Strong metric subregularity of mappings in variational analysis and optimization.

*(English)*Zbl 1376.49048Summary: Although the property of strong metric subregularity of set-valued mappings has been present in the literature under various names and with various (equivalent) definitions for more than two decades, it has attracted much less attention than its older “siblings”, the metric regularity and the strong (metric) regularity. The purpose of this paper is to show that the strong metric subregularity shares the main features of these two most popular regularity properties and is not less instrumental in applications. We show that the strong metric subregularity of a mapping \(F\) acting between metric spaces is stable under perturbations of the form \(f + F\), where \(f\) is a function with a small calmness constant. This result is parallel to the Lyusternik-Graves theorem for metric regularity and to the Robinson theorem for strong regularity, where the perturbations are represented by a function \(f\) with a small Lipschitz constant. Then, we study perturbation stability of the same kind for mappings acting between Banach spaces, where \(f\) is not necessarily differentiable but admits a set-valued derivative-like approximation. Strong metric \(q\)-subregularity is also considered, where \(q\) is a positive real constant appearing as exponent in the definition. Rockafellar’s criterion for strong metric subregularity involving injectivity of the graphical derivative is extended to mappings acting in infinite-dimensional spaces. A sufficient condition for strong metric subregularity is established in terms of surjectivity of the Fréchet coderivative, and it is shown by a counterexample that surjectivity of the limiting coderivative is not a sufficient condition for this property, in general. Then various versions of Newton’s method for solving generalized equations are considered including inexact and semismooth methods, for which superlinear convergence is shown under strong metric subregularity. As applications to optimization, a characterization of the strong metric subregularity of the KKT mapping is obtained, as well as a radius theorem for the optimality mapping of a nonlinear programming problem. Finally, an error estimate is derived for a discrete approximation in optimal control under strong metric subregularity of the mapping involved in the Pontryagin principle.

##### MSC:

49N60 | Regularity of solutions in optimal control |

49J53 | Set-valued and variational analysis |

49M15 | Newton-type methods |

49M37 | Numerical methods based on nonlinear programming |

90C30 | Nonlinear programming |

##### Keywords:

strong metric subregularity; perturbations and approximations; generalized derivatives; Newton’s method; nonlinear programming; optimal control
PDF
BibTeX
Cite

\textit{R. Cibulka} et al., J. Math. Anal. Appl. 457, No. 2, 1247--1282 (2018; Zbl 1376.49048)

Full Text:
DOI

##### References:

[1] | Akhmerov, R. R.; Kamenskii, M. I.; Potapova, A. S.; Rodkina, A. E.; Sadovskii, B. N., Measure of noncompactness and condensing operators, (1992), Birkhäuser Basel · Zbl 0748.47045 |

[2] | Aragón Artacho, F. J.; Geoffroy, M. H., Metric subregularity of the convex subdifferential in Banach spaces, J. Nonlinear Convex Anal., 15, 35-47, (2014) · Zbl 1290.49028 |

[3] | Borwein, J. M., Stability and regular points of inequality systems, J. Optim. Theory Appl., 48, 9-52, (1986) · Zbl 0557.49020 |

[4] | Borwein, J. M.; Lewis, A. S., Convex analysis and nonlinear optimization: theory and examples, (2010), Springer Science & Business Media |

[5] | Cibulka, R.; Dontchev, A. L., A nonsmooth Robinson’s inverse function theorem in Banach spaces, Math. Program., 156, 257-270, (2016) · Zbl 1337.49022 |

[6] | Cibulka, R.; Dontchev, A. L.; Geoffroy, M. H., Inexact Newton methods and dennis-moré theorems for nonsmooth generalized equations, SIAM J. Control Optim., 53, 1003-1019, (2015) · Zbl 1321.49044 |

[7] | Cibulka, R.; Dontchev, A. L.; Veliov, V. M., Lyusternik-Graves theorems for the sum of a Lipschitz function and a set-valued mapping, SIAM J. Control Optim., (2016), in press · Zbl 1372.47070 |

[8] | De Giorgi, E.; Marino, A.; Tosques, M., Problems of evolution in metric spaces and maximal decreasing curve, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 68, 527-533, (2013), Springer Heidelberg, (in Italian). English translation in: E. De Giorgi, Selected Papers |

[9] | Dontchev, A. L., Characterizations of Lipschitz stability in optimization, (Recent Developments in Well-Posed Variational Problems, (1995), Kluwer), 95-115 · Zbl 0856.49019 |

[10] | Dontchev, A. L., Generalizations of the dennis-more theorem, SIAM J. Optim., 22, 821-830, (2012) · Zbl 1255.49028 |

[11] | Dontchev, A. L.; Hager, W. W., Lipschitzian stability in nonlinear control and optimization, SIAM J. Control Optim., 31, 569-603, (1993) · Zbl 0779.49032 |

[12] | Dontchev, A. L.; Lewis, A. S.; Rockafellar, R. T., The radius of metric regularity, Trans. Amer. Math. Soc., 355, 493-517, (2003) · Zbl 1042.49026 |

[13] | Dontchev, A. L.; Malanowski, K., A characterization of Lipschitzian stability in optimal control, (Calculus of Variations and Optimal Control, Haifa, 1998, Chapman & Hall/CRC Res. Notes Math., vol. 411, (2000), Chapman & Hall/CRC Boca Raton, FL), 62-76 · Zbl 0970.49021 |

