Optimality Conditions for Vector Equilibrium Problems on Hadamard Manifolds

The aim of this paper is to obtain Karush-Kuhn-Tucker optimality conditions for weakly efficient solutions to vector equilibrium problems with the addition of constraints in the novel context of Hadamard manifolds as opposed to the classical examples of Banach, normed or Hausdorff spaces. More specifically, classical necessary and sufficient conditions for weakly efficient solutions to the constrained vector optimization problem are presented. As well as some examples. The results presented in this paper generalize results obtained by Gong (2008) and Wei and Gong (2010) and Feng and Qiu (2014) from Hausdorff topological vector spaces, real normed spaces and real Banach spaces to Hadamard manifolds, respectively.

[1]  K. Fan.  A generalization of Tychonoff's fixed point theorem , 1961 .

[2]  V. Bangert.  Totally convex sets in complete Riemannian manifolds , 1981 .

[3]  M. A. Hanson.  On sufficiency of the Kuhn-Tucker conditions , 1981 .

[4]  O. P. Ferreira,et al.  Proximal Point Algorithm On Riemannian Manifolds , 2002 .

[5]  K. Neeb.  A Cartan–Hadamard Theorem for Banach–Finsler Manifolds , 2002 .

[6]  Q. Ansari,et al.  Generalized vector quasi-equilibrium problems with applications , 2003 .

[7]  S. Németh.  Variational inequalities on Hadamard manifolds , 2003 .

[8]  A. Iusem,et al.  New existence results for equilibrium problems , 2003 .

[9]  Shotaro Akaho,et al.  Learning algorithms utilizing quasi-geodesic flows on the Stiefel manifold , 2005, Neurocomputing.

[10]  F. Clarke,et al.  Dini Derivative and a Characterization for Lipschitz and Convex Functions on Riemannian Manifolds , 2006 .

[11]  Robert E. Mahony,et al.  Optimization Algorithms on Matrix Manifolds , 2007 .

[12]  Rama Chellappa,et al.  Statistical analysis on Stiefel and Grassmann manifolds with applications in computer vision , 2008, 2008 IEEE Conference on Computer Vision and Pattern Recognition.

[13]  M. R. Pouryayevali,et al.  Invariant monotone vector fields on Riemannian manifolds , 2009 .

[14]  Suresh Venkatasubramanian,et al.  The geometric median on Riemannian manifolds with application to robust atlas estimation , 2009, NeuroImage.

[15]  X. Gong,et al.  Kuhn-Tucker Optimality Conditions for Vector Equilibrium Problems , 2010 .

[16]  X. Gong.  Scalarization and optimality conditions for vector equilibrium problems , 2010 .

[17]  M. R. Pouryayevali,et al.  Generalized gradients and characterization of epi-Lipschitz sets in Riemannian manifolds , 2011 .

[18]  G. Marino,et al.  Equilibrium problems in Hadamard manifolds , 2012 .

[19]  N. Huang,et al.  ROUGHLY GEODESIC $B$-INVEX AND OPTIMIZATION PROBLEM ON HADAMARD MANIFOLDS , 2013 .

[20]  Lu Wei-zhong,et al.  Optimality Conditions for ε -Vector Equilibrium Problems , 2014 .

[21]  Yanyan Feng,et al.  Optimality conditions for vector equilibrium problems with constraint in Banach spaces , 2014, Optim. Lett..

[22]  Alexandru Krist'aly.  Nash-type equilibria on Riemannian manifolds: a variational approach , 2016, 1602.04157.

[23]  R. Osuna-Gómez,et al.  Optimality and Duality on Riemannian Manifolds , 2018, Taiwanese Journal of Mathematics.

[24]  Amira A. Ishan,et al.  Generalized Geodesic Convexity on Riemannian Manifolds , 2019, Mathematics.