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- ItemConditioning of linear-quadratic two-stage stochastic optimization problems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Emich, Konstantin; Henrion, René; Römisch, WernerIn 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.
- ItemSecond-order subdifferential of 1- and maximum norm(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Emich, KonstantinWe derive formulae for the second-order subdifferential of polyhedral norms. These formulae are fully explicit in terms of initial data. In a first step we rely on the explicit formula for the coderivative of normal cone mapping to polyhedra. Though being explicit, this formula is quite involved and difficult to apply. Therefore, we derive simple formulae for the 1-norm and
- ItemA simple formula for the second-order subdifferential of maximum functions(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Emich, Konstantin; Henrion, RenéWe derive a simple formula for the second-order subdifferential of the maximum of coordinates which allows us to construct this set immediately from its argument and the direction to which it is applied. This formula can be combined with a chain rule recently proved by Mordukhovich and Rockafellar [9] in order to derive a similarly simple formula for the extended partial second-order subdifferential of finite maxima of smooth functions. Analogous formulae can be derived immediately for the full and conventional partial secondorder subdifferentials.