Geometrical interpretation of the predictor-corrector type algorithms in structured optimization problems

Optimization

Volumn 55, Issue 5-6, 2006, Pages 481-503

  Daniilidis, Aris ^a   Hare, Warren ^b   Malick, Jérôme ^c  

^a UNIVERSITAT AUTÒNOMA DE BARCELONA   (Spain)

^b IMPA   (Brazil)

^c INRIA   (France)

Author keywords

Newton type methods; Partly smooth function; Proximal algorithm; Riemannian gradient; U Lagrangian

Indexed keywords

EID: 33750892341     PISSN: 02331934     EISSN: 10294945     Source Type: Journal    
DOI: 10.1080/02331930600815884     Document Type: Article

References (30)

1. Arias, T., Edelman, A. and Smith, S.T., 1999, The geometry of algorithms with orthogonality constraints. SIAM Journal on Matrix Analysis and Applications, 20(2), 303-353.
2. Al-Khayyal, F. and Kyparisis, J., 1991, Finite convergence of algorithms for nonlinear programs and variational inequalities. Journal of Optimization Theory and Applications, 70(2), 319-332.
3. Arnold, V., 1971, On matrices depending on parameters. Russian Mathematical Surveys, 26, 29-43.
4. Burke, J.V. and Moré, J. J., 1988, On the identification of active constraints. SIAM Journal on Numerical Analysis, 25(5), 1197-1211.
5. Burke, J., 1990, On the identification of active constraints. II. The nonconvex case. SIAM Journal on Numerical Analysis, 27(4), 1081-1103.
6. Calamai, P.H. and Moré, J.J., 1987, Projected gradient methods for linearly constrained problems. Mathematical Programming, 39(1), 93-116.
7. do Carmo, M.P., 1992, Riemannian Geometry, Mathematics: Theory and Applications (Boston: Birkhäuser).
8. Dedieu, J.-P., Malajovich, G. and Priouret, P., 2003, Newton's method on Riemannian manifolds: covariant alpha-theory. IMA J. Num. Anal., 23(3), 395-419.
9. Gabay, D., June 1982, Minimizing a differentiable function over a differentiable manifold. Journal of Optimization Theory and Applications, 37(2), 177-219.
10. Hare, W.L., 2004, Nonsmooth Optimization with Smooth Substructure. PhD thesis (Vancouver: Simon Fraser University).
11. Hare, W.L., April 2006, Recent functions and sets of smooth substructure: Relationships and examples. J. Comput. Optim. Appl., 33(2) (To appear).
12. Hare, W.L. and Lewis, A.S., 2004, Identifying active constraints via partial smoothness and prox-regularity. Journal of Convex Analysis, 11(2), 251-266.
13. Lewis, A.S., 2003, Active sets, nonsmoothness and sensitivity. SIAM Journal on Optimization, 13, 702-725.
14. Lemaréchal, C., Oustry, F. and Sagastizábal, C., 2000, The U-Lagrangian of a convex function. Transactions of the American Mathematical Society, 352(2), 711-729.
15. Malick, J. and Miller, S., 2005, Newton methods for nonsmooth convex minimization: connection among U-Lagrangian, Riemannian Newton and SQP methods. Mathematical Programming, 104(3), 609-633.
16. Mifflin, R. and Sagastizábal, C., 1999, VU-decomposition derivatives for convex max-functions. In: Ill-Posed Variational Problems and Regularization Techniques (Trier, 1998), Vol. 477 of Lecture Notes in Econom. and Math. Systems (Berlin: Springer) pp. 167-186.
17. Mifflin, R. and Sagastizábal, C., 2000, Functions with primal-dual gradient structure and U-hessians. In: Nonlinear Optimization and Related Topics (Erice, 1998), Vol. 36 of Applied Optimization, (Dordrecht: Kluwer Acad. Publ.) pp. 219-233.
18. Mifflin, R. and Sagastizábal, C., 2000, On VU-theory for functions with primal-dual gradient structure. SIAM Journal on Optimization, 11(2), 547-571.
19. Mifflin, R. and Sagastizábal, C., 2002, Proximal points are on the fast track. Journal of Convex Analysis, 9(2), 563-579.
20. Mifflin, R. and Sagastizábal, C., 2004, VU-smoothness and proximal point results for some non-convex functions. Optimization Methods & Software, 19(5), 463-478.
21. Mifflin, R. and Sagastizábal, C., 2005, A VU-proximal point algorithm for convex minimization. Math. Programming, 104(3), 583-608.
22. Oustry, F., 1999, The U-Lagrangian of the maximum eigenvalue function. SIAM Journal on Optimization, 9(2), 526-549, 1999.
23. Poliquin, R.A. and Rockafellar, R.T., 1996, Generalized hessian properties of regularized nonsmooth functions. SIAM Journal on Optimization, 6(4), 1121-1137.
24. Poliquin, R.A. and Rockafellar, R.T., 1996, Prox-regular functions in variational analysis. Transactions of the American Mathematical Society, 348(5), 1805-1838.
25. Rockafellar, R.T., 1982, Favorable classes of Lipschitz-continuous functions in subgradient optimization. In: Progress in Nondifferentiable Optimization, Vol. 8 of IIASA Collaborative Proc. Ser. CP-82 (Laxenburg: Internat. Inst. Appl. Systems Anal) pp. 125-143.
26. Tyrrell Rockafellar, R. and Wets, R.J.-B., 1998, Variational Analysis. Number 317 in Grundlehren der mathematischen Wissenschaften. (Berlin: Springer-Verlag).
27. Shapiro, A., 2003, On a class of nonsmooth composite functions. Mathematics of Operations Research, 28, 677-692.
28. Smith, S.T., 1994, Optimization techniques on Riemannian manifolds. Fields Institute Communications, 3, 113-136.
29. Wolfe, P., 1975, A method of conjuguate subgradients. Mathematical Programming, 3, 145-173.
30. Wright, S.J., 1993, Identifiable surfaces in constrained optimization. SIAM Journal on Control and Optimization, 31(4), 1063-1079.