Parabolic optimal control problems with combinatorial switching constraints – Part II: Outer approximation algorithm
arxiv(2022)
摘要
We consider optimal control problems for partial differential equations where
the controls take binary values but vary over the time horizon, they can thus
be seen as dynamic switches. The switching patterns may be subject to
combinatorial constraints such as, e.g., an upper bound on the total number of
switchings or a lower bound on the time between two switchings. In a companion
paper [arXiv:2203.07121], we describe the L^p-closure of the convex hull of
feasible switching patterns as intersection of convex sets derived from
finite-dimensional projections. In this paper, the resulting outer description
is used for the construction of an outer approximation algorithm in function
space, whose iterates are proven to converge strongly in L^2 to the global
minimizer of the convexified optimal control problem. The linear-quadratic
subproblems arising in each iteration of the outer approximation algorithm are
solved by means of a semi-smooth Newton method. A numerical example in two
spatial dimensions illustrates the efficiency of the overall algorithm.
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