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An application of matrix computations to classical second-order optimality conditions

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Abstract

For finite dimensional optimization problems with equality and inequality constraints, a weak constant rank condition (WCR) was introduced by Andreani–Martinez–Schuverdt (AMS) (Optimization 5–6:529–542, 2007) to study classical necessary second-order optimality conditions. However, this condition is not easy to check. Using a polynomial and matrix computation tools, we can substitute it by a weak constant rank condition (WCRQ) for an approximated problem and we present a method for checking points that satisfy WCRQ. We extend the result of (Andreani et al. in Optimization 5–6:529–542, 2007), we show that WCR can be replaced by WCRQ and we prove that these two conditions are independent.

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References

  1. Abdeljaoued, J., Lombardi, H.: Méthode matricielles: Introduction à la complexité algébrique. Springer (2000)

  2. Andreani R., Martinez J.M., Schuverdt M.L.: On second-order optimality conditions for nonlinear programming. Optimization 5–6, 529–542 (2007)

    Article  MathSciNet  Google Scholar 

  3. Anitescu M.: Degenerate nonlinear programming with a quadratic growth condition. SIAM J. Optim. 10(4), 1116–1135 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  4. Arutyunov A.V.: Perturbations of extremum problems with constraints and necessary optimality conditions. J. Sov. Math. 54, 1342–1400 (1991)

    Article  MATH  Google Scholar 

  5. Baccari A.: On the classical necessary second-order optimality conditions. J. Optim. Theory Appl. 123(1), 213–221 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  6. 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 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bazara M.S., Sherali H.D., Shetty C.M.: Nonlinear Programming, Theory and Algorithms. Wiley, New York (1994)

    Google Scholar 

  8. Ben-Tal A., Zowe J.: A unified theory of first and second-order conditions for extremum problems in topological vector spaces. Math. Program. Study 19, 39–76 (1982)

    MathSciNet  Google Scholar 

  9. Bini, D., Pan, V.: Polynomial and Matrix Computations, vol. 1. Birkhauser (1994)

  10. Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer (2000)

  11. Durand, E.: Solutions Numériques des Equations Algébriques, Tome II. Masson-Cie (1961)

  12. Gantmacher, F.R.: Théorie des matrices, Tome I. Dunod (1966)

  13. Gauvin J.: A necessary and sufficient regularity condition to have bounded multipliers in nonconvex programming. Math. Program. 12, 136–138 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  14. Gauvin, J.: Leçons de programmation mathématique. Editions de l’Ecole Polytechnique de Montréal (1995)

  15. Hestenes, M.R.: Optimization Theory, the Finite Dimensional Case. Robert E. Krieger Publshing Company (1981)

  16. Hiriart-Urruty, J.B.: L’optimisation. Que sais-je, Presse Universitaire de France (1996)

  17. Ioffe A.: Necessary and sufficient conditions for a local minimum. 3: second-order conditions and augmented duality. SIAM J. Control Optim. 17, 245–288 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  18. Janin R.: Directional derivative of the marginal function in nonlinear programming. Math. Program. Study 21, 127–138 (1984)

    MathSciNet  Google Scholar 

  19. Lancaster, E., Timenetsky, M.: The Theory of Matrices, 2nd edn. Academic Press (1985)

  20. Mangasarian O.L., Fromovitz S.: The Fritz John necessary optimality conditions in the presence of equality and inequality constraints. J. Math. Anal. Appl. 17, 37–47 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  21. McCormick G.: Nonlinear Programming. Wlley, New York (1983)

    MATH  Google Scholar 

  22. Saint Raymond, J.: Private communication. Université Pierre et Marie Curie (France)

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  1. École Supérieure des Sciences et Techniques de Tunis, 5 Avenue Taha Hussein, 1008, Tunis, Tunisia

    M. Daldoul & A. Baccari

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  1. M. Daldoul
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  2. A. Baccari
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Correspondence to M. Daldoul.

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Daldoul, M., Baccari, A. An application of matrix computations to classical second-order optimality conditions. Optim Lett 3, 547–557 (2009). https://doi.org/10.1007/s11590-009-0134-9

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