Second-order conditions for existence of augmented Lagrange multipliers for eigenvalue composite optimization problems

1. Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. CMS Books in Mathematics. Springer, New York (2011)

   Book  Google Scholar 

2. Bellman, R.: Introduction to Matrix Analysis, 2nd edn. McGraw-Hill, New York (1970)

   MATH  Google Scholar 

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

   Book  MATH  Google Scholar 

4. Cullum, J., Donath, W.E., Wolfe, P.: The minimization of certain nondifferentiable sums of eigenvalues of symmetric matrices. Math. Program. Study 3, 35–55 (1975)

   Article  MathSciNet  Google Scholar 

5. Fernández, D., Solodov, M.V.: Local convergence of exact and inexact augmented Lagrangian methods under the second-order sufficient optimality condition. SIAM J. Optim. 22, 384–407 (2012)

   Article  MathSciNet  MATH  Google Scholar 

6. Fletcher, R.: Semi-definite matrix constraints in optimization. SIAM J. Control Optim. 23, 493–513 (1985)

   Article  MathSciNet  MATH  Google Scholar 

7. Fletcher, R.: Practical Methods of Optimization, 2nd edn. Wiley, Chichester (1987)

   MATH  Google Scholar 

8. Friedland, S.: Convex spectral functions. Linear Multilinear Algebra 9, 299–316 (1981)

   Article  MathSciNet  MATH  Google Scholar 

9. Combettes, P.L., Wajs, V.R.: Signal recovery by proximal forward-backward splitting. Multiscale Model. Simul. 4, 1168–1200 (2005)

   Article  MathSciNet  MATH  Google Scholar 

10. Do, Chi Ngoc: Generalized second-order derivatives of convex function in reflexive Banach spaces. Trans. Amer. Math. Soc. 334, 281–301 (1992)

    Article  MathSciNet  MATH  Google Scholar 

11. Haug, E.J., Rousselet, B.: Design sensitivity analysis in structural mechanics II: eigenvalue variation. J. Struct. Mech. 8, 161–186 (1980)

    Article  MathSciNet  Google Scholar 

12. Hiriart-Urruty, J.-B., Ye, D.: Sensitivity analysis of all eigenvalues of a symmetric matrix. Numer. Math. 70, 45–72 (1995)

    Article  MathSciNet  MATH  Google Scholar 

13. Lewis, A.S.: Derivatives of spectral functions. Math. Oper. Res. 21, 576–588 (1996)

    Article  MathSciNet  MATH  Google Scholar 

14. Lewis, A.S., Sendov, H.S.: Twice differentiable spectral functions. SIAM. J. Matrix Anal. Appl. 23, 368–386 (2001)

    Article  MathSciNet  MATH  Google Scholar 

15. Liu, Y.J., Zhang, L.W.: Convergence analysis of the augmented Lagrangian method for nonlinear second-order cone optimization problem. Nonlinear Anal. 67, 1359–1373 (2007)

    Article  MathSciNet  MATH  Google Scholar 

16. Overton, M.L.: On minimizing the maximum eigenvalue of a symmetric matrix. SIAM J. Matrix Anal. Appl. 9, 256–268 (1988)

    Article  MathSciNet  MATH  Google Scholar 

17. Overton, M.L.: Large-scale optimization of eigenvalues. SIAM J. Optim. 2, 88–120 (1992)

    Article  MathSciNet  MATH  Google Scholar 

18. Overton, M.L., Womersley, R.S.: Optimality conditions and duality theory for minimizing sums of the largest eigenvalues of symmtric matrix. Math. Program. Ser. B 62, 321–357 (1993)

    Article  MathSciNet  MATH  Google Scholar 

19. Poliquin, R.A., Rockafellar, R.T.: Prox-regular functions in variational analysis. Trans. Amer. Math. Soc. 348, 1805–1838 (1996)

    Article  MathSciNet  MATH  Google Scholar 

20. Poliquin, R., Rockafellar, R.T.: Generalized Hessian properties of regularized nonsmooth functions. SIAM J. Optim. 6, 1121–1137 (1996)

    Article  MathSciNet  MATH  Google Scholar 

21. Rockafellar, R.T.: Augmented Lagrange multiplier functions and duality in nonconvex programming. SIAM J. Control 12, 268–285 (1974)

    Article  MathSciNet  MATH  Google Scholar 

22. Rockafellar, R.T.: Conjugate duality and optimization, regional conference series in applied mathematics. SIAM, Philadelphia (1974)

23. Rockafellar, R.T.: First- and second-order epi-differentiability in nonlinear programming. Trans. Amer. Math. Soc. 307, 75–108 (1988)

    Article  MathSciNet  MATH  Google Scholar 

24. Rockafellar, R.T.: Generalized second derivatives of convex functions and saddle function. Trans. Amer. Math. Soc. 322, 51–77 (1990)

    Article  MathSciNet  MATH  Google Scholar 

25. Rockafellar, R.T.: Lagrange multipliers and optimality. SIAM Rev. 35, 183–238 (1993)

    Article  MathSciNet  MATH  Google Scholar 

26. Rockafellar, R.T.: Second-order optimality conditions in nonlinear programming obtained by way of epi-derivatives. Math. Oper. Res. 14, 462–484 (1989)

    Article  MathSciNet  MATH  Google Scholar 

27. Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, New York (1998)

    Book  MATH  Google Scholar 

28. Shapiro, A., Sun, J.: Some properties of the augmented Lagrangian in cone constrained optimization. Math. Oper. Res. 29, 479–491 (2004)

    Article  MathSciNet  MATH  Google Scholar 

29. Shapiro, A.: Perturbation theory of nonlinear programs when the set of optimal solutions is not a singleton. Appl. math. Optim. 18, 215–229 (1988)

    Article  MathSciNet  MATH  Google Scholar 

30. Shapiro, A., Fan, M.K.H.: On eigenvalue optimization. SIAM J. Optim. 3, 552–568 (1995)

    Article  MathSciNet  Google Scholar 

31. Torki, M.: Second-order directional derivatives of all eigenvalues of a symmetric matrix. Nonlinear Anal. Ser. A Methods 46, 1133–1150 (2001)

    Article  MathSciNet  MATH  Google Scholar 

32. Torki, M.: First- and second-order epi-differentiability in eigenvalue optimization. J. Math. Anal. Appl. 234, 391–416 (1999)

    Article  MathSciNet  MATH  Google Scholar 

33. Wielandt, H.: An extremum property of sums of eigenvalues. Proc. Amer. Math. Soc. 6, 106–110 (1955)

    Article  MathSciNet  MATH  Google Scholar 

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