Taiwanese Journal of Mathematics

Optimality and Duality on Riemannian Manifolds

Gabriel Ruiz-Garzón, Rafaela Osuna-Gómez, Antonio Rufián-Lizana, and Beatriz Hernández-Jiménez

Full-text: Open access


Our goal in this paper is to translate results on function classes that are characterized by the property that all the Karush-Kuhn-Tucker points are efficient solutions, obtained in Euclidean spaces to Riemannian manifolds. We give two new characterizations, one for the scalar case and another for the vectorial case, unknown in this subject literature. We also obtain duality results and give examples to illustrate it.

Article information

Taiwanese J. Math., Volume 22, Number 5 (2018), 1245-1259.

Received: 1 October 2017
Revised: 3 January 2018
Accepted: 6 May 2018
First available in Project Euclid: 21 May 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 90C29: Multi-objective and goal programming 53B21: Methods of Riemannian geometry 53C22: Geodesics [See also 58E10] 58E10: Applications to the theory of geodesics (problems in one independent variable) 80M50: Optimization

generalized convexity Riemannian manifolds efficient solutions duality


Ruiz-Garzón, Gabriel; Osuna-Gómez, Rafaela; Rufián-Lizana, Antonio; Hernández-Jiménez, Beatriz. Optimality and Duality on Riemannian Manifolds. Taiwanese J. Math. 22 (2018), no. 5, 1245--1259. doi:10.11650/tjm/180501. https://projecteuclid.org/euclid.twjm/1526889714

