We consider the solution of a system of stochastic generalized equations (SGE) where the underlying functions are mathematical expectation of random set-valued mappings. SGE has many applications such as characterizin...
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We consider the solution of a system of stochastic generalized equations (SGE) where the underlying functions are mathematical expectation of random set-valued mappings. SGE has many applications such as characterizing optimality conditions of a nonsmooth stochastic optimization problem or equilibrium conditions of a stochastic equilibrium problem. We derive quantitative continuity of expected value of the set-valued mapping with respect to the variation of the underlying probability measure in a metric space. This leads to the subsequent qualitative and quantitative stability analysis of solution set mappings of the SGE. Under some metric regularity conditions, we derive Aubin's property of the solution set mapping with respect to the change of probability measure. The established results are applied to stability analysis of stochastic variational inequality, stationary points of classical one-stage and two-stage stochastic minimization problems, two-stage stochastic mathematical programs with equilibrium constraints, and stochastic programs with second order dominance constraints.
The ultimate goal of this paper is to demonstrate that abstract convexity provides a natural language and a suitable framework for the examination of zero duality gap properties and exact multipliers of augmented Lagr...
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The ultimate goal of this paper is to demonstrate that abstract convexity provides a natural language and a suitable framework for the examination of zero duality gap properties and exact multipliers of augmented Lagrangians. We study augmented Lagrangians in a very general setting and formulate the main definitions and facts describing the augmented Lagrangian theory in terms of abstract convexity tools. We illustrate our duality scheme with an application to stochasticsemi- infinite optimization.
In this paper we present a stability analysis of a stochastic optimization problem with stochastic second order dominance constraints. We consider a perturbation of the underlying probability measure in the space of r...
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In this paper we present a stability analysis of a stochastic optimization problem with stochastic second order dominance constraints. We consider a perturbation of the underlying probability measure in the space of regular measures equipped with pseudometric discrepancy distance (Romisch in stochasticprogramming. Elsevier, Amsterdam, pp 483-554, 2003). By exploiting a result on error bounds in semi-infiniteprogramming due to Gugat (Math Program Ser B 88:255-275, 2000), we show under the Slater constraint qualification that the optimal value function is Lipschitz continuous and the optimal solution set mapping is upper semicontinuous with respect to the perturbation of the probability measure. In particular, we consider the case when the probability measure is approximated by an empirical probability measure and show an exponential rate of convergence of the sequence of optimal solutions obtained from solving the approximation problem. The analysis is extended to the stationary points.
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