Characterizations of strong regularity for variational inequalities over polyhedral convex sets

SIAM Journal on Optimization

Volumn 6, Issue 4, 1996, Pages 1087-1105

  Dontchev, A L ^a   Rockafellar, R T ^b  

^a Mathematical Reviews   (United States)

^b UNIVERSITY OF WASHINGTON   (United States)

Author keywords

Aubin property; Lipschitz stability in optimization; Strong regularity; Variational inequality

Indexed keywords

EID: 0030521635     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1052623495284029     Document Type: Article

References (30)

1. J.-P. AUBIN, Lipschitz behavior of solutions to convex minimization problems, Math. Oper. Res., 9 (1984), pp. 87-111.
2. J. F. BONNANS AND A. SULEM, Pseudopower expansion of solutions of generalized equations and constrained optimization problems, Math. Programming, 70 (1995), pp. 123-148.
3. J. M. BORWEIN AND D. M. ZHUANG, Verifiable necessary and sufficient conditions for openness and regularity of set-valued maps, J. Math. Anal. Appl., 134 (1988), pp. 441-459.
4. R. W. COTTLE, JONG-SHI PANG, AND R. E. STONE, The Linear Complementarity Problem, Academic Press, Boston 1992.
5. A. L. DONTCHEV AND W. W. HAGER, An inverse function theorem for set-valued maps, Proc. Amer. Math. Soc., 121 (1994), pp. 481-489.
6. _, Implicit functions, Lipschitz maps and stability in optimization, Math. Oper. Res., 19 (1994), pp. 753-768.
7. B. C. EAVES AND U. G. ROTHBLUM, Relationship of properties of piecewise affine maps over ordered fields, Linear Algebra Appl., 132 (1990), pp. 1-63.
8. M. S. GOWDA, On the continuity of the solution map in linear complementarity problems, SIAM J. Optim., 2 (1992), pp. 619-634.
9. M. S. GOWDA AND R. SZNAJDER, On the Lipschitz properties of polyhedral multifunctions, Math. Programming, 74 (1996), pp. 267-278.
10. M. HESTENES, Calculus of Variations and Optimal Control Theory, John Wiley, New York, 1966.
11. H. TH. JONGEN, T. MÖBERT, J. RÜCKMANN, AND K. TAMMER, Implicit functions and sensitivity of stationary points, Linear Algebra Appl., 95 (1987), pp. 97-109.
12. H. TH. JONGEN, D. KLATTE, AND K. TAMMER, Implicit functions and sensitivity of stationary points, Math. Programming, 19 (1990), pp. 123-138.
13. D. KLATTE AND K. TAMMER, Strong stability of stationary solutions and Karush-Kuhn-Tucker points in nonlinear optimization, Ann. Oper. Res., 27 (1990), pp. 285-308.
14. M. KOJIMA, Strongly stable stationary solutions in nonlinear programming, in Analysis and Computation of Fixed Points, S. M. Robinson, ed., Academic Press, New York, 1988, pp. 93-138.
15. 1,1 optimization, J. Optim. Theory Appl., 70 (1991), pp. 561-582.
16. O. L. MANGASARIAN AND T.-H. SHIAU, Lipschitz continuity of solutions of linear inequalities, programs and complementarity problems, SIAM J. Control Optim., 25 (1987), pp. 582-595.
17. B. MORDUKHOVICH, Stability theory for parametric generalized equations and variational inequalities via nonsmooth analysis, Trans. Amer. Math. Soc., 343 (1994), pp. 609-657.
18. G. S. R. MURTHY, T. PARTHASARATHY, AND M. SABATINI, On Lipschitzian Q-matrices, Technical report 20, Statistical Quality Control and Operations Research Unit, Indian Statistical Institute, Madras, India, July 1994.
19. K. G. MURTY, Linear Complementarity, Linear and Nonlinear Programming, Heldermann Verlag, Berlin, 1988.
20. JONG-SHI PANG, A degree-theoretic approach to parametric nonsmooth equations with multivalued perturbed solution sets, Math. Programming, 62 (1993), pp. 359-383.
21. J.-P. PENOT, Metric regularity, openness and Lipschitz multifunctions, Nonlinear Anal., 13 (1989), pp. 629-643.
22. D. RALPH, A new proof of Robinson's homeomorphism theorem for PL-normal maps, Linear Algebra Appl., 178 (1993), pp. 249-260.
23. S. M. ROBINSON, Strongly regular generalized equations, Math. Oper. Res., 5 (1980), pp. 43-62.
24. _, Local structure of feasible sets in nonlinear programming, Part II: Nondegeneracy, Math. Programming Study, 22 (1984), pp. 217-230.
25. _, An implicit-function theorem for a class of nonsmooth functions, Math. Oper. Res., 16 (1991), pp. 292-309.
26. _, Normal maps induced by linear transformations, Math. Oper. Res., 17 (1992), pp. 691-714.
27. R. T. ROCKAFELLAR, Lipschitz properties of multifunctions, Nonlinear Analysis, 9 (1985), pp. 867-885.
28. _, Proto-differentiability of set-valued mappings and its applications in optimization, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 6 (1989), suppl. pp. 449-482.
29. H. SAMELSON, R. M. THRALL, AND O. WESLER, A partition theorem for Euclidean n-space, Proc. Amer. Math. Soc., 9 (1958), pp. 805-807.
30. S. SCHOLTES, Introduction to piecewise differentiate equations, Institut für Statistik und Mathematische Wirtschaftstheorie, Universität Karlsruhe, Karlsruhe, Germany, May 1994, preprint 53/1994.