Stochastic programming

In the field of mathematical optimization, stochastic programming is a framework for modeling optimization problems that involve uncertainty. A stochastic program is an optimization problem in which some or all problem parameters are uncertain, but follow known probability distributions. This framework contrasts with deterministic optimization, in which all problem parameters are assumed to be known exactly. The goal of stochastic programming is to find a decision which both optimizes some criteria chosen by the decision maker, and appropriately accounts for the uncertainty of the problem parameters. Because many real-world decisions involve uncertainty, stochastic programming has found applications in a broad range of areas ranging from finance to transportation to energy optimization. The basic idea of two-stage stochastic programming is that (optimal) decisions should be based on data available at the time the decisions are made and cannot depend on future observations. The two-stage formulation is widely used in stochastic programming. The general formulation of a two-stage stochastic programming problem is given by: where is the optimal value of the second-stage problem The classical two-stage linear stochastic programming problems can be formulated as where is the optimal value of the second-stage problem In such formulation is the first-stage decision variable vector, is the second-stage decision variable vector, and contains the data of the second-stage problem. In this formulation, at the first stage we have to make a "here-and-now" decision before the realization of the uncertain data , viewed as a random vector, is known. At the second stage, after a realization of becomes available, we optimize our behavior by solving an appropriate optimization problem. At the first stage we optimize (minimize in the above formulation) the cost of the first-stage decision plus the expected cost of the (optimal) second-stage decision.