Minimax Inequalities and Optimization

  1. Manuel Ruiz Galán
Revista:
BEIO, Boletín de Estadística e Investigación Operativa

ISSN: 1889-3805

Año de publicación: 2019

Volumen: 35

Número: 3

Páginas: 198-215

Tipo: Artículo

Otras publicaciones en: BEIO, Boletín de Estadística e Investigación Operativa

Información de financiación

The author would like to thank the referees for their suggestions and careful reading. Research partially supported by project MTM2016-80676-P (AEI/FEDER, UE) and by Junta de Andaluc?a Grant FQM359.

Financiadores

Referencias bibliográficas

  • [1] Brickman, L. (1961). On the field of values of a matrix.Proceedings of theAmerican Mathematical Society,12, 61-66.
  • [2] Chang, S.Y. (2010). Inequalities and Nash equilibria.Nonlinear Analysis,73, 2933-2940.
  • [3] Chinchuluun, A., Pardalos, P.M., Migdalas, A. and Pitsoulis, L. (2008).Pareto optimality, game theory and equilibria.Springer Optimization andits Applications,17, Springer, New York.
  • [4] Dines, L.L. (1944). On the mapping of quadratic forms.Bulletin of theAmerican Mathematical Society,47, 494-498.
  • [5] Fan, K. (1953). Minimax theorems.Proceedings of the National Academy ofSciences of the United States of America,39, 42-47.
  • [6] Gordan, P. (1873). ̈Uber die aufl ̈osung linearer gleichungen mit reelencoefficienten.Mathematische Annalen,6, 23-28.
  • [7] Jeyakumar, V., Lee, G.M. and Li, G.Y. (2009). Alternative theoremsfor quadratic inequality systems and global quadratic optimization.SIAMJournal on Optimization,20, 983-1001.
  • [8] Jeyakumar, V., Lee, G.M. and Linh, N.T.H. (2016). Generalized Farkas’lemma and gap-free duality for minimax DC optimization with polynomialsand robust quadratic optimization.Journal of Global Optimization,64,679-702.
  • [9] Kassay, G. and Kolumb ́an, J. (1996). On a generalized sup-inf problem.Journal of Optimization Theory and Applications,91, 651–670.
  • [10] K ̈onig, H. (1968). ̈Uber das von Neumannsche minimax-theorem.Archivder Mathematik,19, 482-487.
  • [11] Lin, Y.-C. (2016). Minimax inequalities for a couple of non-continuousset-valued mappings.Journal of Nonlinear and Convex Analysis,17,931-943.
  • [12] Mazur, S. and Orlicz, W. (1953). Sur les espaces m ́etriques lin ́eaires II.Studia Mathematica,13, 137-179.
  • [13] Montiel L ́opez, P. and Ruiz Gal ́an, M. (2019). Infinite programming andtheorems of the alternative.Mathematical Methods in the Applied Sciences,https://doi.org/10.1002/mma.5566
  • [14] Montiel L ́opez, P. and Ruiz Gal ́an, M. (2017). Revisiting the Hahn–Banachtheorem and nonlinear infinite programming.Journal of MathematicalAnalysis and Applications,455, 1037-1050.
  • [15] Orihuela, J. and Zapata, J.M. (2017). Stability in locally L0-convex modulesand a conditional version of James’ compactness theorem.Journal ofMathematical Analysis and Applications,452, 1101-1127.
  • [16] Park, S. (2011). New generalizations of basic theorems in the KKM theory.Nonlinear Analysis,74, 3000-3010.
  • [17] P ́olik, I. and Terlaky, T. (2007). A survey of the S-lemma.SIAM Review,49, 371-418.
  • [18] Ricceri, B. (2017). On a minimax theorem: an improvement, a new proofand an overview of its applications.Minimax Theory and its Applications,2, 99-152.
  • [19] Ruiz Gal ́an, M. (2018). Elementary convex techniques for equilibrium,minimax and variational problems.Optimization Letters,12, 137-154.
  • [20] Ruiz Gal ́an, M. (2018). Minimax inequalities in the absence of topologicalassumptions.Minimax Theory and its Applications,3, 81-90.
  • [21] Ruiz Gal ́an, M. (2017). A theorem of the alternative with an arbitrarynumber of inequalities and quadratic programming.Journal of GlobalOptimization,69, 427-442.
  • [22] Ruiz Gal ́an, M. (2016). A sharp Lagrange multiplier theorem for nonlinearprogramming.Journal of Global Optimization,65, 513-530.
  • [23] Ruiz Gal ́an, M. (2016). The Gordan theorem and its implications forminimax theory.Journal of Nonlinear and Convex Analysis,17, 2385-2405.
  • [24] Simons, S.Minimax and monotonicity, Lecture Notes in Mathematics1693, Springer, Berlin, 1998.
  • [25] Syga, M. (2018). Minimax theorems for extended real-valued abstractconvex-concave functions.Journal of Optimization Theory and Applications,176, 306-318.
  • [26] Sion, M. (1958). On general minimax theorems.Pacific Journal ofMathematics,8, 171-176.
  • [27] Tian, G. (2017) Full characterizations of minimax inequality, fixed pointtheorem, saddle point theorem, and KKM principle in arbitrary topologicalspaces.Journal of Fixed Point Theory and Applications,19, 1679-1693.
  • [28] von Neumann, J. (1928). Zur theorie der gesellschaftsspiele.MathematischeAnnalen,100, 295-320.
  • [29] Yuan, Y. (1990). On a subproblem of trust region algorithms for constrainedoptimization.Mathematical Programming,47, 53-63.
  • [30] Q.B. Zhang, Q.B. (2014). General two-function topological minimaxtheorems.Annals of Operational Research,217, 591-598