# zbMATH — the first resource for mathematics

On solvability and regularity of a parametrized version of optimality conditions. (English) Zbl 0834.90120
Summary: We investigate a linear homotopy $$F(\cdot, t)$$ connecting an appropriate smooth equation $$G= 0$$ with Kojima’s (nonsmooth) system $$K= 0$$ describing critical points (primal-dual) of a nonlinear optimization problem (NLP) in finite dimension.
For $$t= 0$$, our system may be seen e.g. as a starting system for an embedding procedure to determine a critical point to NLP. For $$t\approx 1$$, it may be regarded as a regularization of $$K$$. Conditions for regularity (necessary and sufficient) and solvability (sufficient) are studied. Though, formally, they can be given in a unified way, we show that their meaning differs for $$t< 1$$ and $$t= 1$$. Particularly, no MFCQ- like condition must be imposed in order to ensure regularity for $$t< 1$$.

##### MSC:
 90C30 Nonlinear programming 90C31 Sensitivity, stability, parametric optimization
Full Text:
##### References:
 [1] Clarke FH (1976) On the inverse function theorem. Pacific Journ Math 64/1:97-102 · Zbl 0331.26013 [2] Clarke FH (1983) Optimization and nonsmooth analysis. Wiley New York · Zbl 0582.49001 [3] Harker PT, Pang J-S (1990) Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications. Mathematical Programming 48:161-220 · Zbl 0734.90098 · doi:10.1007/BF01582255 [4] Jongen HTh, Klatte D, Tammer K (1988) Implicit functions and sensitivity of stationary points. Preprint 1. Lehrstuhl C für Mathematik RWTH Aachen D-5100 Aachen; Mathematical Programming (1990) 49:123-138 · Zbl 0715.65034 [5] Kojima M (1980) Strongly stable stationary solutions in nonlinear programs. In: Robinson SM (ed) Analysis and Computation of Fixed Points Academic Press New York 93-138 [6] Kummer B (1991) Lipschitzian inverse functions, directional derivatives and application inC 1,1-optimization. Journal of Optimization Theory & Appl 70/3:559-580 [7] Kummer B (1991) An implicit function theorem forC 0.1-equations and parametricC 1,1-optimization. Journal of Mathematical Analysis & Appl 158/1:35-46 · Zbl 0742.49006 · doi:10.1016/0022-247X(91)90264-Z [8] Kummer B, (1988) Newton’s method for nondifferentiable functions. In: Guddat J et al (ed) Advances in math. optimization Akademie Verlag Berlin Ser Mathem Res 45:114-125 [9] Kummer B (1992) On stability and newton-type methods for lipschitzian equations with applications to optimization problems. In: Kall P (Ed) Lecture Notes in Control and Information Science 180; System Modelling and Optimization, Proceedings of the 15th IPIF Conference, Zürich 1991. Springer-Verlag [10] Kummer B (1992) Newton’s method based on generalized derivatives for nonsmooth functions: Convergence analysis. In: Oettli W, Pallaschke D (Eds) Lecture Notes in Economics and Mathematical Systems 382; Advances in Optimization, Proceedings Lambrecht FRG 1991. Springer-Verlag · Zbl 0768.49012 [11] Thibault L (1980) Subdifferentials of compactly Lipschitzian vector-valued functions. Ann Mat Pura Appl 4/125:157-192 · Zbl 0486.46037 · doi:10.1007/BF01789411 [12] Thibault L (1982) On generalized differentials and subdifferentials of Lipschitz vector-valued functions. Nonlinear Analysis Theory Methods Appl 6/10:1037-1053 · Zbl 0492.46036 · doi:10.1016/0362-546X(82)90074-8
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.