Tree-Sparse Modeling and Solution of Multistage Stochastic Programs

Multistage stochastic programs are prototypical for nonlinear programs with an inherent tree structure inducing characteristic sparsity patterns in the KKT systems of interior methods. We present an integrated modeling and solution approach for such tree-sparse programs. Three closely related natural formulations having desirable control-theoretic properties lead to KKT system solution algorithms with linear complexity. Application examples from computational finance and process engineering demonstrate the efficiency of the approach

Power management in a hydro-thermal system under uncertainty by Lagrangian relaxation

We present a dynamic multistage stochastic programming model for the cost-optimal generation of electric power in a hydro-thermal system under uncertainty in load, inflow to reservoirs and prices for fuel and delivery contracts. The stochastic load process is approximated by a scenario tree obtained by adapting a SARIMA model to historical data, using empirical means and variances of simulated scenarios to construct an initial tree, and reducing it by a scenario deletion procedure based on a suitable probability distance. Our model involves many mixed-integer variables and individual power unit constraints, but relatively few coupling constraints. Hence we employ stochastic Lagrangian relaxation that assigns stochastic multipliers to the coupling constraints. Solving the Lagarangian dual by a proximal bundle method leads to successive decomposition into single thermal and hydro unit subproblems that are solved by dynamic programming and a specialized descent algorithm, respectively. The optimal stochastic multipliers are used in Lagrangian heuristics to construct approximately optimal first stage decisions. Numerical results are presented for realistic data from a German power utility, with a time horizon of one week and scenario numbers ranging from 5 to 100. The corresponding optimization problems have up to 200,000 binary and 350,000 continuous variables, and more than 500,000 constraints

Global supply chains of high value low volume products

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