Annals of Operations Research

, Volume 27, Issue 1, pp 143–173 | Cite as

Sensitivity analysis for variational inequalities and nonlinear complementarity problems

  • Jerzy Kyparisis
Part I Theoretical Results for Specific Problem Types

Abstract

This paper surveys the main results in the area of sensitivity analysis for finite-dimensional variational inequality and nonlinear complementarity problems. It provides an overview of Lipschitz continuity and differentiability properties of perturbed solutions for variational inequality problems, defined on both fixed and perturbed sets, and for nonlinear complementarity problems.

Keywords

Sensitivity analysis parametric solution set variational inequalities nonlinear complementarity problems generalized equations 

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References

  1. [1]
    C. Baiocchi and A. Capelo,Variational and Quasivariational Inequalities: Application to Free-Boundary Problems (Wiley, New York, 1984).Google Scholar
  2. [2]
    B. Bank, J. Guddat, D. Klatte, B. Kummer and K. Tammer,Nonlinear Parametric Optimization (Birkhäuser, Basel, 1983).Google Scholar
  3. [3]
    R.W. Cottle, Nonlinear programs with positively bounded Jacobians, SIAM J. Appl. Math. 14 (1966) 147–158.Google Scholar
  4. [4]
    R.W. Cottle, F. Giannessi and J.L. Lions (eds.),Variational Inequalities and Complementarity Problems: Theory and Applications (Wiley, New York, 1980).Google Scholar
  5. [5]
    S. Dafermos, Sensitivity analysis in variational inequalities, Math. Oper. Res. 13 (1988) 421–434.Google Scholar
  6. [6]
    S. Dafermos and A. Nagurney, Sensitivity analysis for the general spatial economic equilibrium problem, Oper. Res. 32 (1984) 1069–1086.Google Scholar
  7. [7]
    S. Dafermos and A. Nagurney, Sensitivity analysis for the asymmetric network equilibrium problem, Math. Programming 28 (1984) 174–184.Google Scholar
  8. [8]
    A.V. Fiacco,Introduction to Sensitivity and Stability Analysis in Nonlinear Programming (Academic Press, New York, 1983).Google Scholar
  9. [9]
    A.V. Fiacco and G.P. McCormick,Nonlinear Programming: Sequential Unconstrained Minimization Techniques (Wiley, New York, 1968).Google Scholar
  10. [10]
    J. Gauvin, A necessary and sufficient regularity condition to have bounded multipliers in nonconvex programming, Math. Programming 12 (1977) 136–138.Google Scholar
  11. [11]
    C.D. Ha, Stability of the linear complementarity problem at a solution point, Math. Programming 31 (1985) 327–338.Google Scholar
  12. [12]
    C.D. Ha, Application of degree theory in stability of the complementarity problem, Math. Oper. Res. 12 (1987) 368–376.Google Scholar
  13. [13]
    P.T. Harker and J.-S. Pang, Existence of optimal solutions to mathematical programs with equilibrium constraints, Oper. Res. Lett. 7 (1988) 61–64.Google Scholar
  14. [14]
    P.T. Harker and J.-S. Pang, Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications, Math. Programming, Series B 48 (1990) forthcoming.Google Scholar
  15. [15]
    P. Hartman and G. Stampacchia, On some nonlinear elliptic differential functional equations, Acta Math. 115 (1966) 153–188.Google Scholar
  16. [16]
    R. Janin, Directional derivative of the marginal function in nonlinear programming, Math. Programming Study 21 (1984) 110–126.Google Scholar
  17. [17]
    K. Jittorntrum, Solution point differentiability without strict complementarity in nonlinear programming, Math. Programming Study 21 (1984) 127–138.Google Scholar
  18. [18]
    S. Karamardian, Generalized complementarity problem, J. Optim. Theory Appl. 8 (1971) 161–167.Google Scholar
  19. [19]
    D. Kinderlehrer and G. Stampacchia,An Introduction to Variational Inequalities and Their Applications (Academic Press, New York, 1980).Google Scholar
  20. [20]
    M. Kojima, Studies on piecewise-linear approximations of piecewise-C1 mappings in fixed points and complementarity theory, Math. Oper. Res. 3 (1978) 17–36.Google Scholar
  21. [21]
    J. Kyparisis, On uniqueness of Kuhn-Tucker multipliers in nonlinear programming, Math. Programming 32 (1985) 242–246.Google Scholar
  22. [22]
    J. Kyparisis, Uniqueness and differentiability of solutions of parametric nonlinear complementarity problems, Math. Programming 36 (1986) 105–113.Google Scholar
  23. [23]
    J. Kyparisis, Sensitivity analysis framework for variational inequalities, Math. Programming 38 (1987) 203–213.Google Scholar
  24. [24]
    J. Kyparisis, Perturbed solutions of variational inequality problems over polyhedral sets, J. Optim. Theory Appl. 57 (1988) 295–305.Google Scholar
  25. [25]
