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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

90C31 Sensitivity, stability, parametric optimization
90C30 Nonlinear programming