Optimization-Constrained Differential Equations with Active Set Changes

Foundational theory is established for nonlinear differential equations with embedded nonlinear optimization problems exhibiting active set changes. Existence, uniqueness, and continuation of solutions are shown, followed by lexicographically smooth (implying Lipschitzian) parametric dependence. The sensitivity theory found here accurately characterizes sensitivity jumps resulting from active set changes via an auxiliary nonsmooth sensitivity system obtained by lexicographic directional differentiation. The results in this article hold under easily verifiable regularity conditions (linear independence of constraints and strong second-order sufficiency), which are shown to imply generalized differentiation index one of a nonsmooth differential-algebraic equation system obtained by replacing the optimization problem with its optimality conditions and recasting the complementarity conditions as nonsmooth algebraic equations. The theory in this article is computationally relevant, allowing for implementation of dynamic optimization strategies (i.e., open-loop optimal control), and recovers (and rigorously formalizes) classical results in the absence of active set changes. Along the way, contributions are made to the theory of piecewise differentiable functions.

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