    J. Kyparisis, Sensitivity analysis for nonlinear programs and variational inequalities with nonunique multipliers, Math. Oper. Res. 15 (1990) 286–298.Google Scholar
  26. [26]
    J. Kyparisis, Solution differentiability for variational inequalities, Math. Programming, Series B 48 (1990) forthcoming.Google Scholar
  27. [27]
    L. McLinden, The complementarity problem for maximal monotone multifunctions, in:Variational Inequalities and Complementarity Problems: Theory and Applications, eds. R.W. Cottle, F. Giannessi and J.L. Lions (Wiley, New York, 1980) pp. 251–270.Google Scholar
  28. [28]
    L. McLinden, An analogue of Moreau's proximation theorem, with application to the nonlinear complementarity problem, Pacific J. Math. 88 (1980) 101–161.Google Scholar
  29. [29]
    L. McLinden, Stable monotone variational inequalities, Math. Programming, Series B 48 (1990) forthcoming.Google Scholar
  30. [30]
    N. Megiddo, On the parametric nonlinear complementarity problem, Math. Programming Study 7 (1978) 142–150.Google Scholar
  31. [31]
    N. Megiddo and M. Kojima, On the existence and uniqueness of solutions in nonlinear complementarity theory, Math. Programming 12 (1977) 110–130.Google Scholar
  32. [32]
    K. Murty,Linear Complementarity, Linear and Nonlinear Programming (Helderman, Berlin, 1988).Google Scholar
  33. [33]
    J.-S. Pang, Two characterization theorems in complementarity theory, Oper. Res. Lett. 7 (1988) 27–31.Google Scholar
  34. [34]
    J.-S. Pang, Solution differentiability and continuation of Newton's method for variational inequality problems over polyhedral sets, J. Optim. Theory Appl. (1990) forthcoming.Google Scholar
  35. [35]
    J.-S. Pang, Newton's method for B-differentiable equations, Math. Oper. Res. 15 (1990) 311–341.Google Scholar
  36. [36]
    A.B. Poore and C.A. Tiahrt, Bifurcation problems in nonlinear parametric programming, Math. Programming 39 (1987) 189–205.Google Scholar
  37. [37]
    Y. Qiu and T.L. Magnanti, Sensitivity analysis for variational inequalities defined on polyhedral sets, Math. Oper. Res. 14 (1989) 410–432.Google Scholar
  38. [38]
    Y. Qiu and T.L. Magnanti, Sensitivity analysis for variational inequalities, Working Paper OR 163-87, Operations Research Center, Massachusetts Institute of Technology, Cambridge, Massachusetts (1987).Google Scholar
  39. [39]
    A. Reinoza, The strong positivity conditions, Math. Oper. Res. 10 (1985) 54–62.Google Scholar
  40. [40]
    S.M. Robinson, Stability theory for systems of inequalities, part II: differentiable nonlinear systems, SIAM J. Num. Anal. 13 (1976) 497–513.Google Scholar
  41. [41]
    S.M. Robinson, Generalized equations and their solutions, part I: basic theory, Math. Programming Study 10 (1979) 128–141.Google Scholar
  42. [42]
    S.M. Robinson, Strongly regular generalized equations, Math. Oper. Res. 5 (1980) 43–62.Google Scholar
  43. [43]
    S.M. Robinson, Generalized equations, in:Mathematical Programming: The State of the Art, eds. A. Bachem, M. Grötschel and B. Korte (Springer, Berlin, 1982) pp. 346–367.Google Scholar
  44. [44]
    S.M. Robinson, Implicit B-differentiability in generalized equations, Technical Summary Report No. 2854, Mathematics Research Center, University of Wisconsin, Madison, Wisconsin (1985).Google Scholar
  45. [45]
    S.M. Robinson, Local structure of feasible sets in nonlinear programming, part III: stability and sensitivity, Math. Programming Study 30 (1987) 45–66.Google Scholar
  46. [46]
    R.T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim. 14 (1976) 877–898.Google Scholar
  47. [47]
    R.T. Rockafellar, Lagrange multipliers and variational inequalities, in:Variational Inequalities and Complementarity Problems: Theory and Applications, eds. R.W. Cottle, F. Giannessi and J.L. Lions (Wiley, New York, 1980) pp. 303–322.Google Scholar
  48. [48]
    A. Shapiro, On concepts of directional differentiability, J. Optim. Theory Appl., forthcoming.Google Scholar
  49. [49]
    R.L. Tobin, Sensitivity analysis for variational inequalities, J. Optim. Theory Appl. 48 (1986) 191–204.Google Scholar
  50. [50]
    R.L. Tobin, Sensitivity analysis for general spatial price equilibria, J. Regional Sci. 27 (1987) 77–101.Google Scholar
  51. [51]
    R.L. Tobin and T.L. Friesz, Sensitivity analysis for equilibrium network flow, Transportation Sci. 22 (1988) 242–250.Google Scholar

Copyright information

© J.C. Baltzer A.G. Scientific Publishing Company 1990

Authors and Affiliations

  • Jerzy Kyparisis
    • 1
  1. 1.Department of Decision Sciences and Information Systems, College of Business AdministrationFlorida International UniversityMiamiUSA

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