This paper deals with multiobjective semi-infinite programming problems on Hadamard manifolds. We establish the sufficient optimality criteria of the considered problem under generalized geodesic convexity assumptions. Moreover, we formulate the Mond-Weir and Wolfe type dual problems and derive the weak, strong and strict converse duality theorems relating the primal and dual problems under generalized geodesic convexity assumptions. Suitable examples have also been given to illustrate the significance of these results. The results presented in this paper extend and generalize the corresponding results in the literature.
Keywords: Semi-infinite programming, multiobjective optimization, optimality, duality, Hadamard manifolds
@article{RO_2022__56_4_2037_0,
author = {Upadhyay, Balendu Bhooshan and Ghosh, Arnav and Mishra, Priyanka and Trean\c{t}\u{a}, Savin},
title = {Optimality conditions and duality for multiobjective semi-infinite programming problems on {Hadamard} manifolds using generalized geodesic convexity},
journal = {RAIRO. Operations Research},
pages = {2037--2065},
year = {2022},
publisher = {EDP-Sciences},
volume = {56},
number = {4},
doi = {10.1051/ro/2022098},
mrnumber = {4450251},
zbl = {1492.90185},
language = {en},
url = {https://www.numdam.org/articles/10.1051/ro/2022098/}
}
TY - JOUR AU - Upadhyay, Balendu Bhooshan AU - Ghosh, Arnav AU - Mishra, Priyanka AU - Treanţă, Savin TI - Optimality conditions and duality for multiobjective semi-infinite programming problems on Hadamard manifolds using generalized geodesic convexity JO - RAIRO. Operations Research PY - 2022 SP - 2037 EP - 2065 VL - 56 IS - 4 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/ro/2022098/ DO - 10.1051/ro/2022098 LA - en ID - RO_2022__56_4_2037_0 ER -
%0 Journal Article %A Upadhyay, Balendu Bhooshan %A Ghosh, Arnav %A Mishra, Priyanka %A Treanţă, Savin %T Optimality conditions and duality for multiobjective semi-infinite programming problems on Hadamard manifolds using generalized geodesic convexity %J RAIRO. Operations Research %D 2022 %P 2037-2065 %V 56 %N 4 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/ro/2022098/ %R 10.1051/ro/2022098 %G en %F RO_2022__56_4_2037_0
Upadhyay, Balendu Bhooshan; Ghosh, Arnav; Mishra, Priyanka; Treanţă, Savin. Optimality conditions and duality for multiobjective semi-infinite programming problems on Hadamard manifolds using generalized geodesic convexity. RAIRO. Operations Research, Tome 56 (2022) no. 4, pp. 2037-2065. doi: 10.1051/ro/2022098
[1] , and , Trust-region methods on Riemannian manifolds. Found. Comput. Math. 7 (2007) 303–330. | MR | Zbl | DOI
[2] , and , Optimization Algorithms on Matrix Manifolds. Princeton University Press, Princeton, NJ (2009). | MR | Zbl
[3] and , Quasi-concave programming. Econom. J. Econom. Soc. 29 (1961) 779–880. | MR | Zbl
[4] , Convexity of the solution set of a pseudoconvex inequality in Riemannian manifolds. Numer. Funct. Anal. Optim. 39 (2018) 588–599. | MR | DOI
[5] , On pseudoconvex functions in Riemanian manifolds. J. Finsler Geom. Appl. 2 (2021) 14–22. | MR | Zbl
[6] and , Characterization of solution sets of convex optimization problems in Riemannian manifolds. Arch. Math. (Basel) 114 (2020) 215–225. | MR | Zbl | DOI
[7] and , Subgradient method for convex feasibility on Riemannian manifolds. J. Optim. Theory Appl. 152 (2012) 773–785. | MR | Zbl | DOI
[8] and , Intrinsic formulation of KKT conditions and constraint qualifications on smooth manifolds. SIAM J. Optim. 29 (2019) 2423–2444. | MR | Zbl | DOI
[9] and , Convex Analysis and Nonlinear Optimization: Theory and Examples. Springer Science & Business Media, NY (2010). | Zbl
