Tree-sparse convex programs

Dynamic stochastic programs are prototypical for optimization problems with an inherent tree structure inducing characteristic sparsity patterns in the KKT systems of interior methods. We propose an integrated modeling and solution approach for such tree-sparse programs. Three closely related natural formulations are theoretically analyzed from a control-theoretic perspective and compared to each other. Associated KKT system solution algorithms with linear complexity are developed and comparisons to other interior approaches and related problem formulations are discussed. 0. INTRODUCTION The current paper studies convex programs with an underlying tree topology, such as discrete-time stochastic control problems. We propose an integrated natural modeling and solution framework for this class of large, tree-sparse optimization problems. Dynamic control problems are characterized by an inherent recourse structure. The proposed modeling is natural in the sense that constraints are categorized according to their control-theoretic interpretation, namely as dynamic equations (in which the recourse structure manifests itself), local constraints, and global constraints; the latter two categories have subcategories covering all kinds of boundary conditions. Furthermore, natural regularity assumptions are associated with each (sub)category of constraints. Our principal interest here is in the algebraic structure of the KKT systems arising in standard interior methods. (Stochastic aspects will be discussed elsewhere in full detail.) Extending earlier work [28, 30, 31, 32] with promising computational results, we develop a thorough theoretical understanding of these tree-sparse linear indefinite KKT systems, with accent on the hierarchical constraint structure of three principal variants differing in the formulation of dynamics. The theoretical analysis leads to natural solution algorithms which combine a dynamic recursion with local projections for local constraints and a Schur complement approach for global constraints, giving linear complexity in the tree size. Other interior approaches for stochastic programs include [2, 6, 9, 11, 13, 20, 23] (twostage LP case) and [5, 12, 19, 27] (linear or convex multistage case). We compare our framework with these approaches and with the generalized linear-quadratic control formulations developed by Rockafellar [24, 25] and Rockafellar and Wets [26]. The material is organized as follows. After recalling basic facts on convex programs, trees, and interior methods, we present in Section 2 the tree-sparse problem classes along with regularity conditions and selected references to examples and applications. A detailed technical comparison is provided in Section 3, and the KKT solution algorithms are discussed in Section 4. Sections 5 and 6 investigate the respective relations of our approach to other interior methods and to generalized linear-quadratic control problems. Final remarks and indications of future research in Section 7 conclude the paper. 2000 Mathematics Subject Classification. 90C15, 90C06, 90C25, 65F50, 15A23 .