Implicit functions, Lipschitz maps, and stability in optimization. (English) Zbl 0835.49019

Dontchev, Asen L.; Hager, William W.

Math. Oper. Res. 19, No. 3, 753-768 (1994).

In the spirit of Robinson’s approach to stability in optimization problems, the authors study the set-valued map \(\Sigma(y)= \{x\in X: 0\in f(x, y)+ F(x)\}\), where \(f: X\times Y\to Z\) is a function and \(F: X\to Z\) is set-valued and \(X\), \(Y\), \(Z\) are normed spaces with \(X\) complete. They prove that \(\Sigma\) is pseudo-Lipschitz, resp. Lipschitz, if \(f\), \(F\) and a certain strong approximation \(g\) to \(f\), assumed to exist, has suitable properties. They show that known theorems of Robinson are a consequence of this result. They apply their theorem to prove that, for a general nonlinear optimization problem in Banach spaces, suitable second-order conditions imply the Lipschitz stability of the optimal solution as well as of the associated Lagrange multipliers.

Reviewer: M.Brokate (Kaiserslautern)

MSC:

49K40	Sensitivity, stability, well-posedness
26B10	Implicit function theorems, Jacobians, transformations with several variables
49J52	Nonsmooth analysis
90C31	Sensitivity, stability, parametric optimization
47H04	Set-valued operators

Keywords:

implicit functions; perturbation; second-order conditions; sufficiency; set-valued map; nonlinear optimization problem; Lipschitz stability; Lagrange multipliers