Geodesic
Duality for non-smooth semidefinite multiobjective programming problems with equilibrium constraints using convexificators
Upadhyay, B. B.1 ; Singh, S. K. 1 ; Stancu-Minasian, I‎. ‎M‎. ‎2
1 Department of Mathematics, ‎Indian Institute of Technology, ‎Patna, India
2 Gheorghe Mihoc-Caius Iacob, ‎Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy, ‎050711 Bucharest, Romania
Journal of nonlinear sciences and its applications, Tome 17 (2024) no. 3, p. 128-149.

Voir la notice de l'article provenant de la source International Scientific Research Publications

‎In this article‎, ‎we investigate the duality theorems for a class of non-smooth semidefinite multiobjective programming problems with equilibrium constraints (in short‎, ‎NSMPEC) via convexificators‎. ‎Utilizing the properties of convexificators‎, ‎we present Wolfe-type (in short‎, ‎WMPEC) and Mond-Weir-type (in short‎, ‎MWMPEC) dual models for the problem NSMPEC‎. ‎Furthermore‎, ‎we establish various duality theorems‎, ‎such as weak‎, ‎strong‎, ‎and strict converse duality theorems relating to the primal problem NSMPEC and the corresponding dual models‎, ‎in terms of convexificators‎. ‎Numerous illustrative examples are furnished to demonstrate the importance of the established results‎. ‎Furthermore‎, ‎we discuss an application of semidefinite multiobjective programming problems in approximating K-means-type clustering problems‎. ‎To the best of our knowledge‎, ‎duality results presented in this paper for NSMPEC using convexificators have not been explored before‎.
DOI : 10.22436/jnsa.017.03.03
Classification : 90C22, 90C46, 90C29
Keywords: Semidefinite programming, multiobjective optimization, duality, equilibrium constraints, convexificators

Upadhyay, B. B. 1 ; Singh, S. K.  1 ; Stancu-Minasian, I‎. ‎M‎. ‎ 2

1 Department of Mathematics, ‎Indian Institute of Technology, ‎Patna, India
2 Gheorghe Mihoc-Caius Iacob, ‎Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy, ‎050711 Bucharest, Romania 
@article{JNSA_2024_17_3_a2,
     author = {Upadhyay, B. B. and Singh, S. K.  and Stancu-Minasian, I‎. ‎M‎. ‎},
     title = {Duality for non-smooth semidefinite multiobjective programming problems with equilibrium constraints using convexificators},
     journal = {Journal of nonlinear sciences and its applications},
     pages = {128-149},
     publisher = {mathdoc},
     volume = {17},
     number = {3},
     year = {2024},
     doi = {10.22436/jnsa.017.03.03},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.22436/jnsa.017.03.03/}
}
TY  - JOUR
AU  - Upadhyay, B. B.
AU  - Singh, S. K. 
AU  - Stancu-Minasian, I‎. ‎M‎. ‎
TI  - Duality for non-smooth semidefinite multiobjective programming problems with equilibrium constraints using convexificators
JO  - Journal of nonlinear sciences and its applications
PY  - 2024
SP  - 128
EP  - 149
VL  - 17
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.22436/jnsa.017.03.03/
DO  - 10.22436/jnsa.017.03.03
LA  - en
ID  - JNSA_2024_17_3_a2
ER  - 
%0 Journal Article
%A Upadhyay, B. B.
%A Singh, S. K. 
%A Stancu-Minasian, I‎. ‎M‎. ‎
%T Duality for non-smooth semidefinite multiobjective programming problems with equilibrium constraints using convexificators
%J Journal of nonlinear sciences and its applications
%D 2024
%P 128-149
%V 17
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.22436/jnsa.017.03.03/
%R 10.22436/jnsa.017.03.03
%G en
%F JNSA_2024_17_3_a2
Upadhyay, B. B.; Singh, S. K. ; Stancu-Minasian, I‎. ‎M‎. ‎. Duality for non-smooth semidefinite multiobjective programming problems with equilibrium constraints using convexificators. Journal of nonlinear sciences and its applications, Tome 17 (2024) no. 3, p. 128-149. doi : 10.22436/jnsa.017.03.03. http://geodesic.mathdoc.fr/articles/10.22436/jnsa.017.03.03/

