×

zbMATH — the first resource for mathematics

Implicit functions and sensitivity of stationary points. (English) Zbl 0715.65034
The space L(D) consists of Lipschitz continuous mappings from D to the Euclidean n-space \(R^ n\), D being an open bounded subset of \(R^ n\). Let F belong to L(D) and suppose that \(\bar x\) solves the equation \(F(x)=0\). If the generalized Jacobian (in some particular sense) of F at \(\bar x\) is nonsingular then it is shown that for G near F (with respect to a natural norm) the system \(G(x)=0\) has a unique solution x(G) in neighbourhood of \(\bar x\) and the mapping x to x(G) is Lipschitz continuous.
Reviewer: V.Subba Rao

MSC:
65H10 Numerical computation of solutions to systems of equations
55M25 Degree, winding number
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] F.M. Clarke,Optimization and Nonsmooth Analysis (Wiley, New York, 1983). · Zbl 0582.49001
[2] K. Deimling,Nichtlineare Gleichungen und Abbildungsgrade (Springer, Berlin-Heidelberg-New York, 1974).
[3] H. Federer,Geometric Measure Theory (Springer, Berlin-Heidelberg-New York, 1969). · Zbl 0176.00801
[4] J. Guddat, Hj. Wacker and W. Zulehner, ”On imbedding and parametric optimization”,Mathematical Programming Study 21 (1984) 79–96. · Zbl 0547.90092
[5] H.Th. Jongen, T. Moebert and K. Tammer, ”On iterated minimization in nonconvex optimization,”Mathematics of Operations Research 11 (1986) 679–691. · Zbl 0626.90080 · doi:10.1287/moor.11.4.679
[6] H.Th. Jongen, T. Moebert, J. Rueckmann and K. Tammer, ”On intertia and Schur complement in optimization,”Linear Algebra and its Applications 95 (1987) 97–109. · Zbl 0642.90091 · doi:10.1016/0024-3795(87)90028-0
[7] 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.
[8] B. Kummer, ”Linearly and nonlinearly perturbed optimization problems,” in: J. Guddat, H.Th. Jongen, B. Kummer and F. Nožička, eds.,Parametric Optimization and Related Topics (Akademie Verlag, Berlin, 1987) pp. 249–267. · Zbl 0657.90092
[9] R. Lehmann, ”On the numerical feasibility of continuation methods for nonlinear programming problems,”Mathematische Operationsforschung und Statistik, Series Optimization 15 (1984) 517–520. · Zbl 0553.90083
[10] J.M. Ortega and W. Rheinboldt,Iterative Solutions of Nonlinear Equations in Several Variables (Academic Press, New York, 1970). · Zbl 0241.65046
[11] D.V. Ouelette, ”Schur complements and statistics,”Linear Algebra and its Applications 36 (1981) 187–295. · Zbl 0455.15012 · doi:10.1016/0024-3795(81)90232-9
[12] C. Richter, ”Ein implementierbares Einbettungsverfahren der nichtlinearen Optimierung,”Mathematische Operationsforschung und Statistik, Series Optimization 15 (1984) 545–553. · Zbl 0555.90091
[13] 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
[14] S.M. Robinson, ”Generalized equations and their solutions, Part II: Applications to nonlinear programming,”Mathematical Programming Study 19 (1982) 200–221. · Zbl 0495.90077
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.