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A theorem of the alternative with an arbitrary number of inequalities and quadratic programming

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Abstract

In this paper we are concerned with a Gordan-type theorem involving an arbitrary number of inequality functions. We not only state its validity under a weak convexity assumption on the functions, but also show it is an optimal result. We discuss generalizations of several recent results on nonlinear quadratic optimization, as well as a formula for the Fenchel conjugate of the supremum of a family of functions, in order to illustrate the applicability of that theorem of the alternative.

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References

  1. Bapat, R.B., Raghavan, T.E.S.: Nonnegative Matrices and Applications. Cambridge University Press, Cambridge (1997)

    Book  MATH  Google Scholar 

  2. Boţ, R.I.: Conjugate Duality in Convex Optimization. Lecture Notes in Economics and Mathematical Systems, vol. 637. Springer, Berlin (2010)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Carvajal, A., Weretka, M.: No-arbitrage, state prices and trade in thin financial markets. Econ. Theor. 50, 223–268 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen, X., Xiang, S.: Newton iterations in implicit time-stepping scheme for differential linear complementarity systems. Math. Program. 138, 579–606 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chuong, T.D.: L-invex-infine functions and applications. Nonlinear Anal. 75, 5044–5052 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dax, A., Sreedharan, V.P.: Theorems of the alternative and duality. J. Optim. Theory Appl. 94, 561–590 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  8. De Martino, D.: Thermodynamics of biochemical networks and duality theorems. Phys. Rev. E 87, 052108 (2013)

  9. Dines, L.L.: On the mapping of quadratic forms. Bull. Am. Math. Soc. 47, 494–498 (1944)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dinh, N., Mo, T.H.: Generalizations of the Hahn–Banach theorem revisited. Taiwan. J. Math. 19, 1285–1304 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. Flores-Bazán, F., Flores-Bazán, F., Vera, C.: Gordan-type alternative theorems and vector optimization revisited. In: Ansari, Q.H., Yao, J.-C. (eds.) Recent Developments in Vector Optimization, pp. 29–59. Springer, Berlin (2012)

  13. Giorgi, G., Guerraggio, A., Thierfelder, J.: Mathematics of Optimization: Smooth and Nonsmooth Case. Elsevier Science B.V, Amsterdam (2004)

    MATH  Google Scholar 

  14. Golikov, A.I., Evtushenko, Y.G.: Theorems of the alternative and their applications in numerical methods. Comput. Math. Math. Phys. 43, 338–358 (2003)

    MathSciNet  MATH  Google Scholar 

  15. Gordan, P.: Uber die auflösung linearer gleichungen mit reelen coefficienten. Math. Ann. 6, 23–28 (1873)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gowdaa, M.S., Ravindranb, G.: On the game-theoretic value of a linear transformation relative to a self-dual cone. Linear Algebra Appl. 469, 440–463 (2015)

    Article  MathSciNet  Google Scholar 

  17. Grzybowski, J., Przybycień, H., Urbański, R.: On Simons’ Version of Hahn–Banach–Lagrange Theorem, Function Spaces X, 99–104, Banach Center Publications 102. Polish Academy of Sciences, Institute of Mathematics, Warsaw (2014)

  18. Hu, S.-L., Huang, Z.-H.: Theorems of the alternative for inequality systems of real polynomials. J. Optim. Theory Appl. 154, 1–16 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. Jakubovič, V.A.: The S-Procedure in Nonlinear Control Theory, vol. 1, pp. 62–77. Vestnik Leningrad University (1971)

  20. Jeyakumar, V.: On optimality conditions in nonsmooth inequality constrained minimization. Numer. Funct. Anal. Optim. 9, 535–546 (1981)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  22. Jeyakumar, V., Lee, G.M., Li, G.Y.: Global optimality conditions for classes of non-convex multi-objective quadratic optimization problems. In: Burachik, R.S., Yao, J.-C. (eds.) Variational Analysis and Generalized Differentiation in Optimization and Control. Springer Optimization and Its Applications, vol. 47, pp. 177–186. Springer, New York (2010)

  23. Jin, Y., Kalantari, B.: A procedure of Chvátal for testing feasibility in linear programming and matrix scaling. Linear Algebra Appl. 416, 795–798 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kassay, G., Kolumbán, J.: On a generalized sup-inf problem. J. Optim. Theory Appl. 91, 651–670 (1996)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  26. König, H.: Sublinear functionals and conical measures. Arch. Math. 77, 56–64 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kuroiwa, D., Lee, G.: On robust convex multiobjective optimization. J. Nonlinear Convex Anal. 15, 1125–1136 (2014)

    MathSciNet  MATH  Google Scholar 

  28. Li Y.: Automatic synthesis of multiple ranking functions with supporting invariants via DISCOVERER. In: 2010 3rd International Conference on Advanced Computer Theory and Engineering (ICACTE 2010), Proceedings, vol. 1, pp. V1285–V1290 (2010)

  29. Luo, Z., Qin, L., Kong, L., Xiu, N.: The nonnegative zero-norm minimization under generalized Z-matrix measurement. J. Optim. Theory Appl. 160, 854–864 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  30. Mangasarian, O.L.: A stable theorem of the alternative: an extension of the Gordan theorem. Linear Algebra Appl. 41, 209–223 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  31. Mazur, S., Orlicz, W.: Sur les espaces métriques linéaires II. Stud. Math. 13, 137–179 (1953)

    Article  MATH  Google Scholar 

  32. Noor, E., Lewis, N.E., Milo, R.: A proof for loop-law constraints in stoichiometric metabolic networks. BMC Syst. Biol. 6, 140 (2012)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  34. Pták, V.: On a theorem of Mazur and Orlicz. Stud. Math. 15, 365–366 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  35. Ruiz Galán, M.: An intrinsic notion of convexity for minimax. J. Convex Anal. 21, 1105–1139 (2014)

    MathSciNet  MATH  Google Scholar 

  36. Ruiz Galán, M.: The Gordan theorem and its implications for minimax theory. J. Nonlinear Convex Anal. 17, 2385–2405 (2016)

    MathSciNet  MATH  Google Scholar 

  37. Simons, S.: Minimax and Monotonicity. Lecture Notes in Mathematics, vol. 1693. Springer, Berlin (1998)

    MATH  Google Scholar 

  38. Simons, S.: The Hahn–Banach–Lagrange theorem. Optimization 56, 149–169 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  39. Stefanescu, A.: A theorem of the alternative and a two-function minimax theorem. J. Appl. Math. 2004(2), 167–177 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  40. Tang, L.P., Zhao, K.Q.: Optimality conditions for a class of composite multiobjective nonsmooth optimization problems. J. Glob. Optim. 57, 399–414 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

Research partially supported by Project MTM2016-80676-P (AEI/FEDER, UE) and by Junta de Andalucía Grant FQM359.

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  1. Department of Applied Mathematics, E.T.S. Ingeniería de Edificación, University of Granada, c/Severo Ochoa s/n, 18071, Granada, Spain

    M. Ruiz Galán

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  1. M. Ruiz Galán
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Ruiz Galán, M. A theorem of the alternative with an arbitrary number of inequalities and quadratic programming. J Glob Optim 69, 427–442 (2017). https://doi.org/10.1007/s10898-017-0525-x

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