[10] , , and , Manopt, a matlab toolbox for optimization on manifolds. J. Mach. Learn. Res. 15 (2014) 1455–1459. | Zbl
[11] , and , Duality, Haar programs, and finite sequence spaces. Proc. Natl. Acad. Sci. USA 48 (1962) 783. | MR | Zbl | DOI
[12] , and , Duality in semi-infinite programs and some works of Haar and Carathéodory. Manag. Sci. 9 (1963) 209–228. | MR | Zbl | DOI
[13] , and , On the theory of semi-infinite programming and a generalization of the Kuhn-Tucker saddle point theorem for arbitrary convex functions. Nav. Res. Logist. Quart. 16 (1969) 41–52. | MR | Zbl | DOI
[14] , Existence results for vector variational inequality problems on Hadamard manifolds. Optim. Lett. 14 (2020) 2395–2411. | MR | Zbl | DOI
[15] , The KKT optimality conditions for optimization problem with interval-valued objective function on Hadamard manifolds. Optimization (2020). | MR | Zbl
[16] and , Nonsmooth semi-infinite multiobjective optimization problems. J. Optim. Theory Appl. 160 (2014) 748–762. | MR | Zbl | DOI
[17] , , and , Convex-and monotone-transformable mathematical programming problems and a proximal-like point method. J. Global Optim. 35 (2006) 53–69. | MR | Zbl | DOI
[18] , Riemannian Geometry. Springer (1992). | MR | Zbl | DOI
[19] , On the variational principle. J. Math. Anal. Appl. 47 (1974) 324–353. | MR | Zbl | DOI
[20] , Necessary optimality and duality for multiobjective semi-infinite programming. J. Theor. Appl. Inf. Technol. 46 (2012) 347–354.
[21] , Optimality and duality for non-smooth multiple objective semi-infinite programming. J. Netw. 8 (2013).
[22] and , Linear semi-infinite programming theory: An updated survey. Eur. J. Oper. Res. 143 (2002) 390–405. | MR | Zbl | DOI
[23] and , Recent contributions to linear semi-infinite optimization: an update. Ann. Oper. Res. 271 (2018) 237–278. | MR | Zbl | DOI
[24] , Über lineare ungleichungen. Acta Sci. Math. (Szeged) 2 (1924) 1–14. | JFM
[25] , Riemannian Geometry and Geometric Analysis. Springer (2008). | MR | Zbl
[26] and , Optimality conditions for non-smooth semi-infinite programming. Optimization 59 (2010) 717–727. | MR | Zbl | DOI
[27] and , Optimality conditions for nonsmooth semi-infinite multiobjective programming. Optim. Lett. 8 (2014) 1517–1528. | MR | Zbl | DOI
[28] and , Nonconvex weak sharp minima on Riemannian manifolds. J. Optim. Theory Appl. 183 (2019) 85–104. | MR | Zbl | DOI
[29] and , Optimality conditions for convex semi-infinite programming problems with finitely representable compact index sets. J. Optim. Theory Appl. 175 (2017) 76–103. | MR | Zbl | DOI
[30] , Introduction to Riemannian Manifolds. Springer (2018). | MR | Zbl | DOI
[31] , , and , Weak sharp minima on Riemannian manifolds. SIAM J. Optim. 21 (2011) 1523–1560. | MR | Zbl | DOI
[32] and , Optimality conditions for nondifferentiable convex semi-infinite programming. Math. Program. 27 (1983) 307–319. | MR | Zbl | DOI
[33] , Theory of Vector Optimization. Springer (1989). | MR | DOI
[34] , Duality in nonlinear programming. Quart. Appl. Math. 20 (1962) 300–302. | MR | Zbl | DOI
[35] , Pseudo-convex functions. J. SIAM Control Ser. A. 3 (1965) 281–290. | MR | Zbl
[36] , Nonlinear Programming. SIAM (1994). | MR | Zbl | DOI
[37] and , Pseudolinear Functions and Optimization. Chapman and Hall/CRC (2019). | Zbl | MR
[38] , and , Duality for nonsmooth semi-infinite programming problems. Optim. Lett. 6 (2012) 261–271. | MR | Zbl | DOI
[39] , Five kinds of monotone vector fields. Pure Appl. Math. 9 (1999) 417–428. | MR | Zbl
