A new sequential optimality condition for constrained optimization and algorithmic consequences

SIAM Journal on Optimization

Volumn 20, Issue 6, 2010, Pages 3533-3554

  Andreani, Roberto ^a   Mart, J M ^a   Svaiter, B F ^b  

^a UNIVERSITY OF CAMPINAS   (Brazil)

^b Institute For Pure Applied Mathematics   (Brazil)

Author keywords

Approximate KKT conditions; Nonlinear programming; Optimality conditions; Stopping Criteria

Indexed keywords

AUGMENTED LAGRANGIAN ALGORITHM; CONSTRAINT QUALIFICATION; FIRST-ORDER; KKT CONDITION; LOCAL MINIMIZER; OPTIMALITY CONDITIONS; OPTIMIZATION PROBLEMS; STOPPING CRITERIA;

ALGORITHMS; CONSTRAINED OPTIMIZATION; LAGRANGE MULTIPLIERS;

NONLINEAR PROGRAMMING;

EID: 79251478775     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/090777189     Document Type: Article

References (27)

1. R. Andreani, E. G. Birgin, J. M. Martínez, and M. L. Schuverdt, On augmented Lagrangian methods with general lower-level constraints, SIAM J. Optim., 18 (2007), pp. 1286-1309.
2. R. Andreani, G. Haeser, and J. M. Martínez, On sequential optimality conditions for smooth constrained optimization, Optimization, to appear.
3. R. Andreani, J. M. Martínez, L. Martínez, and F. S. Yano, Low order-value optimization and applications, J. Global Optim., 43 (2009), pp. 1-10.
4. R. Andreani, J. M. Martínez, and M. L. Schuverdt, On the relation between the constant positive linear dependence condition and quasinormality constraint qualification, J.Optim. Theory Appl., 125 (2005), pp. 473-485.
5. H. Attouch and J. Bolte, On the convergence of the proximal algorithm for nonsmooth functions involving analytic features, Math. Program., 116 (2009), pp. 5-16.
6. H. Attouch, J. Bolte, P. Redont, and A. Soubeyran, Proximal alternating minimization and projection methods for nonconvex problems. An approach based on the Kurdyka- Lojasiewicz inequality, Math. Oper. Res., 35 (2010), pp. 438-457.
7. D. P. Bertsekas, Nonlinear Programming, 2nd ed., Athena Scientific, Belmont, Mass., 1999.
8. E. G. Birgin, C. A. Floudas, and J. M. Martínez, Global minimization using an augmented Lagrangian method with variable lower-level constraints, Math. Program., 125 (2010), pp. 139-162.
9. J. Bolte, A. Daniilidis, and A. S. Lewis, The Lojasiewicz inequality for nonsmooth subanalytic functions with applications to subgradient dynamical systems, SIAM J. Optim., 17 (2007), pp. 1205-1223.
10. J. Bolte, A. Daniilidis, O. Ley, and L. Mazet, Characterizations of Lojasiewicz inequalities and applications: Subgradient flows, talweg, convexity, Trans. Amer. Math. Soc., 362 (2010), pp. 3319-3363.
11. A. R. Conn, N. I. M. Gould, and Ph. L. Toint, Trust-Region Methods, MPS/SIAM Ser. Optim. 1, SIAM, Philadelphia, 2000.
12. A. V. Fiacco and G. P. McCormick, Nonlinear Programming: Sequential Unconstrained Minimization Techniques, Wiley, New York, 1968.
13. A. Fischer and A. Friedlander, A new line search inexact restoration approach for nonlinear programming, Comput. Optim. Appl., 46 (2010), pp. 333-346.
14. M. A. Gomes-Ruggiero, J. M. Martínez, and S. A. Santos, Spectral projected gradient method with inexact restoration for minimization with nonconvex constraints, SIAM J. Sci. Comput., 31 (2009), pp. 1628-1652.
15. C. C. Gonzaga, E.W. Karas, and M. Vanti, A globally convergent filter method for nonlinear rogramming, SIAM J. Optim., 14 (2004), pp. 646-669.
16. M. R. Hestenes, Optimization Theory: The Finite Dimensional Case, Wiley, New York, 1975.
17. 17] A. D. Ioffe, An invitation to tame optimization, SIAM J. Optim., 19 (2009), pp. 1894-1917.
18. 18] S. G. Krantz and H. G. Parks, A Primer on Real Analytic Functions, 2nd ed., Birkhäuser oston, Cambridge, Mass., 2002.
19. K. Kurdyka, On gradients of functions definable in o-minimal structures, Ann. Inst. Fourier Grenoble), 48 (1998), pp. 769-783.
20. S. Lojasiewicz, Sur les trajectories du gradient d'une fonction analytique, in Seminari di Geometria 982-1983, Universitá di Bologna, Istituto di Geometria, Dipartamento di Matematica, 1984, pp. 115-117.
21. O. L. Mangasarian and S. Fromovitz, The Fritz-John necessary optimality conditions in presence of equality and inequality constraints, J. Math. Anal. Appl., 17 (1967), pp. 37-47.
22. J. M. Martínez, Inexact restoration method with Lagrangian tangent decrease and new merit function for nonlinear programming, J. Optim. Theory Appl., 111 (2001), pp. 39-58.
23. J. M. Martínez and E. A. Pilotta, Inexact restoration algorithms for constrained optimization, J. Optim. Theory Appl., 104 (2000), pp. 135-163.
24. J. M. Martínez and B. F. Svaiter, A practical optimality condition without constraint qualifications for nonlinear programming, J. Optim. Theory Appl., 118 (2003), pp. 117-133.
25. J. Nocedal and S. J. Wright, Numerical Optimization, Springer, New York, 1999.
26. L. Qi and Z. Wei, On the constant positive linear dependence condition and its application to SQP methods, SIAM J. Optim., 10 (2000), pp. 963-981.
27. M. L. Schuverdt, Métodos de Lagrangiano Aumentado com Convergência Usandoa Condição de Dependência Linear Positiva Constante Tese de Doutorado Departamento de Matemática Aplicada Universidade Estadual de Campinas 2006.