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Date:
01 Sep 2003
Coderivatives in parametric optimization
 Adam B. Levy,
 Boris S. Mordukhovich
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We consider parametric families of constrained problems in mathematical programming and conduct a local sensitivity analysis for multivalued solution maps. Coderivatives of setvalued mappings are our basic tool to analyze the parametric sensitivity of either stationary points or stationary pointmultiplier pairs associated with parameterized optimization problems. An implicit mapping theorem for coderivatives is one key to this analysis for either of these objects, and in addition, a partial coderivative rule is essential for the analysis of stationary points. We develop general results along both of these lines and apply them to study the parametric sensitivity of stationary points alone, as well as stationary pointmultiplier pairs. Estimates are computed for the coderivative of the stationary point multifunction associated with a general parametric optimization model, and these estimates are refined and augmented by estimates for the coderivative of the stationary pointmultiplier multifunction in the case when the constraints are representable in a special composite form. When combined with existing coderivative formulas, our estimates are entirely computable in terms of the original data of the problem.
This research was partly supported by the National Science Foundation under grant DMS0072179.
References
1.
Aubin, J.P.: Lipschitz behavior of solutions to convex minimization problems. Math. Oper. Res. 9, 87–111 (1984)MathSciNetMATH
2.
Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. SpringerVerlag, New York, 2000
3.
Burke, J.V., Overton, M.L.: Variational analysis of nonLipschitz spectral functions. Math. Program. 90, 317–351 (2001)MathSciNet
4.
Dontchev, A.L., Rockafellar, R.T.: Characterizations of strong regularity for variational inequalities over polyhedral convex sets. SIAM J. Optim. 7, 1087–1105 (1996)MathSciNet
5.
Henrion, R., Römisch, W.: Metric regularity and quantitative stability in stochastic programming with probabilistic constraints. Math. Prog. 84, 55–88 (1999)MathSciNet
6.
Klatte, D., Kummer, B.: Nonsmooth Equations in Optimization. Kluwer Academic Publishers, Boston, 2002
7.
Lewis, A.S.: Nonsmooth analysis of eigenvalues. Math. Prog. 84, 1–24 (1999)MathSciNetMATH
8.
Lucet, Y., Ye, J.J.: Sensitivity analysis of the value function for optimization problems with variational inequality constraints. SIAM J. Control Optim. 40, 699–723 (2001)CrossRefMathSciNetMATH
9.
Luo, Z.Q., Pang, J.S., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge, U.K., 1996
10.
Levy, A.B., Poliquin, R., Rockafellar, R.T.: Stability of locally optimal solutions. SIAM J. Optim. 10, 580–604 (2000)CrossRefMathSciNetMATH
11.
Levy, A.B., Rockafellar, R.T.: Variational conditions and the protodifferentiation of partial subgradient mappings. Nonlinear Analysis, Theory, Methods and Applications 26, 1951–1964 (1996)
12.
Mordukhovich, B.S.: Complete characterization of openness, metric regularity, and Lipschitzian properties of multifunctions. Trans. Am. Math. Soc. 340, 1–35 (1993)MathSciNetMATH
13.
Mordukhovich, B.S.: Generalized differential calculus for nonsmooth and setvalued mappings. J. Math. Anal. Appl. 183, 250–288 (1994)CrossRefMathSciNetMATH
14.
Mordukhovich, B.S.: Stability theory for parametric generalized equations and variational inequalities via nonsmooth analysis. Trans. Am. Math. Soc. 343, 609–658 (1994)MathSciNetMATH
15.
Mordukhovich, B.S., Outrata, J.V.: On secondorder subdifferentials and their applications. SIAM J. Optim. 12, 139–169 (2001)CrossRefMathSciNetMATH
16.
Outrata, J.V.: A generalized mathematical program with equilibrium constraints. SIAM J. Control Optim. 38, 1623–1638 (2000)CrossRefMathSciNetMATH
17.
Poliquin, R., Rockafellar, R.T.: Tilt stability of a local minimum. SIAM J. Optim. 8, 287–299 (1998)CrossRefMathSciNetMATH
18.
Robinson, S.M.: Generalized equations and their solutions, part I: basic theory. Math. Prog. Study 10, 128–141 (1979)MathSciNetMATH
19.
Rockafellar, R.T., Wets, R.J.B.: Variational Analysis. SpringerVerlag, Berlin, 1998
 Title
 Coderivatives in parametric optimization
 Journal

Mathematical Programming
Volume 99, Issue 2 , pp 311327
 Cover Date
 20040301
 DOI
 10.1007/s1010700304520
 Print ISSN
 00255610
 Online ISSN
 14364646
 Publisher
 SpringerVerlag
 Additional Links
 Keywords

 parametric optimization
 variational analysis
 sensitivity
 Lipschitzian stability
 generalized differentiation
 coderivatives
 Industry Sectors
 Authors

 Adam B. Levy ^{(1)}
 Boris S. Mordukhovich ^{(2)}
 Author Affiliations

 1. Department of Mathematics, Bowdoin College, Brunswick, ME, 04011
 2. Department of Mathematics, Wayne State University, Detroit, MI, 48202