A Nonconvex Parametric SQP Method
The real–time iteration scheme described for Nonlinear Model Predictive Control (NMPC) in chapter 4 was based on the idea of repeatedly performing a single iteration of a Newton–type method to compute control feedback. We have seen that it can be combined with the convexification and relaxation method of chapter 2 to develop a new mixed–integer real–time iteration scheme. For Mixed–Integer Optimal Control Problems (MIOCPs) with constraints depending directly on an integer control, that method yields an Nonlinear Program (NLP) with vanishing constraints as seen in chapter 5. In this chapter we develop a new Sequential Quadratic Programming (SQP) method to solve the arising class of NLPs with vanishing constraints in an active–set framework. It generates a sequence of local quadratic subproblems inheriting the vanishing constraints property from the NLP. These local subproblems are referred to as Quadratic Programs with Vanishing Constraints (QPVCs) and are the subproblems to be solved in the mixed–integer real–time iteration scheme. For this purpose, we develop a new active set strategy that finds strongly stationary points of a QPVC with convex objective function but nonconvex feasible set. This active set strategy is based on the idea of using parametric quadratic programming techniques to efficiently move between convex subsets of the QPVC’s nonconvex feasible set. Strongly stationary points of QPVCs are locally optimal, and we develop a heuristic that improves these points to global optimality.
KeywordsSequential Quadratic Programming Sequential Quadratic Programming Method Global Optimality Condition Homotopy Path Dual Parametric
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