Optimization-Constrained Differential Equations with Active Set Changes
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Optimization-Constrained Differential Equations with Active Set Changes Peter Stechlinski1 Received: 30 October 2019 / Accepted: 2 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract 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. Keywords Nonsmooth DAEs · Well-posedness · Sensitivity analysis · Lexicographic derivatives · Piecewise smooth functions Mathematics Subject Classification 49K15 · 49K40 · 49J52 · 34A09
Communicated by Lorenz T. Biegler.
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Peter Stechlinski [email protected] Department of Mathematics and Statistics, University of Maine, Orono, ME, USA
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Journal of Optimization Theory and Applications
1 Introduction This article is concerned with nonlinear ODEs with nonlinear optimization problems embedded. This type of problem arises in, for example, using dynamic flux balance analysis to study biochemical networks [1,2], models of atmospheric aerosol particles [3,4], and control systems theory [5]. Assuming certain regularity conditions, including strict complementarity (i.e., absence of active constraint set changes), the embedded optimization can be replaced by its optimality conditions. This yields a system of differential-algebraic equations (DAEs) with optimality conditions, which are “regular” (in the sense of having classical differentiation index one [6–8]) and thus solvable with advanced simulation software [8–12]. If the number of active constraints changes along a solution trajectory or at initial data under parametric perturbations, then regularity of the DAE system is lost, invalidating standard theory and methods. In this case, the optimality conditions, and therefore the DAE system, include complementarity conditions. Consequently, the notion of differentiation index one is not well-defined and solutions may vary nonsmoot
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