Tree-sparse convex programs

Abstract

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.

On a strategy for operational planning modes pumping station

“…Оперативное планирование режимов работы НС [9], способы оптимизации работы НС [10] рассмо-трены в работах Pulido-Calvo I., Ruuskanen A., Steinbach M. C. [11,12].…”

“…Про-веряем, может ли НА с полностью открытой РЗ обеспе-чить заданные параметры на выходе НС (13), при усло-вии, что НА работает с КПД большим заданного. Если может, то пересчитываем степень открытия РЗ (13), (17), вычисляем целевую функцию F1 (энергозатраты) по формуле (11). Если не может, выбираем другой НА (или их комбинацию).…”

On Shape Optimization with Stochastic Loadings

“…In stochastic optimization, see [56] for a recent textbook, data uncertainty is captured by probability distributions. In finite dimension, there exists a rich theory and methodology of linear models [49,56] and, to lesser extent, linear mixed-integer or nonlinear models [13,46,50,63]. Stochastic optimization in continuous time has been analyzed in stochastic dynamic programming and stochastic control, see [30,19].…”

A high-performance Riccati based solver for tree-structured quadratic programs

“…In this paper, we directly tackle the lower dimensional treestructured QP with an IP method as in Steinbach (2002), and employ a modified Riccati recursion to efficiently compute the search direction. We combine the algorithm with a tailored Linear Algebra (LA) library, especially suited for matrix dimensions that are common in most MPC problems.…”

“…In the pioneering work of (Steinbach, 2002), an Interior Point (IP) method is used with a generic framework to recursively factorize the KKT matrix of a wide class of tree-structured optimization problems. This framework is closely related to the developments of this paper.…”