[2] N. Huy Hung, H. Ngoc Tuan, and N. Van Tuyen, On approximate quasi Pareto
solutions in nonsmooth semi-infinite interval-valued vector optimization problems,
Applicable Analysis, vol. 102, no. 9, pp. 2432–2448, 2023.
[3] A. Dwivedi, S. K. Mishra, J. Shi, and V. Laha, On optimality and duality for
approximate solutions in nonsmooth interval-valued multiobjective semi-infinite pro
gramming, Applied Analysis and Optimization, vol. 8, no. 2, pp. 145–165, 2024.
[4] M. Jennane, E. M. Kalmoun, and L. Lafhim, Optimality conditions for nonsmooth
interval-valued and multiobjective semi-infinite programming, RAIRO-Operations
Research, vol. 55, no. 1, pp. 1–11, 2021.
[5] L.T. Tung, Karush-Kuhn-Tucker optimality conditions and duality for a semi-infinite
programming with multiple interval-valued objective functions, Journal of Nonlinear
Functional Analysis, vol. 2019, pp. 1–21, 2019.
[6] T. Antczak, and A. Farajzadeh, On nondifferentiable semi-infinite multiobjective
programming with interval-valued functions, Journal of Industrial & Management
Optimization, vol. 19, no. 8, 2023.
[7] P. Mehlitz, Stationarity conditions and constraint qualifications for mathematical
programs with switching constraints: with applications to either-or-constrained pro
gramming, Mathematical Programming, vol. 181, no. 1, pp. 149–186, 2020.
[8] G. Li, and L. Guo, Mordukhovich stationarity for mathematical programs with
switching constraints under weak constraint qualifications, Optimization, vol. 72, no.
7, pp. 1817–1838, 2023.
[9] M. Gugat, Optimal switching boundary control of a string to rest in finite time,
ZAMM-Journal of Applied Mathematics and Mechanics, vol. 88, no. 4, pp. 283–305,
2008.
[10] T. I. Seidman, Optimal control of a diffusion/reaction/switching system, Evolution
Equations and Control Theory, vol. 2, no. 4, pp. 723–731, 2013.
[11] L. Wang, and Q. Yan, Time optimal controls of semilinear heat equation with
switching control, Journal of Optimization Theory and Applications, vol. 165, pp.
263–278, 2015.
[12] F. M. Hante, and S. Sager, Relaxation methods for mixed-integer optimal control
of partial differential equations, Computational Optimization and Applications, vol.
55, pp. 197–225, 2013.
[13] C. Clason, A. Rund, K. Kunisch, and R. C. Barnard, A convex penalty for switching
control of partial differential equations, Systems & Control Letters, vol. 89, pp.
66–73, 2016.
15
[14] C. Clason, A. Rund, and K. Kunisch, Nonconvex penalization of switching control
of partial differential equations, Systems & Control Letters, vol. 106, pp. 1–8, 2017.
[15] V. Laha, H. N. Singh, and S. K. Mishra, On quasidifferentiable mathematical
programming problems with vanishing, Convex Optimization—Theory, Algorithms
and Applications: RTCOTAA-2020, Patna, India, October 29–31, Springer Nature,
vol. 476, pp. 423, 2025.
[16] S. K. Mishra, V. Singh, V. Laha, and R. N. Mohapatra, On constraint qualifications
for multiobjective optimization problems with vanishing constraints, Optimization
Methods, Theory and Applications, pp. 95–135, 2015, Springer.
[17] V. Laha, V. Singh, Y. Pandey, and S. K. Mishra, Nonsmooth mathematical programs
with vanishing constraints in Banach spaces, in High-Dimensional Optimization and
Probability: With a View Towards Data Science, pp. 395–417, 2022, Springer.
[18] V. Laha, R. Kumar, H. N. Singh, and S. K. Mishra, On minimax programming
with vanishing constraints, in Optimization, Variational Analysis and Applications:
IFSOVAA-2020, Varanasi, India, February 2–4, Springer, pp. 247–263, 2021.