[14] | Dontchev, A. L.; Rockafellar, R. T., Characterizations of Lipschitz stability in nonlinear programming, (Mathematical Programming with Data Perturbations, Lect. Notes Pure Appl. Math., vol. 195, (1998), Dekker New York), 65-82 · Zbl 0891.90146 |

[15] | Dontchev, A. L.; Rockafellar, R. T., Regularity and conditioning of solution mappings in variational analysis, Set-Valued Anal., 12, 79-109, (2004) · Zbl 1046.49021 |

[16] | Dontchev, A. L.; Rockafellar, R. T., Implicit functions and solution mappings. A view from variational analysis, (2014), Springer · Zbl 1046.49021 |

[17] | Drusvyatskiy, D.; Ioffe, A. D., Quadratic growth and critical point stability of semi-algebraic functions, Math. Program. Ser. A, 153, 635-653, (2015) · Zbl 1325.49021 |

[18] | Drusvyatskiy, D.; Mordukhovich, B. S.; Nghia, T. T., Second-order growth, tilt stability, and metric regularity of the subdifferential, J. Convex Anal., 21, 1165-1192, (2014) · Zbl 1311.49035 |

[19] | Fabian, M.; Preiss, D., A generalization of the interior mapping theorem of Clarke and pourciau, Comment. Math. Univ. Carolin., 28, 311-324, (1987) · Zbl 0625.46052 |

[20] | Facchinei, F.; Pang, J.-S., Finite-dimensional variational inequalities and complementarity problems, vol. II, (2003), Springer-Verlag New York · Zbl 1062.90002 |

[21] | Gfrerer, H., First order and second order characterizations of metric subregularity and calmness of constraint set mappings, SIAM J. Optim., 21, 1439-1474, (2011) · Zbl 1254.90246 |

[22] | Gowda, M. S., Inverse and implicit function theorems for H-differentiable and semismooth functions, Optim. Methods Softw., 19, 443-461, (2004) · Zbl 1099.49018 |

[23] | Ioffe, A. D., Nonsmooth analysis: differential calculus of nondifferentiable mappings, Trans. Amer. Math. Soc., 266, 1-56, (1981) · Zbl 0651.58007 |

[24] | Ioffe, A. D., Metric regularity and subdifferential calculus, Russian Math. Surveys, 55, 501-558, (2000) · Zbl 0979.49017 |

[25] | Izmailov, A. F., Strongly regular nonsmooth generalized equations, Math. Program., 147, 581-590, (2014) · Zbl 1301.49043 |

[26] | Klatte, D.; Kummer, B., Generalized kojima-functions and Lipschitz stability of critical points, Comput. Optim. Appl., 13, 61-85, (1999) · Zbl 1017.90104 |

[27] | Klatte, D.; Kummer, B., Strong Lipschitz stability of stationary solutions for nonlinear programs and variational inequalities, SIAM J. Optim., 16, 96-119, (2005) · Zbl 1097.90058 |

[28] | Klatte, D.; Kummer, B., Nonsmooth equations in optimization. regularity, calculus, methods and applications, Nonconvex Optim. Appl., vol. 60, (2002), Kluwer Dordrecht · Zbl 1173.49300 |

[29] | Kruger, A. Y., Error bounds and metric subregularity, Optimization, 64, 49-79, (2015) · Zbl 1311.49043 |

[30] | Kyparisis, J., On uniqueness of Kuhn-Tucker multipliers in nonlinear programming, Math. Program., 32, 242-246, (1985) · Zbl 0566.90085 |

[31] | Mordukhovich, B. S.; Ouyang, W., Higher-order metric subregularity and its applications, J. Global Optim., 63, 777-795, (2015) · Zbl 1329.49029 |

[32] | Penot, J.-P., Calculus without derivatives, (2013), Springer New York · Zbl 1264.49014 |

[33] | Quincampoix, M.; Veliov, V. M., Metric regularity and stability of optimal control problems for linear systems, SIAM J. Control Optim., 51, 4118-4137, (2013) · Zbl 1279.49014 |

[34] | Robinson, S. M., Strongly regular generalized equations, Math. Oper. Res., 5, 43-62, (1980) · Zbl 0437.90094 |

[35] | Robinson, S. M., Some continuity properties of polyhedral multifunctions, Math. Program. Stud., 14, 206-214, (1981) · Zbl 0449.90090 |

[36] | Rockafellar, R. T., Proto-differentiability of set-valued mappings and its applications in optimization, Analyse non linéaire, Perpignan, 1987, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6, suppl., 448-482, (1989) · Zbl 0674.90082 |

[37] | Uderzo, A., A strong metric subregularity analysis of nonsmooth mappings via steepest displacement rate, J. Optim. Theory Appl., 171, 573-599, (2016) · Zbl 1349.49022 |

[38] | Wang, J. J.; Song, W., Characterization of quadratic growth of extended-real-valued functions, J. Inequal. Appl., 2016, (2016) |

[39] | Zolezzi, T., On the distance theorem in quadratic optimization, J. Convex Anal., 9, 693-700, (2002) · Zbl 1034.49030 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.