Open Access
RAIRO-Oper. Res.
Volume 56, Number 4, July-August 2022
Page(s) 2037 - 2065
Published online 11 July 2022
  • P.A. Absil, C.G. Baker and K.A. Gallivan, Trust-region methods on Riemannian manifolds. Found. Comput. Math. 7 (2007) 303–330. [CrossRef] [MathSciNet] [Google Scholar]
  • P.A. Absil, R. Mahony and R. Sepulchre, Optimization Algorithms on Matrix Manifolds. Princeton University Press, Princeton, NJ (2009). [Google Scholar]
  • K.J. Arrow and A.C. Enthoven, Quasi-concave programming. Econom. J. Econom. Soc. 29 (1961) 779–880. [Google Scholar]
  • A. Barani, Convexity of the solution set of a pseudoconvex inequality in Riemannian manifolds. Numer. Funct. Anal. Optim. 39 (2018) 588–599. [CrossRef] [MathSciNet] [Google Scholar]
  • A. Barani, On pseudoconvex functions in Riemanian manifolds. J. Finsler Geom. Appl. 2 (2021) 14–22. [Google Scholar]
  • A. Barani and S. Hosseini, Characterization of solution sets of convex optimization problems in Riemannian manifolds. Arch. Math. (Basel) 114 (2020) 215–225. [CrossRef] [MathSciNet] [Google Scholar]
  • G.C. Bento and J.G. Melo, Subgradient method for convex feasibility on Riemannian manifolds. J. Optim. Theory Appl. 152 (2012) 773–785. [CrossRef] [MathSciNet] [Google Scholar]
  • R. Bergmann and R. Herzog, Intrinsic formulation of KKT conditions and constraint qualifications on smooth manifolds. SIAM J. Optim. 29 (2019) 2423–2444. [CrossRef] [MathSciNet] [Google Scholar]
  • J. Borwein and A.S. Lewis, Convex Analysis and Nonlinear Optimization: Theory and Examples. Springer Science & Business Media, NY (2010). [Google Scholar]
  • N. Boumal, B. Mishra, P.A. Absil and R. Sepulchre, Manopt, a matlab toolbox for optimization on manifolds. J. Mach. Learn. Res. 15 (2014) 1455–1459. [Google Scholar]
  • A. Charnes, W. Cooper and K. Kortanek, Duality, Haar programs, and finite sequence spaces. Proc. Natl. Acad. Sci. USA 48 (1962) 783. [CrossRef] [PubMed] [Google Scholar]
  • A. Charnes, W. Cooper and K. Kortanek, Duality in semi-infinite programs and some works of Haar and Carathéodory. Manag. Sci. 9 (1963) 209–228. [CrossRef] [Google Scholar]
  • A. Charnes, W. Cooper and K. Kortanek, 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. [CrossRef] [Google Scholar]
  • S.L. Chen, Existence results for vector variational inequality problems on Hadamard manifolds. Optim. Lett. 14 (2020) 2395–2411. [CrossRef] [MathSciNet] [Google Scholar]
  • S.L. Chen, The KKT optimality conditions for optimization problem with interval-valued objective function on Hadamard manifolds. Optimization (2020). [Google Scholar]
  • T.D. Chuong and D.S. Kim, Nonsmooth semi-infinite multiobjective optimization problems. J. Optim. Theory Appl. 160 (2014) 748–762. [Google Scholar]
  • J.X. Da Cruz Neto, O.P. Ferreira, L.R. Lucambio Perez and S.Z. Németh, Convex-and monotone-transformable mathematical programming problems and a proximal-like point method. J. Global Optim. 35 (2006) 53–69. [CrossRef] [MathSciNet] [Google Scholar]
  • M.P. Do Carmo, Riemannian Geometry. Springer (1992). [CrossRef] [Google Scholar]
  • I. Ekeland, On the variational principle. J. Math. Anal. Appl. 47 (1974) 324–353. [Google Scholar]
  • X. Gao, Necessary optimality and duality for multiobjective semi-infinite programming. J. Theor. Appl. Inf. Technol. 46 (2012) 347–354. [Google Scholar]
  • X. Gao, Optimality and duality for non-smooth multiple objective semi-infinite programming. J. Netw. 8 (2013). [Google Scholar]
