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Solution differentiability for variational inequalities. (English) Zbl 0727.90082

The main result of the paper is a characterization of Fréchet- differentiability of the sensitivity function of a perturbed variational inequality problem on a polyhedral set in \(R^ n\). The assumption of strong regularity gives more generality to an earlier result of J. S. Pang [see J. Optimization Theory Appl. 66, No.1, 121-135 (1990; Zbl 0681.49011)]. The theorem is specialized for the nonlinear complementarity problem. Most attention is given to variational inequalities on perturbed sets and general parametric nonlinear programming problems. The author shows how the results relate to more familiar assumptions, used in several other papers.
Reviewer: E.Iwanow (Wien)

MSC:

90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
90C31 Sensitivity, stability, parametric optimization
49J40 Variational inequalities
90C30 Nonlinear programming
49J50 Fréchet and Gateaux differentiability in optimization

Citations:

Zbl 0681.49011
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References:

[1] S. Dafermos, ”Sensitivity analysis in variational inequalities,”Mathematics of Operations Research 13 (1988) 421–434. · Zbl 0674.49007 · doi:10.1287/moor.13.3.421
[2] A.L. Dontchev and H.Th. Jongen, ”On the regularity of the Kuhn-Tucker curve,”SIAM Journal of Control and Optimization 24 (1986) 169–176. · Zbl 0598.90086 · doi:10.1137/0324009
[3] A.V. Fiacco, ”Sensitivity analysis for nonlinear programming using penalty methods,”Mathematical Programming 10 (1976) 287–311. · Zbl 0357.90064 · doi:10.1007/BF01580677
[4] A.V. Fiacco,Introduction to Sensitivity and Stability Analysis in Nonlinear Programming (Academic Press, New York, 1983). · Zbl 0543.90075
[5] A.V. Fiacco and G.P. McCormick,Nonlinear Programming: Sequential Unconstrained Minimization Techniques (Wiley, New York, 1968). · Zbl 0193.18805
[6] J. Gauvin and R. Janin, ”Directional behaviour of optimal solutions in nonlinear mathematical programming problems,”Mathematics of Operations Research 13 (1988) 629–649. · Zbl 0665.90086 · doi:10.1287/moor.13.4.629
[7] P.T. Harker and J.-S. Pang, ”Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications,”Mathematical Programming (Series B) 48 (1990) 161–220, this issue. · Zbl 0734.90098 · doi:10.1007/BF01582255
[8] K. Jittorntrum, ”Solution point differentiability without strict complementarity in nonlinear programming,”Mathematical Programming Study 21 (1984) 127–138. · Zbl 0571.90080
[9] M. Kojima, ”Strongly stable stationary solutions in nonlinear programs,” in: S.M. Robinson, ed.,Analysis and Computation of Fixed Points (Academic Press, New York, 1980) pp. 93–138.
[10] J. Kyparisis, ”On uniqueness of Kuhn-Tucker multipliers in nonlinear programming,”Mathematical Programming 32 (1985) 242–246. · Zbl 0566.90085 · doi:10.1007/BF01586095
[11] J. Kyparisis, ”Uniqueness and differentiability of solutions of parametric nonlinear complementarity prolbems,”Mathematical Programming 36 (1986) 105–113. · Zbl 0613.90096 · doi:10.1007/BF02591993
[12] J. Kyparisis, ”Sensitivity analysis framework for variational inequalities,”Mathematical Programming 38 (1987) 203–213. · doi:10.1007/BF02604641
[13] J. Kyparisis, ”Perturbed solutions of variational inequality problems over polyhedral sets,”Journal of Optimization Theory and Applications 57 (1988) 295–305. · Zbl 0621.49004 · doi:10.1007/BF00938541
[14] J. Kyparisis, ”Sensitivity analysis for nonlinear programs and variational inequalities with nonunique multipliers,”Mathematics of Operations Research 15 (1990) 286–298. · Zbl 0708.90086 · doi:10.1287/moor.15.2.286
[15] J.-S. Pang, ”Two characterization theorems in complementarity theory,”Operations Reserach Letters 7 (1988) 27–31. · Zbl 0643.90087 · doi:10.1016/0167-6377(88)90048-X
[16] J.-S. Pang, ”Solution differentiability and continuation of Newton’s method for variational inequality problems over polyhedral sets,”Journal of Optimization Theory and Applications, forthcoming. · Zbl 0681.49011
[17] A.B. Poore and C.A. Tiahrt, ”Bifurcation problems in nonlinear parametric programming,”Mathematical Programming 39 (1987) 189–205. · Zbl 0639.90084 · doi:10.1007/BF02592952
[18] Y. Qiu and T.L. Magnanti, ”Sensitivity analysis for variational inequalities defined on polyhedral sets,”Mathematics of Operations Research 14 (1989) 410–432. · Zbl 0698.90069 · doi:10.1287/moor.14.3.410
[19] Y. Qiu and T.L. Magnanti, ”Sensitivity analysis for variational inequalities,” Working Paper OR 163-87, Operations Research Center, Massachusetts Institute of Technology (Cambridge, MA, 1987).
[20] A. Reinoza, ”The strong positivity conditions,”Mathematics of Operations Research 10 (1985) 54–62. · Zbl 0568.90080 · doi:10.1287/moor.10.1.54
[21] S.M. Robinson, ”Generalized equations and their solutions, Part I: Basic theory,”Mathemiatical Programming Study 10 (1979) 128–141. · Zbl 0404.90093
[22] S.M. Robinson, ”Strongly regular generalized equations,”Mathematics of Operations Research 5 (1980) 43–62. · Zbl 0437.90094 · doi:10.1287/moor.5.1.43
[23] S.M. Robinson, ”Implicit B-differentiability in generalized equations,” Technical Summary Report No. 2854, Mathematics Research Center, University of Wisconsin (Madison, WI, 1985).
[24] S.M. Robinson, ”Local strucutre of feasible sets in nonlinear programming, Part III: Stability and Sensitivity,”Mathematical Programming Study 30 (1987) 45–66. · Zbl 0629.90079
[25] A. Shapiro, ”Second order sensitivity analysis and asymptotic theory of parametrized nonlinear programs,”Mathematical Programming 33 (1985) 280–299. · Zbl 0579.90088 · doi:10.1007/BF01584378
[26] A. Shapiro, ”Sensitivity analysis of nonlinear programs and differentiability properties of metric projections,”SIAM Journal of Control and Optimization 26 (1988) 628–645. · Zbl 0647.90089 · doi:10.1137/0326037
[27] R.L. Tobin, ”Sensitivity analysis for variational inequalities,”Journal of Optimization Theory and Applications 48 (1986) 191–204. · Zbl 0557.49004
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