Entropic Penalty Approach to Stochastic Programming

摘要

A new decision-theoretic approach to Nonlinear Programming Problems with stochastic constraints is introduced. The Stochastic Program (SP) is replaced by a Deterministic Program (DP) in which a term is added to the objective function to penalize solutions which are not feasible in the mean. The special feature of the author's approach is the choice of the penalty function P sub E, which is given in terms if the relative entropy functional, and is accordingly called entropic penalty. It is shown that P sub E has properties which make it suitable to treat stochastic programs. Some of these properties are derived via a dual representation independent. The dual representation is also used to express the Deterministric Problem (DP) as a saddle function problem. For problems in which the randomness occurs in the rhs of the constraints, it 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 sub E and the induced Approximate Deterministic Program are considered.