Export citation


  • P.-A. Absil, R. Mahony and R. Sepulchre, Optimization Algorithms on Matrix Manifolds, Princeton University Press, Princeton, NJ, 2008.
  • R. P. Agarwal, I. Ahmad, A. Iqbal and S. Ali, Generalized invex sets and preinvex functions on Riemannnian manifolds, Taiwanese J. Math. 16 (2012), no. 5, 1719–1732.
  • A. Barani and M. R. Pouryayevali, Invex sets and preinvex functions on Riemannian manifolds, J. Math. Anal. Appl. 328 (2007), no. 2, 767–779.
  • ––––, Invariant monotone vector fields on Riemannian manifolds, Nonlinear Anal. 70 (2009), no. 5, 1850–1861.
  • D. Bartl, Farkas' Lemma, other theorems of the alternative, and linear programming in infinite-dimensional spaces: a purely linear-algebraic approach, Linear Multilinear Algebra 55 (2007), no. 4, 327–353.
  • A. Ben-Israel and B. Mond, What is invexity?, J. Austral. Math. Soc. Ser. B 28 (1986), no. 1, 1–9.
  • G. C. Bento and J. X. Cruz Neto, A subgradient method for multiobjective optimization on Riemannian manifolds, J. Optim. Theory Appl. 159 (2013), no. 1, 125–137.
  • S.-I. Chen, N.-J. Huang and D. O'Regan, Geodesic $B$-preinvex functions and multiobjective optimization problems on Riemannian manifolds, J. Appl. Math. 2014 (2014), Art. ID 524698, 12 pp.
  • V. Colao, G. López, G. Marino and V. Martín-Márquez, Equilibrium problems in Hadamard manifolds, J. Math. Anal. Appl. 388 (2012), no. 1, 61–77.
  • B. D. Craven and B. M. Glover, Invex functions and duality, J. Austral. Math. Soc. Ser. A 39 (1985), no. 1, 1–20.
  • G. Dahl, J. M. Leinaas, J. Myrheim and E. Ovrum, A tensor product matrix approximation problem in quantum physics, Linear Algebra Appl. 420 (2007), no. 2-3, 711–725.
  • P. T. Fletcher, S. Venkatasubramanian and S. Joshi, The geometric median on Riemannian manifolds with application to robust atlas estimation, NeuroImage 45 (2009), no. 1, S143–S152.
  • M. A. Hanson, On sufficiency of the Kuhn-Tucker conditons, J. Math. Anal. Appl. 80 (1981), no. 2, 545–550.
  • S. Hosseini and M. R. Pouryayevali, Nonsmooth optimization techniques on Riemannian manifolds, J. Optim. Theory Appl. 158 (2013), no. 2, 328–342.
  • A. Iqbal, I. Ahmad and S. Ali, Strong geodesic $\alpha$-preinvesity and invariant $\alpha$-monotonicity on Riemannian manifolds, Numer. Funct. Anal. Optim. 31 (2010), no. 12, 1342–1361.
  • M. A. Khan, I. Ahamd and F. R. Al-Solamy, Geodesic $r$-preinvex functions on Riemannian manifolds, J. Inequal. Appl. 2014, 2014:144, 11 pp.
  • A. Kristály, Nash-type equilibria on Riemannian manifolds: a variational approach, J. Math. Pures Appl. (9) 101 (2014), no. 5, 660–688.
  • H. Li, D. Chen, H. Zhang, C. Wu and X. Wang, Hamiltonian analysis of a hydro-energy generation system in the transient of sudden load increasing, Applied Energy 185 (2017), 244–253.
  • C. Li, G. López and V. Martín-Márquez, Iterative algorithms for nonexpansive mappings on Hadamard manifolds, Taiwanese J. Math. 14 (2010), no. 2, 541–559.
  • Y. Ma, J. Košecká and S. Sastry, Optimization criteria and geometric algorithms for motion and structure estimation, Int. J. Computer Vision 44 (2001), no. 3, 219–249.
  • J. H. Manton, A centroide (Karcher mean) approach to the joint approximate diagonalisation problem: the real symmetric case, Digital Signal Processing 16 (2006), no. 5, 468–478.
  • D. M. Martin, The essence of invexity, J. Optim. Theory Appl. 47 (1985), no. 1, 65–76.
  • Y. Nishimori and S. Akaho, Learning algorithms utilizing quasi-geodesic flows on the Stiefel manifold, Neurocoputing 67 (2005), 106–135.
  • R. Osuna-Gómez, A. Beato-Moreno and A. Rufián-Lizana, Generalized convexity in multiobjective programming, J. Math. Anal. Appl. 233 (1999), no. 1, 205–220.
  • R. Osuna-Gómez, A. Rufián-Lizana and P. Ruiz-Canales, Invex functions and generalized convexity in multiobjective programming, J. Optim. Theory Appl. 98 (1998), no. 3, 651–661.
  • R. Pini, Convexity along curves and invexity, Optimization 29 (1994), no. 4, 301– \!\!309.
  • T. Rapcsák, Geodesic convexity in nonlinear optimization, J. Optim. Theory Appl. 69 (1991), no. 1, 169–183.
  • G. Ruiz-Garzón, R. Osuna-Gómez and A. Rufián-Lizana, Generalized invex monotonicity, European J. Oper. Res. 144 (2003), no. 3, 501–512.
  • P. Turaga, A. Veeraraghavan and R. Chellappa, Statistical analysis on Stiefel and Grassmann manifolds with applications in computer vision, IEEE Conference on Computer Vision and Pattern Recognition, 2008.
  • C. Udrişte, Convex Functions and Optimization Methods on Riemannian Manifolds, Mathematics and its Applications 297, Kluwer Academic, Dordrecht, 1994.
  • J. M. Wang, G. López, V. Martín-Márquez and C. Li, Monotone and accretive vector fields on Riemannian manifolds, J. Optim. Theory Appl. 146 (2010), no. 3, 691–708.
  • B. Xu, D. Chen, H. Zhang and R. Zhou, Dynamic analysis and modeling of a novel fractional-order hydro-turbine-generator unit, Nonlinear Dynam. 81 (2015), no. 3, 1263–1274.
  • H. Zhang, D. Chen, C. Wu, X. Wang, J.-M. Lee and K.-H. Jung, Dynamic modeling and dynamical analysis of pump-turbines in S-shaped regions during runaway operation, Energy Conversion and Management 138 (2017), no. 15, 375–382.
  • L.-W. Zhou and N.-J. Huang, Existence of solutions for vector optimization on Hadamard manifolds, J. Optim. Theory Appl. 157 (2013), no. 1, 44–53.
  • ––––, Roughly geodesic $B$-invex and optimization problem on Hadamard manifolds, Taiwanese J. Math. 17 (2013), no. 3, 833–855.