  • M.A. Goberna and M.A. Lopez, Linear semi-infinite programming theory: An updated survey. Eur. J. Oper. Res. 143 (2002) 390–405. [CrossRef] [Google Scholar]
  • M.A. Goberna and M.A. López, Recent contributions to linear semi-infinite optimization: an update. Ann. Oper. Res. 271 (2018) 237–278. [CrossRef] [MathSciNet] [Google Scholar]
  • A. Haar, Über lineare ungleichungen. Acta Sci. Math. (Szeged) 2 (1924) 1–14. [Google Scholar]
  • J. Jost, Riemannian Geometry and Geometric Analysis. Springer (2008). [Google Scholar]
  • N. Kanzi and S. Nobakhtian, Optimality conditions for non-smooth semi-infinite programming. Optimization 59 (2010) 717–727. [CrossRef] [MathSciNet] [Google Scholar]
  • N. Kanzi and S. Nobakhtian, Optimality conditions for nonsmooth semi-infinite multiobjective programming. Optim. Lett. 8 (2014) 1517–1528. [CrossRef] [MathSciNet] [Google Scholar]
  • M.M. Karkhaneei and N. Mahdavi-Amiri, Nonconvex weak sharp minima on Riemannian manifolds. J. Optim. Theory Appl. 183 (2019) 85–104. [CrossRef] [MathSciNet] [Google Scholar]
  • O. Kostyukova and T. Tchemisova, Optimality conditions for convex semi-infinite programming problems with finitely representable compact index sets. J. Optim. Theory Appl. 175 (2017) 76–103. [Google Scholar]
  • J.M. Lee, Introduction to Riemannian Manifolds. Springer (2018). [CrossRef] [Google Scholar]
  • C. Li, B.S. Mordukhovich, J. Wang and J.C. Yao, Weak sharp minima on Riemannian manifolds. SIAM J. Optim. 21 (2011) 1523–1560. [CrossRef] [MathSciNet] [Google Scholar]
  • M.A. López and E. Vercher, Optimality conditions for nondifferentiable convex semi-infinite programming. Math. Program. 27 (1983) 307–319. [CrossRef] [Google Scholar]
  • D.T. Luc, Theory of Vector Optimization. Springer (1989). [Google Scholar]
  • O.L. Mangasarian, Duality in nonlinear programming. Quart. Appl. Math. 20 (1962) 300–302. [Google Scholar]
  • O.L. Mangasarian, Pseudo-convex functions. J. SIAM Control Ser. A. 3 (1965) 281–290. [MathSciNet] [Google Scholar]
  • O.L. Mangasarian, Nonlinear Programming. SIAM (1994). [CrossRef] [Google Scholar]
  • S.K. Mishra and B.B. Upadhyay, Pseudolinear Functions and Optimization. Chapman and Hall/CRC (2019). [Google Scholar]
  • S.K. Mishra, M. Jaiswal and L.T.H. An, Duality for nonsmooth semi-infinite programming problems. Optim. Lett. 6 (2012) 261–271. [CrossRef] [MathSciNet] [Google Scholar]
  • S. Németh, Five kinds of monotone vector fields. Pure Appl. Math. 9 (1999) 417–428. [Google Scholar]
  • T.H. Pham, Optimality conditions and duality for multiobjective semi-infinite programming with data uncertainty via Mordukhovich subdifferential. Yugosl. J. Oper. Res. 31 (2021) 495–514. [CrossRef] [MathSciNet] [Google Scholar]
  • E.A. Papa Quiroz and P.R. Oliveira, Proximal point methods for quasiconvex and convex functions with Bregman distances on Hadamard manifolds. J. Convex Anal. 16 (2009) 49–69. [MathSciNet] [Google Scholar]
  • E.A. Papa Quiroz and P.R. Oliveira, Full convergence of the proximal point method for quasiconvex functions on Hadamard manifolds. ESAIM: Control. Optim. Cal. Var. 18 (2012) 483–500. [CrossRef] [EDP Sciences] [Google Scholar]
  • E.A. Papa Quiroz, E.M. Quispe and P.R. Oliveira, Steepest descent method with a generalized Armijo search for quasiconvex functions on Riemannian manifolds. J. Math. Anal. Appl. 341 (2008) 467–477. [CrossRef] [MathSciNet] [Google Scholar]
