Abstract
Our paper deals with the interrelation of optimization methods and Lipschitz stability of multifunctions in arbitrary Banach spaces. Roughly speaking, we show that linear convergence of several first order methods and Lipschitz stability mean the same. Particularly, we characterize calmness and the Aubin property by uniformly (with respect to certain starting points) linear convergence of descent methods and approximate projection methods. So we obtain, e.g., solution methods (for solving equations or variational problems) which require calmness only. The relations of these methods to several known basic algorithms are discussed, and errors in the subroutines as well as deformations of the given mappings are permitted. We also recall how such deformations are related to standard algorithms like barrier, penalty or regularization methods in optimization.
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References
Aubin J.-P. and Ekeland I. (1984). Applied Nonlinear Analysis. Wiley, New York
Bank B., Guddat J., Klatte D., Kummer B. and Tammer K. (1982). Non-Linear Parametric Optimization. Akademie-Verlag, Berlin
Bonnans J.F. and Shapiro A. (2000). Perturbation Analysis of Optimization Problems. Springer, New York
Burke J.V. (1991). Calmness and exact penalization. SIAM J. Control Optim. 29: 493–497
Burke, J.V., Deng, S.: Weak sharp minima revisited, Part III: Error bounds for differentiable convex inclusions. Math. Programm. Published online (2007)
Clarke F.H. (1983). Optimization and Nonsmooth Analysis. Wiley, New York
Cominetti R. (1990). Metric regularity, tangent sets and second-order optimality conditions. Appl. Math. Optimiz. 21: 265–287
Dempe S. (2002). Foundations of Bilevel Programming. Kluwer, Dordrecht
Dolecki S. and Rolewicz S. (1979). Exact penalties for local minima. SIAM J. Control Optim. 17: 596–606
Dontchev A. (1996). Local convergence of the Newton method for generalized equations. Comptes Rendus de l’Acédemie des Sciences de Paris 332: 327–331
Dontchev A. and Rockafellar R.T. (2004). Regularity and conditioning of solution mappings in variational analysis. Set-Valued Anal. 12: 79–109
Ekeland I. (1974). On the variational principle. J. Math. Anal. Appl. 47: 324–353
Facchinei F. and Pang J.-S. (2003). Finite-Dimensional Variational Inequalities and Complementary Problems, Vol I and Vol II. Springer, New York
Fusek P. (2001). Isolated zeros of Lipschitzian metrically regular Rn functions. Optimization 49: 425–446
Graves L.M. (1950). Some mapping theorems. Duke Math. J. 17: 11–114
Grossmann C., Klatte D. and Kummer B. (2004). Convergence of primal-dual solutions for the nonconvex log-barrier method without LICQ. Kybernetika 40: 571–584
Henrion R. and Outrata J. (2001). A subdifferential condition for calmness of multifunctions. J. Math. Anal. Appl. 258: 110–130
Henrion R. and Outrata R. (2005). Calmness of constraint systems with applications. Math. Program. Ser. B 104: 437–464
Ioffe A.D. (2000). Metric regularity and subdifferential calculus. Russ. Math. Surv. 55: 501–558
Klatte D. and Kummer B. (2002). Constrained minima and Lipschitzian penalties in metric spaces. SIAM J. Optim. 13(2): 619–633
Klatte D. and Kummer B. (2002). Nonsmooth Equations in Optimization—Regularity, Calculus, Methods and Applications. Kluwer, Dordrecht
Klatte D. and Kummer B. (2005). Strong Lipschitz stability of stationary solutions for nonlinear programs and variational inequalities. SIAM J. Optim. 16: 96–119
Klatte D. and Kummer B. (2006). Stability of inclusions: Characterizations via suitable Lipschitz functions and algorithms. Optimization 55: 627–660
Kummer B. (1998). Lipschitzian and pseudo-Lipschitzian inverse functions and applications to nonlinear programming. In: Fiacco, A.V. (eds) Mathematical Programming with Data Perturbations, pp 201–222. Marcel Dekker, New York
Kummer B. (1999). Metric regularity: characterizations, nonsmooth variations and successive approximation. Optimization 46: 247–281
Kummer B. (2000). Inverse functions of pseudo regular mappings and regularity conditions. Math. Program. Ser. B 88: 313–339
Li W. (1997). Abadie’s constraint qualification, metric regularity and error bounds for differentiable convex inequalities. SIAM J. Optim. 7: 966–978
Lyusternik L. (1934). Conditional extrema of functions. Math. Sbornik 41: 390–401
Mordukhovich B.S. (1988). Approximation Methods in Problems of Optimization and Control (in Russian). Nauka, Moscow
Outrata J., Kočvara M. and Zowe J. (1998). Nonsmooth Approach to Optimization Problems with Equilibrium Constraints. Kluwer, Dordrecht
Robinson S.M. (1976). Stability theorems for systems of inequalities. Part II: Differentiable nonlinear systems. SIAM J. Numer. Anal. 13: 497–513
Robinson S.M. (1979). Generalized equations and their solutions, Part I: Basic theory. Math. Program. Study 10: 128–141
Robinson S.M. (1980). Strongly regular generalized equations. Math. Oper. Res. 5: 43–62
Robinson S.M. (1982). Generalized equations and their solutions Part II: Applications to nonlinear programming. Math. Program. Study 19: 200–221
Robinson S.M. (2003). Variational conditions with smooth constraints: structure and analysis. Math. Program. 97: 245–265
Rockafellar R.T. and Wets R.J.-B. (1998). Variational Analysis. Springer, Berlin
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This paper is dedicated to Professor Stephen M. Robinson on the occasion of his 65th birthday.
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Klatte, D., Kummer, B. Optimization methods and stability of inclusions in Banach spaces. Math. Program. 117, 305–330 (2009). https://doi.org/10.1007/s10107-007-0174-9
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DOI: https://doi.org/10.1007/s10107-007-0174-9
Keywords
- Generalized equation
- Variational inequality
- Perturbation
- Regularization
- Stability criteria
- Metric regularity
- Calmness
- Approximate projections
- Penalization
- Successive approximation
- Newton’s method