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An index condition in input optimization. (English) Zbl 0655.90075
The author studies the following parametric optimization problem: $(P,\theta)_{x} f^ 0(x,\theta)\quad s.t.\quad f^ i(x,\theta)\leq 0,\quad i=1,...,m,\quad \theta \in I.$ Here for $$j=0,1,..,m$$ the functions $$f^ j:$$ $$R^ n\times R^ p\to R$$ satisfy continuity and differentiability assumptions, $$f^ j(.,\theta)$$ and $$f^ j(x,\cdot)$$ are convex, and I is a convex subset of $$R^ p$$. For a given “input” $$\theta \in R^ p$$ let $$\tilde f(\theta)$$ denote the optimal value of $$(P,\theta)$$ (with $$\theta\in I$$ cancelled).
The author establishes a necessary condition for local minima of $$\tilde f(\theta)$$ as well as a marginal formula, both with respect to certain subsets of I (regions of stability). The results are derived under a constraint qualification called index condition which replaces a hypothesis (“Property U”) used by the author in a former paper [Math. Program. 31, 245-268 (1985; Zbl 0589.90068)].
Reviewer: W.Schirotzek

##### MSC:
 90C31 Sensitivity, stability, parametric optimization 90C30 Nonlinear programming