[19] V. Laha and L. Pandey, On mathematical programs with equilibrium constraints
under data uncertainty, in Proc. Int. Conf. on Nonlinear Applied Analysis and
Optimization, pp. 283–300, 2021, Springer.
[20] V. Laha and H. N. Singh, On quasidifferentiable mathematical programs with equi
librium constraints, Computational Management Science, vol. 20, no. 1, pp. 30,
2023.
[21] R. Pandey, Y. Pandey, and V. Singh, Optimality conditions for mathematical pro
gramming problem with equilibrium constraints in terms of tangential subdifferen
tiable, Communications in Combinatorics and Optimization, pp. 1–22, 2025.
[22] Y. C. Liang, and J. J. Ye, Optimality conditions and exact penalty for mathe
matical programs with switching constraints, Journal of Optimization Theory and
Applications, vol. 190, no. 1, pp. 1–31, 2021.
[23] V. Shikhman, Topological approach to mathematical programs with switching con
straints, Set-Valued and Variational Analysis, vol. 30, no. 2, pp. 335–354, 2022.
[24] C. Kanzow, P. Mehlitz, and D. Steck, Relaxation schemes for mathematical pro
grammes with switching constraints, Optimization Methods and Software, vol. 36,
no. 6, pp. 1223–1258, 2021.
[25] F. G. Shabankareh, N. Kanzi, K. Fallahi, and J. Izadi, Stationarity in nonsmooth
optimization with switching constraints, Iranian Journal of Science and Technology,
Transactions A: Science, vol. 46, no. 3, pp. 907–915, 2022.
[26] Y. Pandey, and V. Singh, On Constraint qualifications for multiobjective optimiza
tion problems with switching constraints, in Optimization, Variational Analysis and
Applications: IFSOVAA-2020, Varanasi, India, February 2–4, pp. 283–306, 2021.
16
[27] M. Jennane, E. M. Kalmoun, and L. Lafhim, On nonsmooth multiobjective semi
infinite programming with switching constraints, Journal of Applied & Numerical
Optimization, vol. 5, no. 1, 2023.
[28] M. Jennane, and E. M. Kalmoun, On nonsmooth multiobjective semi-infinite pro
gramming with switching constraints using tangential subdifferentials, Statistics, Op
timization & Information Computing, vol. 11, no. 1, pp. 22–28, 2023.
[29] B. B. Upadhyay, and A. Ghosh, Optimality conditions and duality for multiob
jective semi-infinite optimization problems with switching constraints on Hadamard
manifolds, Positivity, vol. 28, no. 4, pp. 49, 2024.
[30] B. N. Pshenichnyi, Necessary conditions for an extremum, CRC Press, 2020.
[31] C. Lemar´echal, An introduction to the theory of nonsmooth optimization, Optimiza
tion, vol. 17, no. 6, pp. 827–858, 1986.
[32] J. E. Mart´ınez-Legaz, Optimality conditions for pseudoconvex minimization over
convex sets defined by tangentially convex constraints, Optimization Letters, vol. 9,
no. 5, pp. 1017–1023, 2015.
[33] Y. Ishizuka, Optimality conditions for directionally differentiable multi-objective
programming problems, Journal of Optimization Theory and Applications, vol. 72,
pp. 91–111, 1992.
[34] B. Jim´enez, and V. Novo, Optimality conditions in directionally differentiable Pareto
problems with a set constraint via tangent cones, Numerical Functional Analysis and
Optimization, vol. 24, no. 5-6, pp. 557–574, 2003.
[35] X.J. Long, J. Liu, and N.J. Huang, Characterizing the solution set for nonconvex
semi-infinite programs involving tangential subdifferentials, Numerical Functional
Analysis and Optimization, vol. 42, no. 3, pp. 279–297, 2021.
