Clustering-based preconditioning for stochastic programs

Abstract

We present a clustering-based preconditioning strategy for KKT systems arising in stochastic programming within an interior-point framework. The key idea is to perform adaptive clustering of scenarios (inside-the-solver) based on their influence on the problem at hand. This approach thus contrasts with existing (outside-the-solver) approaches that cluster scenarios based on problem data alone. We derive spectral and error properties for the preconditioner and demonstrate that scenario compression rates of up to 94 % can be obtained, leading to dramatic computational savings. In addition, we demonstrate that the proposed preconditioner can avoid scalability issues of Schur decomposition in problems with large first-stage dimensionality.

Hierarchical MPC schemes for periodic systems using stochastic programming

“…This problem is equivalent to P m and, because {d ξ } m ξ=1 is i.i.d., P m is an statistical approximation of P ∞ . In the SP literature problem, (5) is known as a sample average approximation (SAA). We define the solution set of P m as S m ⊆ W. A solution of P m is used to update the targets for the next period w m+1 = (x 0,m+1 , η m+1 ).…”

“…This is, in fact, is also a limitation of statistical approximation schemes for SP. To circumvent this issue, one can use clustering techniques that seek to compress the realization space to maintain a tractable approximation [5]. Such techniques are based on the observation that data realizations tend to be redundant and only a small subset actually impacts the cost.…”

Parallel Solution of Robust Nonlinear Model Predictive Control Problems in Batch Crystallization

“…Lubin et al [12] forms the Schur system as a by-product of a sparse factorization and factorizes the Schur system in parallel. Cao et al [13] performs adaptive clustering of scenarios inside-the-solver and forms a sparse compressed representation of the large Karush-Kuhn-Tucker(KKT) system as a preconditioner. The matrix that needs to be factorized in this approach is much smaller than the full-space KKT system and more sparse than the Schur system.…”

On Geometrical Properties of Preconditioners in IPMs for Classes of Block-Angular Problems

“…It is worth noting that computing approximate Newton directions by PCG does not destroy the good convergence properties of IPMs, as shown in [20]. There is an extensive literature on the use of PCG within IPMs for other types of problems (e.g., [3,4,9,18,25,28] to mention a few).…”

“…It is worth noting that computing approximate Newton directions by PCG does not destroy the good convergence properties of IPMs, as shown in [20]. There is an extensive literature on the use of PCG within IPMs for other types of problems (e.g., [3,4,9,18,25,28] to mention a few).However, it is not yet clear why for some classes of block-angular problems the above approach may be very efficient (see, for instance, the results of [13,12]), while it may need a large number of PCG iterations in others. The spectral radius may be used to monitor the good or bad behaviour, but an ex-ante explanation has to be found in the structural information of the block-angular constraints matrix.…”