Multiperiod design and planning with interior point methods

Abstract

This work develops an efficient decomposition algorithm for solving multiperiod design problems (MPD) using interior point (JP) methods within a reduced Hessian successive quadratic programming (rSQP) framework. MPD problems often result when multiple planning scenarios or discretized uncertainty descriptions are incorporated in an optimization study. Each uncertain or planning scenario is expressed as a separate period in MPD and all of these are linked by a small set of design variables. The limiting factor in solving MPD problems is a disproportionate increase in computational resources and decrease in solution robustness, with an increase in the number of periods. However, efficient decomposition strategies exist that exploit the block bordered diagonal structure of these problems and provide a linear increase in computational resources with the growth in periods. This was proposed in the (MPD–SQP) algorithm for general nonlinear MPD problems by Varvarezos, Biegler and Grossmann, 1994. On the other hand, the MPD/SQP algorithm employs an active set strategy for solving the quadratic programming (QP) sub-problem and this is combinatorial in the number of active constraints. Also, it needs to address a potentially different structure with each update of the active set. This consideration usually leads MPD–SQP to adopt an early termination for the QP problem, and this often requires additional SQP iterations. In order to solve the QP completely at each iteration, interior point methods have advantages as the number of updates is independent of the number of active constraints and we deal with a fixed structure throughout the solution procedure. Incorporating these concepts leads to the MPD–rISQP algorithm. The resulting algorithm maintains the nice features of the MPD–SQP algorithm, such as a range and null space decomposition within each period and a periodic problem decomposition. Moreover, the MPD–rISQP algorithm proposed here employs a primal–dual path following interior point method to address the QP. This permits a rigorous solution to the QP with an effort that is linear with the number of periods and independent of the active set. This algorithm is demonstrated on four example problems with up to 16 000 variables.