Minimizing and stationary sequences of constrained optimization problems

SIAM Journal on Control and Optimization

Volumn 36, Issue 6, 1998, Pages 1908-1936

  Chou, Chin Cheng ^a   Ng, Kung Fu ^b   Pang, Jong Shi ^c  

^a UNIVERSITÉ DE PERPIGNAN   (France)

^b CHINESE UNIVERSITY OF HONG KONG   (Hong Kong)

^c JOHNS HOPKINS UNIVERSITY   (United States)

Author keywords

Approximate optimality conditions; Convergence of algorithms; Convex analysis; Error bounds; Minimizing sequences; Quadratic splines; Residual functions; Stationary sequences; Weak sharp minima; Well posed optimization problems

Indexed keywords

ALGORITHMS; ASYMPTOTIC STABILITY; CONSTRAINT THEORY; CONVERGENCE OF NUMERICAL METHODS; FUNCTIONS; ITERATIVE METHODS; SET THEORY;

CONVEX ANALYSIS; ERROR BOUNDS; MINIMIZING SEQUENCES; QUADRATIC SPLINES; RESIDUAL FUNCTIONS; STATIONARY SEQUENCES; WEAK SHARP MINIMA;

OPTIMIZATION;

EID: 0032203674     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0363012995292548     Document Type: Article

References (40)

