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Theorems of the Alternative Revisited and Some Extensions of Yuan’s Lemma

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Abstract

In this paper we deal with theorems of the alternative for systems of inequalities. Precisely, we give a formulation that encompasses several known alternative theorems and unifies their treatment under a minimax equality assumption which is fulfilled by a whole panoply of families of functions. We also give extensions of Yuan’s lemma to systems with several general functions provided that they fulfill a certain minimax theorem, and to some particular systems of several quadratic functions.

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References

  1. Baccari, A., Trad, A.: On the classical necessary second-order optimality conditions in the presence of equality and inequality constraints. SIAM J. Optim. 15 (2), 394–408 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Brickman, L.: On the field of values of a matrix. Proc. Amer. Math. Soc. 12(1), 61–66 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chen, X., Yuan, Y.: A note on quadratic forms. Math Program. 86(1), 187–197 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  4. Crouzeix, J.P., Martínez-Legaz, J.E., Seeger, A.: An alternative theorem for quadratic forms and extensions. Linear Algebra Appl. 215, 121–134 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  5. Dines, L.L.: On the mapping of quadratic forms. Bull. Amer. Math. Soc. 47(6), 494–498 (1941)

    Article  MathSciNet  MATH  Google Scholar 

  6. Fan, K.: Minimax theorems. Proc. Natl. Acad. Sci. 39(1), 42–47 (1953)

    Article  MathSciNet  MATH  Google Scholar 

  7. Fan, K.: Existence theorems and extreme solutions for inequalities concerning convex functions or linear transformations. Math. Z. 68, 205–216 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  8. Fan, K., Glicksberg, I., Hoffman, A.J.: Systems of inequalities involving convex functions. Proc. Amer. Math Soc. 8(3), 617–622 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fuchssteiner, B., König, H.: New versions of Hahn-Banach theorem. In: General Inequalities 2 (Beckenbach, Ed.) International Series of Numerical Mathematics, vol. 47, pp 255–266. Basel, Birkhäuser (1978)

  10. Geraghty, M.A., Lin, B.-L.: On a minimax theorem of Terkelsen. Bull. Inst. Math. Acad. Sinica 11, 343–347 (1983)

    MathSciNet  MATH  Google Scholar 

  11. Geraghty, M.A., Lin, B.-L.: Topological minimax theorems. Proc. Amer. Math. Soc. 91(3), 377–380 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hanson, M.A.: On sufficiency of the Kuhn-Tucker conditions. J. Math. Anal. Appl. 80(2), 545–550 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hayashi, M., Komiya, H.: Perfect duality for convexlike programs. J. Optim Theory Appl. 38(2), 179–189 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hiriart-Urruty, J.-B., Torki, M.: Permanently going back and forth between the “quadratic world” and the “convexity world” in optimization. Appl. Math. Optim. 45(2), 169–184 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  15. Illés, T., Kassay, G.: Farkas type theorems for generalized convexities. PU.M.A. 5(2), 225–239 (1994)

    MathSciNet  MATH  Google Scholar 

  16. Irle, A.: A general minimax theorem. Z. Oper. Res. 29(7), 229–247 (1985)

    MathSciNet  MATH  Google Scholar 

  17. Jeyakumar, V.: Convexlike alternative theorems and mathematical programming. Optimization 16(5), 643–652 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  18. Jeyakumar, V., Lee, G.M., Li, G.Y.: Alternative theorems for quadratic inequality systems and global quadratic optimization. SIAM J. Optim. 20 (2), 983–1001 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. König, H.: Über das von Neumannsche minimax-theorem. Arch. Math. 19(5), 482–487 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  20. König, H.: A general minimax theorem based on connectedness. Arch. Math. 59(1), 55–64 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  21. König, H.: Addendum to “A general minimax theorem based on connectedness”. Arch. Math. 64(2), 139–143 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  22. Li, G.Y.: A note on nonconvex minimax theorem with separable homogeneous polynomials. J. Optim. Theory Appl. 150(1), 194–203 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. Lin, B.-L., Cheng, C.-Z.: A minimax theorem involving weakly downward functions. Acta. Math. Hungar. 87(4), 287–293 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  24. Mangasarian, O.L.: Nonlinear Programming. Society for industrial and applied mathematics philadelphia (1994)

