Preconditioned Conjugate Gradients in an Interior Point Method for Two-stage Stochastic Programming

We develop a variant of an interior point method for solving two-stage stochastic linear programming problems. The problems are solved in a deterministic equivalent form in which the first stage variables appear as dense columns. To avoid their degrading influence on the adjacency structure AA (and the Cholesky factor) an iterative method is applied to compute orthogonal projections. Conjugate gradient algorithm with a structure-exploiting preconditioner is used. The method has been applied to solve real–life stochastic optimization problems. Preliminary computational results show the feasibility of the approach for problems with up to 80 independent scenarios (a deterministic equivalent linear program has 14001 constraints and 63690 variables).