Reducing Two-Stage Probabilistic Optimization Problems with Discrete Distribution of Random Data to Mixed-Integer Programming Problems*

Reduction of the bilevel stochastic optimization problem with quantile objective function to a mixed‐integer problem

“…In , a method to reduce single‐stage stochastic programming problems to mixed‐integer optimization problems is considered for the case of a discrete distribution of the random parameters. In , this method is applied to two‐stage stochastic programming problems. This method cannot be directly applied to the bilevel stochastic programming problem because this problem has another structure.…”

“…In , this method is applied to two‐stage stochastic programming problems. This method cannot be directly applied to the bilevel stochastic programming problem because this problem has another structure. To apply it, we first use the Karush–Kuhn–Tucker (KKT) conditions for the follower's problem to transform the bilevel optimization problem into a single level one.…”

“…To apply it, we first use the Karush–Kuhn–Tucker (KKT) conditions for the follower's problem to transform the bilevel optimization problem into a single level one. Then, using a method similar to the one suggested in , we can reduce the original problem to a mixed‐integer mathematical optimization problem. We prove equivalence of both problems.…”

Comparison of two algorithms for solving a two‐stage bilinear stochastic programming problem with quantile criterion

“…The problem considered there is also reduced to a mixed integer linear programming problem. A more general two‐stage problem of the quantile optimization with a non‐linear objective function is considered in Norkin et al . It is worth noting that the last problem has a high dimension.…”