  • E.A. Papa Quiroz, N. Baygorrea Cusihuallpa and N. Maculan, Inexact proximal point methods for multiobjective quasiconvex minimization on hadamard manifolds. J. Optim. Theory Appl. 186 (2020) 879–898. [CrossRef] [MathSciNet] [Google Scholar]
  • M. Rahimi and M. Soleimani-Damaneh, Isolated efficiency in nonsmooth semi-infinite multi-objective programming. Optimization 67 (2018) 1923–1947. [CrossRef] [MathSciNet] [Google Scholar]
  • T. Rapcsák, Smooth Nonlinear Optimization in ℝn. Springer Science & Business Media (2013). [Google Scholar]
  • G. Ruiz-Garzón, R. Osuna-Gómez and J. Ruiz-Zapatero, Necessary and sufficient optimality conditions for vector equilibrium problems on Hadamard manifolds. Symmetry 11 (2019) 1037. [CrossRef] [Google Scholar]
  • T. Sakai, Riemannian Geometry. American Mathematical Society (1996). [Google Scholar]
  • A. Shahi and S.K. Mishra, On geodesic convex and generalized geodesic convex functions over Riemannian manifolds. AIP Conf. Proc. 1975 (2018) 030006. [CrossRef] [Google Scholar]
  • O. Stein and G. Still, Solving semi-infinite optimization problems with interior point techniques. SIAM J. Control Optim. 42 (2003) 769–788. [CrossRef] [MathSciNet] [Google Scholar]
  • G.J. Tang and N.J. Huang, Korpelevich’s method for variational inequality problems on Hadamard manifolds. J. Global Optim. 54 (2012) 493–509. [CrossRef] [MathSciNet] [Google Scholar]
  • G.J. Tang, L.W. Zhou and N.J. Huang, The proximal point algorithm for pseudomonotone variational inequalities on Hadamard manifolds. Optim. Lett. 7 (2013) 779–790. [CrossRef] [MathSciNet] [Google Scholar]
  • S. Treanţă, P. Mishra and B.B. Upadhyay, Minty variational principle for nonsmooth interval-valued vector optimization problems on hadamard manifolds. Mathematics 10 (2022) 523. [CrossRef] [Google Scholar]
  • L.T. Tung, 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. [CrossRef] [MathSciNet] [Google Scholar]
  • L.T. Tung, Karush-Kuhn-Tucker optimality conditions and duality for multiobjective semi-infinite programming with vanishing constraints. Ann. Oper. Res. (2020) 1–28. [Google Scholar]
  • L.T. Tung and D.H. Tam, Optimality conditions and duality for multiobjective semi-infinite programming on Hadamard manifolds. Bull. Iranian Math. Soc. (2021) 1–29. [Google Scholar]
  • B.B. Upadhyay, S.K. Mishra and S.K. Porwal, Explicitly geodesic B-preinvex functions on Riemannian Manifolds. Trans. Math. Program. Appl. 2 (2015) 1–14. [Google Scholar]
  • B.B. Upadhyay, I.M. Stancu Minasian, P. Mishra and R.N. Mohapatra, 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. [Google Scholar]
  • C. Udrişte, Convex Functions and Optimization Methods on Riemannian Manifolds. Springer Science & Business Media (2013). [Google Scholar]
  • T. Weir and B. Mond, Generalised convexity and duality in multiple objective programming. Bull. Aust. Math. Soc. 39 (1989) 287–299. [CrossRef] [Google Scholar]
  • P. Wolfe, A duality theorem for non-linear programming. Quart. Appl. Math. 19 (1961) 239–244. [CrossRef] [MathSciNet] [Google Scholar]
  • W.H. Yang, L.H. Zhang and R. Song, Optimality conditions for the nonlinear programming problems on Riemannian manifolds. Pac. J. Optim. 10 (2014) 415–434. [MathSciNet] [Google Scholar]
  • Q. Zhang, Optimality conditions and duality for semi-infinite programming involving B-arcwise connected functions. J. Global Optim. 45 (2009) 615–629. [CrossRef] [MathSciNet] [Google Scholar]

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