[36] J. Liu, X.J. Long, and X.K. Sun, Characterizing robust optimal solution sets for
nonconvex uncertain semi-infinite programming problems involving tangential subd
ifferentials, Journal of Global Optimization, vol. 87, no. 2, pp. 481–501, 2023.
[37] G. Giorgi, B. Jimenez, and V. Novo, Minimum principle-type optimality conditions
for Pareto problems, International Journal of Pure and Applied Mathematics, vol.
10, pp. 51–68, 2004.
[38] L. T. Tung, Strong Karush–Kuhn–Tucker optimality conditions for multiobjec
tive semi-infinite programming via tangential subdifferential, RAIRO-Operations
Research, vol. 52, no. 4-5, pp. 1019–1041, 2018.
[39] L. T. Tung, Karush-Kuhn-Tucker optimality conditions and duality for multiobjec
tive semi-infinite programming via tangential subdifferentials, Numerical Functional
Analysis and Optimization, vol. 41, no. 6, pp. 659–684, 2020.
[40] J. Liu, X.J. Long, and N.J. Huang, Approximate optimality conditions and mixed type
duality for semi-infinite multiobjective programming problems involving tangential
subdifferentials, Journal of Industrial & Management Optimization, vol. 19, no. 9,
2023.
17
[41] X.J. Long, W. Zhang, G.H. Li, and J.W. Peng, Approximate quasiefficient solu
tions of nonsmooth multiobjective semi-infinite programming problems in terms of
tangential subdifferentials, Journal of Nonlinear and Convex Analysis, vol. 25, pp.
2487–2506, 2024.
[42] R. E. Moore, Interval analysis, Prentice-Hall, Englewood Cliffs, NJ, vol. 4, pp.
8–13, 1966.
[43] G. Alefeld, and J. Herzberger, Introduction to interval computation, Academic
Press, New York, 2012.
[44] R. E. Moore, Methods and applications of interval analysis, SIAM, Philadelphia,
1979.
[45] L. T. Tung, Karush–Kuhn–Tucker optimality conditions and duality for convex
semi-infinite programming with multiple interval-valued objective functions, Journal
of Applied Mathematics and Computing, vol. 62, pp. 67–91, 2020.
[46] B. B. Upadhyay, L. Li, and P. Mishra, Nonsmooth interval-valued multiobjective op
timization problems and generalized variational inequalities on Hadamard manifolds,
Applied Set-Valued Analysis & Optimization, vol. 5, no. 1, 2023.
[47] G. Giorgi, B. Jim´enez, and V. Novo, On constraint qualifications in directionally
differentiable multiobjective optimization problems, RAIRO-Operations Research,
vol. 38, no. 3, pp. 255–274, 2004.
[48] G. Giorgi, and S. Koml´osi, Dini derivatives in optimization—Part I, Rivista di
matematica per le scienze economiche e sociali, vol. 15, no. 1, pp. 3–30, 1992.
[49] V. Laha and A. Dwivedi, On approximate strong KKT points of nonsmooth interval
valued multiobjective optimization problems using convexificators, The Journal of
Analysis, vol. 32, no. 1, pp. 219–242, 2024.
[50] S. K. Mishra, V. Singh, and V. Laha, On duality for mathematical programs with
vanishing constraints, Annals of Operations Research, vol. 243, pp. 249–272, 2016.
[51] P. Sachan and V. Laha, On multiobjective optimization problems involving higher
order strong convexity using directional convexificators, OPSEARCH, pp. 1–24, 2024,
Springer.
[52] R. N. Mohapatra, P. Sachan, and V. Laha, Optimality conditions for mathematical
programs with vanishing constraints using directional convexificators, Axioms, vol.
13, no. 8, pp. 516, 2024.
[53] V. Laha, A. Dwivedi, and P. Jaiswal, On quasidifferentiable interval-valued multi
objective optimization, Annals of Operations Research, pp. 1–31, 2025.
[1] N. A. Gadhi, and M. El Idrissi, Necessary optimality conditions for a multiobjective
semi-infinite interval-valued programming problem, Optimization Letters, vol. 16,
no. 2, pp. 653–666, 2022.
