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.

Several stochastic programming methods have been developed:

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: $${\displaystyle \min _{x\in X}\{g(x)=f(x)+E_{\xi }[Q(x,\xi )]\}}$$ where ${\displaystyle Q(x,\xi )}$ is the optimal value of the second-stage problem $${\displaystyle \min _{y}\{q(y,\xi )\,|\,T(\xi )x+W(\xi )y=h(\xi )\}.}$$

The classical two-stage linear stochastic programming problems can be formulated as $${\displaystyle {\begin{array}{llr}\min \limits _{x\in \mathbb {R} ^{n}}&g(x)=c^{T}x+E_{\xi }[Q(x,\xi )]&\\{\text{subject to}}&Ax=b&\\&x\geq 0&\end{array}}}$$

where ${\displaystyle Q(x,\xi )}$ is the optimal value of the second-stage problem $${\displaystyle {\begin{array}{llr}\min \limits _{y\in \mathbb {R} ^{m}}&q(\xi )^{T}y&\\{\text{subject to}}&T(\xi )x+W(\xi )y=h(\xi )&\\&y\geq 0&\end{array}}}$$

In such formulation ${\displaystyle x\in \mathbb {R} ^{n}}$ is the first-stage decision variable vector, ${\displaystyle y\in \mathbb {R} ^{m}}$ is the second-stage decision variable vector, and ${\displaystyle \xi (q,T,W,h)}$ contains the data of the second-stage problem. In this formulation, at the first stage we have to make a "here-and-now" decision ${\displaystyle x}$ before the realization of the uncertain data ${\displaystyle \xi }$, viewed as a random vector, is known. At the second stage, after a realization of ${\displaystyle \xi }$ 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 ${\displaystyle c^{T}x}$ of the first-stage decision plus the expected cost of the (optimal) second-stage decision. We can view the second-stage problem simply as an optimization problem which describes our supposedly optimal behavior when the uncertain data is revealed, or we can consider its solution as a recourse action where the term ${\displaystyle Wy}$ compensates for a possible inconsistency of the system ${\displaystyle Tx\leq h}$ and ${\displaystyle q^{T}y}$ is the cost of this recourse action.

The considered two-stage problem is linear because the objective functions and the constraints are linear. Conceptually this is not essential and one can consider more general two-stage stochastic programs. For example, if the first-stage problem is integer, one could add integrality constraints to the first-stage problem so that the feasible set is discrete. Non-linear objectives and constraints could also be incorporated if needed.