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A stochastic approach to stability in stochastic programming. (English) Zbl 0824.90107
Summary: A random optimization problem \[ \min_{x\in \Gamma_ 0(\omega)} f_ 0(x, \omega),\qquad \omega\in \Omega,\tag{P\(_ 0\)} \] is approximated by a sequence of random surrogate problems \((P_ n)_{n\in \mathbb{N}}\) with \[ \min_{x\in \Gamma_ n(\omega)} f_ n(x, \omega),\qquad \omega\in\Omega\tag{P\(_ n\)} \] (\([\Omega, \Sigma, P]\) a given probability space). We investigate the convergence almost surely and in probability of the optimal values and the solution sets. The results can be regarded as random versions of well-known stability statements of parametric programming. Semicontinuous convergence (almost surely, in probability) of sequences of random functions is a crucial assumption in this framework and will be investigated in more detail.

MSC:
90C15 Stochastic programming
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