[1] Alawode, K. O.; Jubril, A. M.; Kehinde, L. O.; Ogunbona, P. O. Semidefinite programming solution of economic dispatch problem with non-smooth, non-convex cost functions, Electric Power Syst. Res., Volume 164 (2018), pp. 178-187 | DOI

[2] Antczak, T.; Mishra, S. K.; Upadhyay, B. B. First order duality for a new class of nonconvex semi-infinite minimax fractional programming problems, J. Adv. Math. Stud., Volume 9 (2016), pp. 132-162 | Zbl

[3] Ardali, A. Ansari; Movahedian, N.; Nobakhtian, S. Optimality conditions for non-smooth mathematical programs with equilibrium constraints, using convexificators, Optimization, Volume 65 (2016), pp. 67-85 | Zbl | DOI

[4] Ardali, A. Ansari; Zeinali, M.; Shirdel, G. H. Some Pareto optimality results for non-smooth multiobjective optimization problems with equilibrium constraints, Int. J. Nonlinear Anal. Appl., Volume 13 (2022), pp. 2185-2196 | DOI

[5] Boyd, S.; Ghaoui, L. El; Feron, E.; Balakrishnan, V. Linear Matrix Inequalities in System and Control Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1994 | DOI

[6] Branke, J.; Deb, K.; Miettinen, K.; nski, R. Slowi´ Multiobjective Optimization: Interactive and Evolutionary Approaches, Springer, Berlin, 2008 | Zbl

[7] Britz, W.; Ferris, M.; Kuhn, A. Modeling water allocating institutions based on multiple optimization problems with equilibrium constraints, Environ. Model. Softw., Volume 46 (2013), pp. 196-207 | DOI

[8] Burer, S.; Vandenbussche, D. A finite branch-and-bound algorithm for nonconvex quadratic programming via semidefinite relaxations, Math. Program., Volume 113 (2008), pp. 259-282 | Zbl | DOI

[9] Clarke, F. H. Optimization and non-smooth analysis, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1990 | DOI

[10] Demyanov, V. F. Convexification and concavification of positively homogeneous functions by the same family of linear functions, Technical Report, University of Pisa, Italy (1994), pp. 1-11

[11] Demyanov, V. F.; Jeyakumar, V. Hunting for a smaller convex subdifferential, J. Global Optim., Volume 10 (1997), pp. 305-326 | Zbl | DOI

[12] Dutta, J.; Chandra, S. Convexifactors, generalized convexity, and optimality conditions, J. Optim. Theory Appl., Volume 113 (2002), pp. 41-64 | Zbl | DOI

[13] Dutta, J.; Chandra, S. Convexifactors, generalized convexity and vector optimization, Optimization, Volume 53 (2004), pp. 77-94 | Zbl | DOI

[14] Flegel, M. L.; Kanzow, C. Abadie-type constraint qualification for mathematical programs with equilibrium constraints, J. Optim. Theory Appl., Volume 124 (2005), pp. 595-614 | DOI | Zbl

[15] Flegel, M. L.; Kanzow, C. On the Guignard constraint qualification for mathematical programs with equilibrium constraints, Optimization, Volume 54 (2005), pp. 517-534 | Zbl | DOI

[16] Forsgren, A. Optimality conditions for nonconvex semidefinite programming, Math. Program., Volume 88 (2000), pp. 105-128 | DOI | Zbl

[17] Fukushima, M.; Pang, J.-S. Some feasibility issues in mathematical programs with equilibrium constraints, SIAM J. Optim.,, Volume 18 (1998), pp. 673-681 | Zbl | DOI

