Stochastic programming

Stochastic programming is an approach in mathematical programming that allows to take into account the uncertainty in optimization models.

While deterministic optimization problems are formulated using specified parameters, real application problems usually contain some unknown parameters. When parameters are known only within certain limits, one approach to solving such problems is called robust optimization. This approach is to find a solution that is valid for all such data and, in a sense, optimal.

Models of stochastic programming have a similar form, but use knowledge of probability distributions for data or their estimates. The goal here is to find some solution that is valid for all (or almost all) possible data values ​​and maximize the expectation of a certain function of solutions and random variables. In general, such models are formulated, solved analytically or numerically, and their results are analyzed to provide useful information for decision makers.

The most widely used and well studied two-stage linear models of stochastic programming. Here, the decision maker takes some action in the first stage, after which a random event occurs that affects the outcome of the decision of the first stage. At the second stage, a corrective decision can then be taken that compensates for any undesirable effects as a result of the decision of the first stage.

The optimal solution of such a model is the only solution of the first stage and a set of corrective decisions (decision rules) determining what action should be taken at the second stage in response to each random result.