Stochastic Programming: Achievements and Open Problems

As is well known, many applied problems may be modelled as linear programs of the standard type (1.1)$$ subject{\kern 1pt} to\left. {\begin{array}{*{20}{c}} {\min {{c}^{T}}x} \\ {Ax = b} \\ {x \geqslant 0,} \\ \end{array} } \right\} $$ where $$ A \in {{\mathbb{R}}^{{mxn}}},b \in {{\mathbb{R}}^{m}},c \in {{\mathbb{R}}^{n}} $$ are fixed data and $$ x \in {{\mathbb{R}}^{n}} $$ is to be determined as a vector of optimal decisions satisfying the given constraints. However, as was observed already more than 40 years ago, it may also happen in applications, that modelling the linear structures as in (1.1) is justified, but not all of the data in A,b, c are fixed (and known) before the decision on x has to be taken. If those data happen to be random variables with known distributions, we face a problem of stochastic linear programming (SLP). Among the first papers discussing those problems are Beale [2], Charnes—Cooper [7], Dantzig [8], Dantzig—Madansky [9] and Tintner [33].