1. A. AUSLENDER, Convergence of stationary sequences for variational inequalities with maximal monotone operators, Appl. Math. Optim., 28 (1993), pp. 161-172.
2. A. AUSLENDER, R. COMINETTI, AND J.-P. CROUZEIX, Convex functions with unbounded level sets and applications to duality theory, SIAM J. Optim., 3 (1993), pp. 669-687.
3. A. AUSLENDER AND J.-P. CROUZEIX, Well-behaved asymptotical convex functions, Analyse non-linéare, 6 (1989), pp. 101-121.
4. A. BRONSTEDT AND R.T. ROCKAFELLAR, On the subdifferentiability of convex functions, Proc. Amer. Math. Soc., 16 (1965), pp. 605-611.
5. J.V. BURKE AND M.C. FERRIS, Weak sharp minima in mathematical programming, SIAM J. Control Optim., 31 (1993), pp. 1340-1359.
6. F.H. CLARKE, Optimization and Nonsmooth Analysis, John Wiley, New York, 1983.
7. R. DEVILLE, G. GODEFROY, AND V. ZIZLER, A smooth variational principle with, applications to Hamilton-Jacobi equations in infinite dimensions, J. Fund. Anal., 111 (1993), pp. 197-212.
8. A.L. DONTCHEV AND T. ZOLEZZI, Well-Posed Optimization Problems, Lecture Notes in Math. 1543, Springer-Verlag, Berlin, 1991.
9. I. EKELAND, On the variational principle, J. Math. Anal. Appl., 47 (1974), pp. 324-353.
10. M.C. FERRIS, Weak Sharp Minima and Penalty Functions in Mathematical Programming, Technical report 779, Computer Science Department, University of Wisconsin, Madison, WI, 1988.
11. M.C. FERRIS, Finite termination of the proximal point algorithm, Math. Programming, 50 (1991), pp. 359-366.
12. T. FUJISAWA AND E.S. KUH, Piecewise linear theory of nonlinear networks, SIAM J. Appl. Math., 22 (1972), pp. 307-328.
13. M. FUKUSHIMA AND J.S. PANG, Minimizing and stationary sequences of merit functions for nonlinear complementarity problems and variational inequalities, in Complementarity and Variational Problems: State of the Art, M.C. Ferris and J.S. Pang, eds., SIAM, Philadelphia, 1997, pp. 91-104.
14. K.C. KIWIEL AND K. MURTY, Convergence of the steepest descent method for minimizing quasiconvex functions, J. Optim. Theory Appl., 89 (1996), pp. 221-226.
15. B. LEMAIRE, Bonne position, conditionnement, et bon comportement asymptotique, Exposé 5, Seminaire D'Analyse Convexe, Université de Montpellier, 1992.
16. J.P. HIRIART-URRUTY AND C. LEMARÉCHAL, Convex Analysis and Minimization Algorithms, Vols. I and II, Springer-Verlag, Berlin, 1993.
17. E.S. LEVITIN AND B.T. POLYAK, Convergence of minimizing sequences in conditional extremum problems, Soviet Math. Dokl., 7 (1966), pp. 764-767.
18. A.S. LEWIS AND J.S. PANG, Error bounds for convex inequality systems, in Generalized Convexity, Generalized Monotonicity: Recent Results, in Proceedings of the Fifth Symposium on Generalized Convexity, Luminy, June 1996, J.P. Crouzeix, J.-E. Martinez-Legaz, and M. Volle, eds., Kluwer Academic Publishers, Dordrecht, 1997, pp. 75-110.
19. W. LI, Error bounds for piecewise convex quadratic programs and applications, SIAM J. Control Optim., 33 (1995), pp. 1510-1529.
20. W. Li, private e-mail communication, 1997.
21. R. LUCCHETTI AND J. REVALSKI, EDS., Recent Developments in Well-Posed Variational Problems, Kluwer Academic Publishers, Dordrecht, 1995.
22. X.D. LUO AND Z.Q. LUO, Extension of Huffman's error bound to polynomial systems, SIAM J. Optim., 4 (1994), pp. 383-392.
23. Z.Q. LUO AND J.S. PANG, Error bounds for analytic systems and their applications, Math. Programming, 67 (1994), pp. 1-28.
24. O.L. MANGASARIAN, A simple characterization of solution sets of convex programs, Oper. Res. Lett., 7 (1988), pp. 21-26.
25. O.L. MANGASARIAN, A condition number for differentiable convex inequalities, Math. Oper. Res., 10 (1985), pp. 175-179.
26. O.L. MANGASARIAN, Error Bounds for Nondifferentiable Convex Inequalities Under a Strong Slater Constraint Qualification, Mathematical Programming Technical Report 96-04, Computer Science Department, University of Wisconsin, Madison, WI, 1996.
27. O.L. MANGASARIAN AND J. REN, New improved error bounds for the linear complementarity problem, Math. Programming, 66 (1994) pp. 241-257.
28. J.S. PANG, A posteriori error bounds for the linearly constrained variational inequality problem, Math. Oper. Res., 12 (1987), pp. 474-484.
29. J.S. PANG, Error bounds in mathematical programming, Math. Programming Ser. B, 79 (1997), pp. 299-332.
30. J.S. PANG, S.P. HAN, AND N. RANGARAJ, Minimization of locally Lipschitzian functions, SIAM J. Optim., 1 (1991), pp. 57-82.
31. J.P. REVALSKI, Various aspects of well-posedness of optimization problems, in Recent Developments in Well-Posed Variational Problems, Kluwer Academic Publishers, Dordrecht, 1995, pp. 229-256.
32. S.M. ROBINSON, Normal maps induced by linear transformations, Math. Oper. Res., 17 (1992), pp. 691-714.
33. R.T. ROCKAFELLAR, Convex Analysis, Princeton University Press, Princeton, NJ, 1970.
34. p-programming, European J. Oper. Res., 22 (1985), pp. 70-100.
35. M.J. TODD, On convergence properties of algorithms for unconstrained minimization, IMA J. Numer. Anal., 9 (1989), pp. 435-441.
36. A.N. TYKHONOV, On the stability of the functional optimization problem, USSR Comput. Math. Math. Phys., 6 (1966), pp. 28-33.
37. T. WANG AND J.S. PANG, Global error bounds for convex quadratic inequality systems, Optimization, 31 (1994), pp. 1-12.
38. Z. WEI, L. QI, AND H. JIANG, Some Convergence Properties of Descent Methods, manuscript, School of Mathematics, University of New South Wales, Sydney, 1995.
39. F. WU AND S. WM, A modified Frank-Wolfe algorithm and its convergence properties, Acta Math. Appl. Sinica, 11 (1995), pp. 286-291.
40. S. Wu, Convergence properties of descent methods for unconstrained minimization, Optimization, 26 (1992), pp. 229-237.