  25. Martinez-Legaz, J.E., Seeger, A.: Yuan’s alternative theorem and the maximization of the minimum eigenvalue function. J. Optim. Theory Appl. 82(1), 159–167 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  26. Mosconi, S.J.N.: A differential characterisation of the minimax inequality. J. Convex Anal. 19(1), 185–199 (2012)

    MathSciNet  MATH  Google Scholar 

  27. Pólik, I., Terlaky, T.: A survey of the S-lemma. SIAM Rev. 49 (3), 371–418 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  28. Polyak, B.T.: Convexity of quadratic transformations and its use in control and optimization. J. Optim. Theory Appl. 99(3), 553–583 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  29. Pomerol, J.-C.: Inequality systems and minimax theorems. J. Math. Anal. Appl. 103(1), 263–292 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  30. Ricceri, B.: A strict minimax inequality criterion and some of its consequences. Positivity 16(3), 455–470 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  31. Ricceri, B., Simons, S.: Minimax Theory and Applications. Nonconvex Optimization and Its Applications, Ricceri, B., Simons, S. (eds.), vol. 26. Springer, Netherlands (1998)

  32. Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1971)

    MATH  Google Scholar 

  33. Simons, S.: An upward-downward minimax theorem. Arch. Math. 55(3), 275–279 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  34. Simons, S.: A flexible minimax theorem. Acta Math. Hungar. 63 (2), 119–132 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  35. Simons, S.: Addendum to “A flexible minimax theorem”. Acta. Math. Hungar. 69(4), 359–360 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  36. Simons, S.: Minimax theorems and their proofs. In: Du, D.-Z., Pardalos, P.M. (eds.) Minimax and Applications, Nonconvex Optimization and Its Applications, vol. 4, pp 1–23. Kluwer Academic Publishers, Dordrecht (1995)

  37. Sion, M.: On general minimax theorems. Pac. J. Optim. 8(1), 171–176 (1958)

    MathSciNet  MATH  Google Scholar 

  38. Stefănescu, A.: A general min-max theorem. Optimization 16(4), 497–504 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  39. Stefănescu, A.: Alternative and minimax theorems beyond vector spaces. J. Math. Anal. Appl. 264(2), 450–464 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  40. Stoer, J., Witzgall, C.: Convexity and Optimization in Finite Dimensions, vol. I. Springer, Heidelberg (1970)

    Book  MATH  Google Scholar 

  41. Tardella, F.: On the image of a constrained extremum problem and some applications to the existence of a minimum. J. Optim. Theory Appl. 60(1), 93–104 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  42. Tuy, H.: A new topological minimax theorem with application. J. Global Optim. 50(3), 371–378 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  43. von Neumann, J.: Zur theorie der gesellschaftsspiele. Math. Ann. 100(1), 295–320 (1928)

    Article  MathSciNet  MATH  Google Scholar 

  44. Weir, T., Jeyakumar, V.: A class of nonconvex functions and mathematical programming. Bull. Aust. Math. Soc. 38(2), 177–189 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  45. Yakubovich, V.A.: S-procedure in nonlinear control theory. Vestnik Leningrad. Univ. 4, 73–93 (1977)

    Google Scholar 

  46. Yuan, Y.: On a subproblem of trust region algorithms for constrained optimization. Math. Program. 47(1-3), 53–63 (1990)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The author is grateful to anonymous referees for their careful reading of the manuscript and for their helpful suggestions and corrections.

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  1. Hassan First University of Settat, Faculté des Sciences et Techniques, Laboratoire MISI, Settat, Morocco

    Ahmed Roubi

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  1. Ahmed Roubi
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Correspondence to Ahmed Roubi.

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Roubi, A. Theorems of the Alternative Revisited and Some Extensions of Yuan’s Lemma. Set-Valued Var. Anal 30, 1213–1232 (2022). https://doi.org/10.1007/s11228-022-00641-4

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