[2] N. Huy Hung, H. Ngoc Tuan, and N. Van Tuyen, On approximate quasi Pareto
solutions in nonsmooth semi-infinite interval-valued vector optimization problems,
Applicable Analysis, vol. 102, no. 9, pp. 2432–2448, 2023.
[3] A. Dwivedi, S. K. Mishra, J. Shi, and V. Laha, On optimality and duality for
approximate solutions in nonsmooth interval-valued multiobjective semi-infinite pro
gramming, Applied Analysis and Optimization, vol. 8, no. 2, pp. 145–165, 2024.
[4] M. Jennane, E. M. Kalmoun, and L. Lafhim, Optimality conditions for nonsmooth
interval-valued and multiobjective semi-infinite programming, RAIRO-Operations
Research, vol. 55, no. 1, pp. 1–11, 2021.
[5] L.T. Tung, Karush-Kuhn-Tucker optimality conditions and duality for a semi-infinite
programming with multiple interval-valued objective functions, Journal of Nonlinear
Functional Analysis, vol. 2019, pp. 1–21, 2019.
[6] T. Antczak, and A. Farajzadeh, On nondifferentiable semi-infinite multiobjective
programming with interval-valued functions, Journal of Industrial & Management
Optimization, vol. 19, no. 8, 2023.
[7] P. Mehlitz, Stationarity conditions and constraint qualifications for mathematical
programs with switching constraints: with applications to either-or-constrained pro
gramming, Mathematical Programming, vol. 181, no. 1, pp. 149–186, 2020.
[8] G. Li, and L. Guo, Mordukhovich stationarity for mathematical programs with
switching constraints under weak constraint qualifications, Optimization, vol. 72, no.
7, pp. 1817–1838, 2023.
[9] M. Gugat, Optimal switching boundary control of a string to rest in finite time,
ZAMM-Journal of Applied Mathematics and Mechanics, vol. 88, no. 4, pp. 283–305,
2008.
[10] T. I. Seidman, Optimal control of a diffusion/reaction/switching system, Evolution
Equations and Control Theory, vol. 2, no. 4, pp. 723–731, 2013.
[11] L. Wang, and Q. Yan, Time optimal controls of semilinear heat equation with
switching control, Journal of Optimization Theory and Applications, vol. 165, pp.
263–278, 2015.
[12] F. M. Hante, and S. Sager, Relaxation methods for mixed-integer optimal control
of partial differential equations, Computational Optimization and Applications, vol.
55, pp. 197–225, 2013.
[13] C. Clason, A. Rund, K. Kunisch, and R. C. Barnard, A convex penalty for switching
control of partial differential equations, Systems & Control Letters, vol. 89, pp.
66–73, 2016.
15
[14] C. Clason, A. Rund, and K. Kunisch, Nonconvex penalization of switching control
of partial differential equations, Systems & Control Letters, vol. 106, pp. 1–8, 2017.
[15] V. Laha, H. N. Singh, and S. K. Mishra, On quasidifferentiable mathematical
programming problems with vanishing, Convex Optimization—Theory, Algorithms
and Applications: RTCOTAA-2020, Patna, India, October 29–31, Springer Nature,
vol. 476, pp. 423, 2025.
[16] S. K. Mishra, V. Singh, V. Laha, and R. N. Mohapatra, On constraint qualifications
for multiobjective optimization problems with vanishing constraints, Optimization
Methods, Theory and Applications, pp. 95–135, 2015, Springer.
[17] V. Laha, V. Singh, Y. Pandey, and S. K. Mishra, Nonsmooth mathematical programs
with vanishing constraints in Banach spaces, in High-Dimensional Optimization and
Probability: With a View Towards Data Science, pp. 395–417, 2022, Springer.
[18] V. Laha, R. Kumar, H. N. Singh, and S. K. Mishra, On minimax programming
with vanishing constraints, in Optimization, Variational Analysis and Applications:
IFSOVAA-2020, Varanasi, India, February 2–4, Springer, pp. 247–263, 2021.
