Some new existence, sensitivity and stability results for the nonlinear complementarity problem

Rubén López

ESAIM: Control, Optimisation and Calculus of Variations (2008)

  • Volume: 14, Issue: 4, page 744-758
  • ISSN: 1292-8119

Abstract

top
In this work we study the nonlinear complementarity problem on the nonnegative orthant. This is done by approximating its equivalent variational-inequality-formulation by a sequence of variational inequalities with nested compact domains. This approach yields simultaneously existence, sensitivity, and stability results. By introducing new classes of functions and a suitable metric for performing the approximation, we provide bounds for the asymptotic set of the solution set and coercive existence results, which extend and generalize most of the existing ones from the literature. Such results are given in terms of some sets called coercive existence sets, which we also employ for obtaining new sensitivity and stability results. Topological properties of the solution-set-mapping and bounds for it are also established. Finally, we deal with the piecewise affine case.

How to cite

top

López, Rubén. "Some new existence, sensitivity and stability results for the nonlinear complementarity problem." ESAIM: Control, Optimisation and Calculus of Variations 14.4 (2008): 744-758. <http://eudml.org/doc/250361>.

@article{López2008,
abstract = { In this work we study the nonlinear complementarity problem on the nonnegative orthant. This is done by approximating its equivalent variational-inequality-formulation by a sequence of variational inequalities with nested compact domains. This approach yields simultaneously existence, sensitivity, and stability results. By introducing new classes of functions and a suitable metric for performing the approximation, we provide bounds for the asymptotic set of the solution set and coercive existence results, which extend and generalize most of the existing ones from the literature. Such results are given in terms of some sets called coercive existence sets, which we also employ for obtaining new sensitivity and stability results. Topological properties of the solution-set-mapping and bounds for it are also established. Finally, we deal with the piecewise affine case. },
author = {López, Rubén},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Nonlinear complementarity problem; variational inequality; asymptotic analysis; sensitivity analysis; nonlinear complementarity problem},
language = {eng},
month = {1},
number = {4},
pages = {744-758},
publisher = {EDP Sciences},
title = {Some new existence, sensitivity and stability results for the nonlinear complementarity problem},
url = {http://eudml.org/doc/250361},
volume = {14},
year = {2008},
}

TY - JOUR
AU - López, Rubén
TI - Some new existence, sensitivity and stability results for the nonlinear complementarity problem
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2008/1//
PB - EDP Sciences
VL - 14
IS - 4
SP - 744
EP - 758
AB - In this work we study the nonlinear complementarity problem on the nonnegative orthant. This is done by approximating its equivalent variational-inequality-formulation by a sequence of variational inequalities with nested compact domains. This approach yields simultaneously existence, sensitivity, and stability results. By introducing new classes of functions and a suitable metric for performing the approximation, we provide bounds for the asymptotic set of the solution set and coercive existence results, which extend and generalize most of the existing ones from the literature. Such results are given in terms of some sets called coercive existence sets, which we also employ for obtaining new sensitivity and stability results. Topological properties of the solution-set-mapping and bounds for it are also established. Finally, we deal with the piecewise affine case.
LA - eng
KW - Nonlinear complementarity problem; variational inequality; asymptotic analysis; sensitivity analysis; nonlinear complementarity problem
UR - http://eudml.org/doc/250361
ER -

