On relations between chance constrained and penalty function problems under discrete distributions

We extend the theory of penalty functions to stochastic programming problems with nonlinear inequality constraints dependent on a random vector with known distribution. We show that the problems with penalty objective, penalty constraints and chance constraints are asymptotically equivalent under discretely distributed random parts. This is a complementary result to Branda (Kybernetika 48(1):105–122, <CitationRef CitationID="CR3">2012a</CitationRef>), Branda and Dupačová (Ann Oper Res 193(1):3–19, <CitationRef CitationID="CR6">2012</CitationRef>), and Ermoliev et al. (Ann Oper Res 99:207–225, <CitationRef CitationID="CR12">2000</CitationRef>) where the theorems were restricted to continuous distributions only. We propose bounds on optimal values and convergence of optimal solutions. Moreover, we apply exact penalization under modified calmness property to improve the results. Copyright Springer-Verlag Berlin Heidelberg 2013