[18] Ghosh, A.; Upadhyay, B. B.; Stancu-Minasian, I. M. Pareto efficiency criteria and duality for multiobjective fractional programming problems with equilibrium constraints on Hadamard manifolds, Mathematics, Volume 11 (2023), pp. 1-28 | DOI

[19] Ghosh, A.; Upadhyay, B. B.; Stancu-Minasian, I. M. Constraint qualifications for multiobjective programming problems on Hadamard manifolds, Aust. J. Math. Anal. Appl., Volume 20 (2023), pp. 1-17 | Zbl

[20] Gil-Gonz´alez, W.; Montoya, O. D.; Holgu´ın, E.; Garces, A.; na, L. F. Grisales-Nore ˜ Economic dispatch of energy storage systems in DC microgrids employing a semidefinite programming model, J. Energy Storage, Volume 21 (2019), pp. 1-8 | DOI

[21] Goemans, M. X. Semidefinite programming in combinatorial optimization, Math. Program., Volume 79 (1997), pp. 143-161 | DOI

[22] Golestani, M.; Nobakhtian, S. Optimality conditions for non-smooth semidefinite programming via convexificators, Positivity, Volume 19 (2019), pp. 221-236 | Zbl | DOI

[23] Guo, L.; Lin, G.-H.; Zhao, J. Wolfe-type duality for mathematical programs with equilibrium constraints, Acta Math. Appl. Sin. Engl. Ser., Volume 35 (2019), pp. 532-540 | Zbl | DOI

[24] Harker, P.T.; Pang, J.-S. Existence of optimal solutions to mathematical programs with equilibrium constraints, Oper. Res. Lett., Volume 7 (1988), pp. 61-64 | DOI | Zbl

[25] Ioffe, A. D. Approximate subdifferentials and applications. II, Mathematika, Volume 33 (1986), pp. 111-128 | DOI | Zbl

[26] Jeyakumar, V.; Luc, D. T. Non-smooth calculus, minimality, and monotonicity of convexificators, J. Optim. Theory Appl., Volume 10 (1999), pp. 599-621 | DOI | Zbl

[27] Joshi, B. C.; Mishra, S. K.; Pankaj On non-smooth mathematical programs with equilibrium constraints using generalized convexity, Yugosl. J. Oper. Res., Volume 29 (2019), pp. 449-463 | Zbl | DOI

[28] Lafhim, L.; Kalmoun, E. M. Optimality conditions for mathematical programs with equilibrium constraints using directional convexificators, Optimization, Volume 72 (2023), pp. 1363-1383 | DOI | Zbl

[29] Lai, K. K.; Hassan, M.; Singh, S. K.; Maurya, J. K.; Mishra, S. K. Semidefinite multiobjective mathematical programming problems with vanishing constraints using convexificators, Fractal Fract., Volume 6 (2022), pp. 1-19 | DOI

[30] Lewis, A. S.; Overton, M. L. Eigenvalue optimization, Cambridge Univ. Press, Cambridge, Volume 5 (1996), pp. 149-190 | DOI

[31] Luo, Z.-Q.; Pang, J.-S.; Ralph, D. Mathematical Programs With Equilibrium Constraints, Cambridge University Press, Cambridge, 1996

[32] Luu, D. V. Convexificators and necessary conditions for efficiency, Optimization, Volume 63 (2014), pp. 321-335 | DOI | Zbl

[33] Luu, D. V. Necessary and sufficient conditions for efficiency via convexificators, J. Optim. Theory Appl., Volume 160 (2014), pp. 510-526 | Zbl | DOI

[34] Mangasarian, O. L. Nonlinear Programming, McGraw-Hill, New York, 1969

[35] Michel, P.; Penot, J.-P. A generalized derivative for calm and stable functions, Differ. Integral Equ., Volume 5 (1992), pp. 433-454 | Zbl

