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

Abstract

The paper is devoted to solving the two-stage problem of stochastic programming with quantile criterion. It is assumed that the loss function is bilinear in random parameters and strategies, and the random vector has a normal distribution. Two algorithms are suggested to solve the problem, and they are compared. The first algorithm is based on the reduction of the original stochastic problem to a mixed integer linear programming problem. The second algorithm is based on the reduction of the problem to a sequence of convex programming problems. Performance characteristics of both the algorithms are illustrated by an example. A modification of both the algorithms is suggested to reduce the computing time. The new algorithm uses the solution obtained by the second algorithm as a starting point for the first algorithm.

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

“…In the following theorem, it is shown that the optimization problems (4) and (15)- (20) are equivalent in the following sense [17]:…”

“…is an optimal solution of problem (15)-(20); 2. for every optimal solution ( , u, y 1 , … , y N , 1 , … , N , 1 , … , N ) of problem (15)- (20), there exists y(⋅) such that (u, y(⋅)) is an optimal solution of problem (4); 3. the optimal objective function values of problems (4) and (15)- (20) are equal, and they are attained simultaneously in both problems; 4. if the optimal objective function values of problems (4) and (15)- (20) are not attained, then minimizing sequences of the variable u of both problems coincide.…”

“…Denote by the corresponding objective function value in problem (4). Using the triplet (u, y(⋅), (⋅)), let us construct a feasible solution of the problem (15)- (20), which provides values of the objective function not greater than . Let ∈ {1, · · · , N} and̂= ,û = u,…”

“…These algorithms exploit different methods to solve the problem: branch-andbounds procedures, decomposition, and cuts. Some ways to simplify problem (15)- (20) in a particular case are given in Section 4. In general, these algorithms stop with solutions that may be far away from the global optimum.…”

“…According to the proof, the pair (u * , y * (⋅)) provides an upper bound * of the objective function value of (4). But * is the optimal objective value in problems (4) and (15)- (20) because of the equivalence of the considered problems. Therefore, (u * , y * (⋅)) is an optimal solution of (4).…”