The Entropic Penalty Approach to Stochastic Programming

A penalty-type decision-theoretic approach to Nonlinear Programming Problems with stochastic constraints is introduced. The Stochastic Program (SP) is replaced by a Deterministic Program (DP) by adding a term to the objective function to penalize solutions which are not “feasible in the mean.” The special feature of our approach is the choice of the penalty function P_E, which is given in terms of the relative entropy functional, and is accordingly called entropic penalty. It is shown that P_E has properties which make it suitable to treat stochastic programs. Some of these properties are derived via a dual representation of the entropic penalty which also enable one to compute P_E more easily, in particular if the constraints in (SP) are stochastically independent. The dual representation is also used to express the Deterministic Problem (DP) as a saddle function problem. For problems in which the randomness occurs in the rhs of the constraints, it is shown that the dual problem of (DP) is equivalent to Expected Utility Maximization of the classical Lagrangian dual function of (SP), with the utility being of the constant-risk-aversion type. Finally, mean-variance approximations of P_E and the induced Approximating Deterministic Program are considered.