Linear Algebra for Block Structured QPs
In this chapter we survey linear algebra techniques for efficiently solving Quadratic Programs (QPs) and Quadratic Programs with Vanishing Constraints (QPVCs) with block structure due to direct multiple shooting. After a review of existing techniques, we present a novel algorithmic approach tailored to QPs with many control parameters due to the application of outer convexification. Our approach consists of a block structured factorization of the Karush–Kuhn–Tucker (KKT) systems that completes in O(mn) operations without generating any fill–in. It is derived from a combined null–space range–space method due to . All operations on the KKT factorization required for the parametric active set algorithm for QPVC of chapter 6 are presented in detail, and efficient implementations are discussed. We investigate the run time complexity of this algorithm, the number of floating–point operations required for the individual steps of the presented factorization and backsolve, and give details on the memory requirements.
KeywordsNull Space Cholesky Decomposition Runtime Complexity Point Constraint Linear Algebra Operation
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