[36] Miettinen, K. Kluwer Academic Publishers, Boston, MA, 1999

[37] Mishra, S. K.; Kumar, R.; Laha, V.; Maurya, J. K. Optimality and duality for semidefinite multiobjective programming problems using convexificators, J. Appl. Numer. Optim., Volume 4 (2022), pp. 103-118 | DOI

[38] Mishra, S. K.; Upadhyay, B. B. Efficiency and duality in non-smooth multiobjective fractional programming involving -pseudolinear functions, Yugosl. J. Oper. Res., Volume 22 (2012), pp. 3-18 | DOI

[39] Mond, M.; Weir, T. Generallized Concavity and Duality, in Generalized concavity in optimization and economics, Academic Press, New York, 1981

[40] Mordukhovich, B. S.; Shao, Y. H. On nonconvex subdifferential calculus in Banach spaces, J. Convex Anal., Volume 2 (1995), pp. 211-227 | Zbl

[41] Outrata, J. V. Optimality conditions for a class of mathematical programs with equilibrium constraints, Math. Oper. Res., Volume 24 (1999), pp. 627-644 | DOI | Zbl

[42] Pandey, Y.; Mishra, S. K. Duality for non-smooth optimization problems with equilibrium constraints, using convexificators, J. Optim. Theory Appl., Volume 171 (2016), pp. 694-707 | Zbl | DOI

[43] Peng, J.; Wei, Y. Approximating K-means-type clustering via semidefinite programming, SIAM J. Optim., Volume 18 (2007), pp. 186-205 | Zbl | DOI

[44] Raghunathan, A. U.; Biegler, L. T. Mathematical programs with equilibrium constraints (mpecs) in process engineering, Comput. Chem. Eng., Volume 27 (2003), pp. 1381-1392 | DOI

[45] Ralph, D. Mathematical programs with complementarity constraints in traffic and telecommunications networks, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., Volume 336 (2008), pp. 1973-1987 | DOI | Zbl

[46] Rimpi; Lalitha, C. S. Constraint qualifications in terms of convexificators for non-smooth programming problems with mixed constraints, Optimization, Volume 72 (2023), pp. 2019-2038 | Zbl | DOI

[47] Roux, P.; Voronin, Y.-L.; Sankaranarayanan, S. Validating numerical semidefinite programming solvers for polynomial invariants, Form. Methods Syst. Des., Volume 53 (2018), pp. 286-312 | Zbl | DOI

[48] Saini, S.; Kailey, N. Duality results in terms of convexifactors for a bilevel multiobjective problem, Filomat, Volume 38 (2024), pp. 2015-2022

[49] Scheel, H.; Scholtes, S. Mathematical programs with complementarity constraints: Stationarity, optimality, and sensitivity, Math. Oper. Res., Volume 25 (2000), pp. 1-22 | DOI | Zbl

[50] Shapiro, A. First and second order analysis of nonlinear semidefinite programs, Math. Program., Volume 77 (1997), pp. 301-320 | DOI | Zbl

[51] Singh, K. V. K.; Maurya, J. K.; Mishra, S. K. Duality in multiobjective mathematical programs with equilibrium constraints, Int. J. Appl. Comput. Math., Volume 7 (2021), pp. 1-15 | DOI | Zbl

[52] Sun, J. On methods for solving nonlinear semidefinite optimization problems, Numer. Algebra Control Optim., Volume 1 (2011), pp. 1-14 | DOI | Zbl

[53] Sun, D.; Sun, J.; Zhang, L. The rate of convergence of the augmented Lagrangian method for nonlinear semidefinite programming, Math. Program., Volume 114 (2008), pp. 349-391 | DOI | Zbl

[54] Treant¸˘a, S.; Upadhyay, B. B.; Ghosh, A.; Nonlaopon, K. Optimality conditions for multiobjective mathematical programming problems with equilibrium constraints on Hadamard manifolds, Mathematics, Volume 10 (2022), pp. 1-20 | DOI

