Tree-Sparse Modeling and Solution of Multistage Stochastic Programs

The lecture presents an integrated modeling and solution framework aiming at the robust and efficient solution of very large instances of tree-sparse programs. This wide class of nonlinear programs (NLP) is characterized by an underlying tree topology. It includes, in particular, dynamic stochastic programs in scenario tree formulation, multistage stochastic programs, where the objective and constraints depend on random future events with known probability distributions. The basic principle behind the overall concept lies in a suitable nesting of generic and problem-specific algorithmic layers, each handling separate aspects. The large, tree-sparse NLP is tackled by a primal-dual interior method or, as in [1], by an SQP method using a primal-dual interior method as QP solver. These generic methods handle inequalities and, if applicable, nonlinearity and nonconvexity. On the bottom level, the central computational step in every interior iteration consists in calculating a Newton direction from the indefinite KKT system representing suitable linearizations of the perturbed Karush–Kuhn– Tucker optimality conditions. Principal features of the tree-sparse approach include the theoretical analysis and algorithmic exploitation of the structure of such KKT systems, based on their natural interpretation as linear-quadratic control problems. This leads to the distinction of generic block sparsity characterizing the entire class of tree-sparse problems, and sub-block sparsity specific to individual instances. The resulting KKT solution algorithm handles the generic block structure according to the general analysis, and the sub-block structure by local sparse matrix techniques [2,3,4]. The tree-sparse framework generalizes earlier work, providing a unified formulation for multistage stochastic programs and various other problems sharing a similar KKT structure, like trajectory optimization problems or the spatial dynamics of multibody systems in descriptor form. Stochastic programming is the primary application field and source of motivation for the generalization. Here the scenario tree reflects the underlying information structure, or nonanticipativity requirements: every node represents a decision that may depend on the past (the path to the node) but may not anticipate specific realizations of the future. Given the robust overall framework