[40] , Optimality conditions and duality for multiobjective semi-infinite programming with data uncertainty via Mordukhovich subdifferential. Yugosl. J. Oper. Res. 31 (2021) 495–514. | MR | DOI
[41] and , Proximal point methods for quasiconvex and convex functions with Bregman distances on Hadamard manifolds. J. Convex Anal. 16 (2009) 49–69. | MR | Zbl
[42] and , Full convergence of the proximal point method for quasiconvex functions on Hadamard manifolds. ESAIM: Control. Optim. Cal. Var. 18 (2012) 483–500. | MR | Zbl | Numdam
[43] , and , Steepest descent method with a generalized Armijo search for quasiconvex functions on Riemannian manifolds. J. Math. Anal. Appl. 341 (2008) 467–477. | MR | Zbl | DOI
[44] , and , Inexact proximal point methods for multiobjective quasiconvex minimization on hadamard manifolds. J. Optim. Theory Appl. 186 (2020) 879–898. | MR | Zbl | DOI
[45] and , Isolated efficiency in nonsmooth semi-infinite multi-objective programming. Optimization 67 (2018) 1923–1947. | MR | Zbl | DOI
[46] , Smooth Nonlinear Optimization in . Springer Science & Business Media (2013). | MR | Zbl
[47] , and , Necessary and sufficient optimality conditions for vector equilibrium problems on Hadamard manifolds. Symmetry 11 (2019) 1037. | DOI
[48] , Riemannian Geometry. American Mathematical Society (1996). | MR | Zbl | DOI
[49] and , On geodesic convex and generalized geodesic convex functions over Riemannian manifolds. AIP Conf. Proc. 1975 (2018) 030006. | DOI
[50] and , Solving semi-infinite optimization problems with interior point techniques. SIAM J. Control Optim. 42 (2003) 769–788. | MR | Zbl | DOI
[51] and , Korpelevich’s method for variational inequality problems on Hadamard manifolds. J. Global Optim. 54 (2012) 493–509. | MR | Zbl | DOI
[52] , and , The proximal point algorithm for pseudomonotone variational inequalities on Hadamard manifolds. Optim. Lett. 7 (2013) 779–790. | MR | Zbl | DOI
[53] , and , Minty variational principle for nonsmooth interval-valued vector optimization problems on hadamard manifolds. Mathematics 10 (2022) 523. | DOI
[54] , Karush-Kuhn-Tucker optimality conditions and duality for convex semi-infinite programming with multiple interval-valued objective functions. J. Appl. Math. Comput. 62 (2020) 67–91. | MR | Zbl | DOI
[55] , Karush-Kuhn-Tucker optimality conditions and duality for multiobjective semi-infinite programming with vanishing constraints. Ann. Oper. Res. (2020) 1–28. | MR | Zbl
[56] and , Optimality conditions and duality for multiobjective semi-infinite programming on Hadamard manifolds. Bull. Iranian Math. Soc. (2021) 1–29. | MR | Zbl
[57] , and , Explicitly geodesic B-preinvex functions on Riemannian Manifolds. Trans. Math. Program. Appl. 2 (2015) 1–14.
[58] , , and , On generalized vector variational inequalities and nonsmooth vector optimization problems on hadamard manifolds involving geodesic approximate convexity. Adv. Nonlinear Var. Inequal. 25 (2022) 1–25.
[59] , Convex Functions and Optimization Methods on Riemannian Manifolds. Springer Science & Business Media (2013). | MR | Zbl
[60] and , Generalised convexity and duality in multiple objective programming. Bull. Aust. Math. Soc. 39 (1989) 287–299. | MR | Zbl | DOI
[61] , A duality theorem for non-linear programming. Quart. Appl. Math. 19 (1961) 239–244. | MR | Zbl | DOI
[62] , and , Optimality conditions for the nonlinear programming problems on Riemannian manifolds. Pac. J. Optim. 10 (2014) 415–434. | MR | Zbl
[63] , Optimality conditions and duality for semi-infinite programming involving B-arcwise connected functions. J. Global Optim. 45 (2009) 615–629. | MR | Zbl | DOI
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