Conditioning of linear-quadratic two-stage stochastic optimization problems

Abstract

In this paper a condition number for linear-quadratic two-stage stochastic optimization problems is introduced as the Lipschitz modulus of the multifunction assigning to a (discrete) probability distribution the solution set of the problem. Being the outer norm of the Mordukhovich coderivative of this multifunction, the condition number can be estimated from above explicitly in terms of the problem data by applying appropriate calculus rules. Here, a chain rule for the extended partial second-order subdifferential recently proved by Mordukhovich and Rockafellar plays a crucial role. The obtained results are illustrated for the example of two-stage stochastic optimization problems with simple recourse.

Full Stability in Finite-Dimensional Optimization

“…• Other classes of mathematical programs. Besides the classes of mathematical programs listed above, the second-order construction D * N Θ in (5.17) has been constructively calculated for the underlying sets Ω in (5.1), which are not in the discussed forms; see, e.g., [4,10,13,14,25,29,31] and the references therein. These results can be incorporated in the framework of (5.17) and thus allow us to provide via Theorem 5.6 complete characterizations of the equivalent stability properties (i)-(iii) entirely in terms of initial data of the corresponding mathematical programs under the nondegeneracy condition (A1).…”