[19] V. Laha and L. Pandey, On mathematical programs with equilibrium constraints
under data uncertainty, in Proc. Int. Conf. on Nonlinear Applied Analysis and
Optimization, pp. 283–300, 2021, Springer.
[20] V. Laha and H. N. Singh, On quasidifferentiable mathematical programs with equi
librium constraints, Computational Management Science, vol. 20, no. 1, pp. 30,
2023.
[21] R. Pandey, Y. Pandey, and V. Singh, Optimality conditions for mathematical pro
gramming problem with equilibrium constraints in terms of tangential subdifferen
tiable, Communications in Combinatorics and Optimization, pp. 1–22, 2025.
[22] Y. C. Liang, and J. J. Ye, Optimality conditions and exact penalty for mathe
matical programs with switching constraints, Journal of Optimization Theory and
Applications, vol. 190, no. 1, pp. 1–31, 2021.
[23] V. Shikhman, Topological approach to mathematical programs with switching con
straints, Set-Valued and Variational Analysis, vol. 30, no. 2, pp. 335–354, 2022.
[24] C. Kanzow, P. Mehlitz, and D. Steck, Relaxation schemes for mathematical pro
grammes with switching constraints, Optimization Methods and Software, vol. 36,
no. 6, pp. 1223–1258, 2021.
[25] F. G. Shabankareh, N. Kanzi, K. Fallahi, and J. Izadi, Stationarity in nonsmooth
optimization with switching constraints, Iranian Journal of Science and Technology,
Transactions A: Science, vol. 46, no. 3, pp. 907–915, 2022.
[26] Y. Pandey, and V. Singh, On Constraint qualifications for multiobjective optimiza
tion problems with switching constraints, in Optimization, Variational Analysis and
Applications: IFSOVAA-2020, Varanasi, India, February 2–4, pp. 283–306, 2021.
16
[27] M. Jennane, E. M. Kalmoun, and L. Lafhim, On nonsmooth multiobjective semi
infinite programming with switching constraints, Journal of Applied & Numerical
Optimization, vol. 5, no. 1, 2023.
[28] M. Jennane, and E. M. Kalmoun, On nonsmooth multiobjective semi-infinite pro
gramming with switching constraints using tangential subdifferentials, Statistics, Op
timization & Information Computing, vol. 11, no. 1, pp. 22–28, 2023.
[29] B. B. Upadhyay, and A. Ghosh, Optimality conditions and duality for multiob
jective semi-infinite optimization problems with switching constraints on Hadamard
manifolds, Positivity, vol. 28, no. 4, pp. 49, 2024.
[30] B. N. Pshenichnyi, Necessary conditions for an extremum, CRC Press, 2020.
[31] C. Lemar´echal, An introduction to the theory of nonsmooth optimization, Optimiza
tion, vol. 17, no. 6, pp. 827–858, 1986.
[32] J. E. Mart´ınez-Legaz, Optimality conditions for pseudoconvex minimization over
convex sets defined by tangentially convex constraints, Optimization Letters, vol. 9,
no. 5, pp. 1017–1023, 2015.
[33] Y. Ishizuka, Optimality conditions for directionally differentiable multi-objective
programming problems, Journal of Optimization Theory and Applications, vol. 72,
pp. 91–111, 1992.
[34] B. Jim´enez, and V. Novo, Optimality conditions in directionally differentiable Pareto
problems with a set constraint via tangent cones, Numerical Functional Analysis and
Optimization, vol. 24, no. 5-6, pp. 557–574, 2003.
[35] X.J. Long, J. Liu, and N.J. Huang, Characterizing the solution set for nonconvex
semi-infinite programs involving tangential subdifferentials, Numerical Functional
Analysis and Optimization, vol. 42, no. 3, pp. 279–297, 2021.
