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 (2012a), Branda and Dupačová (2012), and Ermoliev et al. (2000) 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.