[55] Treiman, J. S. The linear nonconvex generalized gradient and Lagrange multipliers, SIAM J. Optim., Volume 5 (1995), pp. 670-680 | Zbl | DOI

[56] Upadhyay, B. B.; Ghosh, A.; Stancu-Minasian, I. M. Duality for multiobjective programming problems with equilibrium constraints on Hadamard manifolds under generalized geodesic convexity, WSEAS Trans. Math., Volume 22 (2023), pp. 259-270

[57] Upadhyay, B. B.; Mishra, S. K. Non-smooth semi-infinite minmax programming involving generalized ( , )-invexity, J. Syst. Sci. Complex., Volume 28 (2015), pp. 857-875 | DOI

[58] Upadhyay, B. B.; Mishra, P. On interval-valued multiobjective programming problems and vector variational-like inequalities using limiting subdifferential, In: S. Patnaik, K. Tajeddini and V. Jain, (eds), Computational Management. Modeling and Optimization in Science and Technologies, Springer, Cham (2021), pp. 325-343 | DOI

[59] Upadhyay, B. B.; Mishra, P.; Mohapatra, R. N.; Mishra, S. K. On the applications of non-smooth vector optimization problems to solve generalized vector variational inequalities using convexificators, In: Adv. Intell. Syst. Comput., Springer, Cham, Switzerland, 2019 | Zbl | DOI

[60] Upadhyay, B. B.; Mohapatra, R. N. Sufficient optimality conditions and duality for mathematical programming problems with equilibrium constraints, Comm. Appl. Nonlinear Anal., Volume 25 (2018), pp. 68-84

[61] Upadhyay, B. B.; Singh, S. K. Optimality and duality for non-smooth semidefinite multiobjective fractional programming problems using convexificators, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys., Volume 85 (2023), pp. 79-90

[62] Upadhyay, B. B.; Singh, S. K.; Stancu-Minasian, I. M. Constraint qualifications and optimality conditions for nonsmooth semidefinite multiobjective programming problems with equilibrium constraints using convexificators, submitted in RAIRO Oper. Res. (2023)

[63] Upadhyay, B. B.; Stancu-Minasian, I. M.; Mishra, P. On relations between non-smooth interval-valued multiobjective programming problems and generalized Stampacchia vector variational inequalities, Optimization, Volume 72 (2023), pp. 2635-2659 | Zbl | DOI

[64] Upadhyay, B. B.; Treant¸˘a, S.; Mishra, P. On Minty variational principle for non-smooth multiobjective optimization problems on Hadamard manifolds, Optimization, Volume 72 (2023), pp. 3081-3100 | DOI | Zbl

[65] Vandenberghe, L.; Boyd, S. Applications of semidefinite programming, Appl. Numer. Math., Volume 29 (1999), pp. 283-299 | DOI

[66] Parys, B. P. G. Van; Goulart, P. J.; Kuhn, D. Generalized Gauss inequalities via semidefinite programming, Math. Program., Volume 156 (2016), pp. 271-302 | Zbl | DOI

[67] Wolfe, P. A duality theorem for nonlinear programming, Q. Appl. Math., Volume 19 (1961), pp. 239-244

[68] Yamashita, H.; Yabe, H. A survey of numerical methods for nonlinear semidefinite programming, J. Oper. Res. Soc. Japan, Volume 58 (2015), pp. 24-60 | Zbl | DOI

[69] Ye, J. J. Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints, J. Math. Anal. Appl., Volume 37 (2005), pp. 350-369 | Zbl | DOI

[70] Yu, G.; Li, S.; Pan, X.; Han, W. Optimality of approximate quasi-weakly efficient solutions for vector equilibrium problems via convexificators, Asia-Pac. J. Oper. Res., Volume 39 (2022), pp. 1-16 | Zbl | DOI

Cité par Sources :