A simple formula for the second-order subdifferential of maximum functions

We 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 second-order subdifferentials

Conditioning of linear-quadratic two-stage stochastic optimization problems

In this paper a condition number for linear-quadratic two-stage stochastic optimization problemsis introduced as the Lipschitz modulus of the multifunction assigning to a (discrete) probabilitydistribution the solution set of the problem. Being the outer norm of the Mordukhovich coderivativeof this multifunction, the condition number can be estimated from above explicitly in terms of theproblem data by applying appropriate calculus rules. Here, a chain rule for the extended partialsecond-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 problemswith simple recourse

Conditioning of linear-quadratic two-stage stochastic optimization problems ∗

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. Keywords: Stochastic optimization, two-stage linear-quadratic problems, conditioning, coderivative calculus, simpl