References

top
  1. J.-P. Aubin and H. Frankowska, Set-Valued Analysis. Birkhäuser, Boston (1990).  
  2. A. Auslender and M. Teboulle, Asymptotic Cones and Functions in Optimization and Variational Inequalities. Springer, Berlin (2003).  
  3. R.W. Cottle, J.S. Pang and R.E. Stone, The Linear Complementarity Problem. Academic Press, New York (1992).  
  4. J.P. Crouzeix, Pseudomonotone variational inequality problems: Existence of solutions. Math. Program.78 (1997) 305–314.  
  5. S. Dafermos, Sensitivity analysis in variational inequalities. Math. Oper. Res.13 (1988) 421–434.  
  6. R. Doverspike, Some perturbation results for the linear complementarity problem. Math. Program.23 (1982) 181–192.  
  7. F. Facchinei and J.S. Pang, Total stability of variational inequalities. Technical Report 09–98, Dipartimento di Informatica e Sistematica, Università Degli Stuti di Roma “La Sapienza” (1998).  
  8. F. Facchinei and J.S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems I. Springer, New York (2003).  
  9. F. Flores-Bazán and R. López, The linear complementarity problem under asymptotic analysis. Math. Oper. Res.30 (2005) 73–90.  
  10. F. Flores-Bazán and R. López, Characterizing Q-matrices beyong L-matrices. J. Optim. Theory Appl.127 (2005) 447–457.  
  11. F. Flores-Bazán and R. López, Asymptotic analysis, existence and sensitivity results for a class of multivalued complementarity problems. ESAIM: COCV12 (2006) 271–293.  
  12. M.S. Gowda, Complementarity problems over locally compact cones. SIAM J. Control Optim.27 (1989) 836–841.  
  13. M.S. Gowda and J.S. Pang, On solution stability of the linear complementarity problems. Math. Oper. Res.17 (1992) 77–83.  
  14. M.S. Gowda and J.S. Pang, Some existence results for multivalued complementarity problems. Math. Oper. Res.17 (1992) 657–669.  
  15. M.S. Gowda and J.S. Pang, The basic theorem of complementarity revisited. Math. Program.58 (1993) 161–177.  
  16. M.S. Gowda and J.S. Pang, On the boundedness and stability to the affine variational inequality problem. SIAM J. Control Optim.32 (1994) 421–441.  
  17. M.S. Gowda and R. Sznajder, On the Lipschitzian properties of polyhedral multifunctions. Math. Program.74 (1996) 267–278.  
  18. C.D. Ha, Application of degree theory in stability of the complementarity problem. Math. Oper. Res.12 (1987) 368–376.  
  19. P.T. Harker and J.S. Pang, Finite-dimensional variational and nonlinear complementarity problems: A survey of theory, algorithms and applications. Math. Program.48 (1990) 161–220.  
  20. W.W. Hogan, Point-to-set maps in mathematical programming. SIAM Rev.15 (1973) 591–603.  
  21. G. Isac, The numerical range theory and boundedness of solutions of the complementarity problem. J. Math. Anal. Appl.143 (1989) 235–251.  
  22. G. Isac, Pseudo-monotone complementarity problems in Hilbert space. J. Optim. Theory Appl.75 (1992) 281–295.  
  23. S. Karamardian, Generalized complementarity problem. J. Optim. Theory Appl.8 (1971) 161–168.  
  24. S. Karamardian, Complementarity problems over cones with monotone and pseudomonotone maps. J. Optim. Theory Appl.18 (1976) 445–454.  
  25. D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1980).  
  26. J. Kyparisis, Sensitivity analysis for variational inequalities and complementarity problems. Ann. Oper. Res.27 (1990) 143–174.  
  27. O.L. Mangasarian, Characterizations of bounded solutions of linear complementarity problems. Math. Program. Study19 (1982) 153–166.  
  28. O.L. Mangasarian and L. McLinden, Simple bounds for solutions of monotone complementarity problems and convex programs. Math. Program.32 (1985) 32–40.  
  29. N. Megiddo, A monotone complementarity problem with feasible solutions but no complementarity solutions. Math. Program.12 (1977) 131–132.  
  30. N. Megiddo, On the parametric nonlinear complementarity problem. Math. Program. Study7 (1978) 142–150.  
  31. J.J. Moré, Coercivity conditions in nonlinear complementarity problems. SIAM Rev.17 (1974) 1–16.  
  32. J.S. Pang, Complementarity problems, in Nonconvex Optimization and its Applications: Handbook of Global Optimization, R. Horst and P.M. Pardalos Eds., Kluwer, Dordrecht (1995).  
  33. S.M. Robinson, Some continuity properties of polyhedral multifunctions. Math. Program. Study14 (1981) 206–214.  
  34. R.T. Rockafellar and R.J. Wets, Variational Analysis. Springer, Berlin (1998).  
  35. R.L. Tobin, Sensitivity analysis for complementarity problems. J. Optim. Theory Appl.48 (1986) 191–204.  
  36. S.W. Xiang and Y.H. Zhou, Continuity properties of solutions of vector optimization. Nonlinear Anal.64 (2006) 2496–2506.  
  37. Y. Zhao, Existence of a solution to nonlinear variational inequality under generalized positive homogeneity. Oper. Res. Lett.25 (1999) 231–239.  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.