[36] J. Liu, X.J. Long, and X.K. Sun, Characterizing robust optimal solution sets for
nonconvex uncertain semi-infinite programming problems involving tangential subd
ifferentials, Journal of Global Optimization, vol. 87, no. 2, pp. 481–501, 2023.
[37] G. Giorgi, B. Jimenez, and V. Novo, Minimum principle-type optimality conditions
for Pareto problems, International Journal of Pure and Applied Mathematics, vol.
10, pp. 51–68, 2004.
[38] L. T. Tung, Strong Karush–Kuhn–Tucker optimality conditions for multiobjec
tive semi-infinite programming via tangential subdifferential, RAIRO-Operations
Research, vol. 52, no. 4-5, pp. 1019–1041, 2018.
[39] L. T. Tung, Karush-Kuhn-Tucker optimality conditions and duality for multiobjec
tive semi-infinite programming via tangential subdifferentials, Numerical Functional
Analysis and Optimization, vol. 41, no. 6, pp. 659–684, 2020.
[40] J. Liu, X.J. Long, and N.J. Huang, Approximate optimality conditions and mixed type
duality for semi-infinite multiobjective programming problems involving tangential
subdifferentials, Journal of Industrial & Management Optimization, vol. 19, no. 9,
2023.
17
[41] X.J. Long, W. Zhang, G.H. Li, and J.W. Peng, Approximate quasiefficient solu
tions of nonsmooth multiobjective semi-infinite programming problems in terms of
tangential subdifferentials, Journal of Nonlinear and Convex Analysis, vol. 25, pp.
2487–2506, 2024.
[42] R. E. Moore, Interval analysis, Prentice-Hall, Englewood Cliffs, NJ, vol. 4, pp.
8–13, 1966.
[43] G. Alefeld, and J. Herzberger, Introduction to interval computation, Academic
Press, New York, 2012.
[44] R. E. Moore, Methods and applications of interval analysis, SIAM, Philadelphia,
1979.
[45] L. T. Tung, Karush–Kuhn–Tucker optimality conditions and duality for convex
semi-infinite programming with multiple interval-valued objective functions, Journal
of Applied Mathematics and Computing, vol. 62, pp. 67–91, 2020.
[46] B. B. Upadhyay, L. Li, and P. Mishra, Nonsmooth interval-valued multiobjective op
timization problems and generalized variational inequalities on Hadamard manifolds,
Applied Set-Valued Analysis & Optimization, vol. 5, no. 1, 2023.
[47] G. Giorgi, B. Jim´enez, and V. Novo, On constraint qualifications in directionally
differentiable multiobjective optimization problems, RAIRO-Operations Research,
vol. 38, no. 3, pp. 255–274, 2004.
[48] G. Giorgi, and S. Koml´osi, Dini derivatives in optimization—Part I, Rivista di
matematica per le scienze economiche e sociali, vol. 15, no. 1, pp. 3–30, 1992.
[49] V. Laha and A. Dwivedi, On approximate strong KKT points of nonsmooth interval
valued multiobjective optimization problems using convexificators, The Journal of
Analysis, vol. 32, no. 1, pp. 219–242, 2024.
[50] S. K. Mishra, V. Singh, and V. Laha, On duality for mathematical programs with
vanishing constraints, Annals of Operations Research, vol. 243, pp. 249–272, 2016.
[51] P. Sachan and V. Laha, On multiobjective optimization problems involving higher
order strong convexity using directional convexificators, OPSEARCH, pp. 1–24, 2024,
Springer.
[52] R. N. Mohapatra, P. Sachan, and V. Laha, Optimality conditions for mathematical
programs with vanishing constraints using directional convexificators, Axioms, vol.
13, no. 8, pp. 516, 2024.
[53] V. Laha, A. Dwivedi, and P. Jaiswal, On quasidifferentiable interval-valued multi
objective optimization, Annals of Operations Research, pp. 1–31, 2025.
Published
2025-12-21
Section
Mathematics and Computing - Innovations and Applications (ICMSC-2025)
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