Convergence of Finite-Dimensional Approximations for Mixed-Integer Optimization with Differential Equations

  • We consider a direct approach to solve mixed-integer nonlinear optimization problems with constraints depending on initial and terminal conditions of an ordinary differential equation. In order to obtain a finite-dimensional problem, the dynamics are approximated using discretization methods. In the framework of general one-step methods, we provide sufficient conditions for the convergence of this approach in the sense of the corresponding optimal values. The results are obtained by considering the discretized problem as a parametric mixed-integer nonlinear optimization problem in finite dimensions, where the maximum step size for discretizing the dynamics is the parameter. In this setting, we prove the continuity of the optimal value function under a stability assumption for the integer feasible set and second-order conditions from nonlinear optimization. We address the necessity of the conditions on the example of pipe sizing problems for gas networks.

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Metadaten
Author:Falk M. Hante, Martin Schmidt
Parent Title (German):Control and Cybernetics
Document Type:Article
Language:English
Date of Publication (online):2018/12/05
Date of first Publication:2018/12/06
Release Date:2018/12/06
Tag:Lipschitz continuity; Mixed-integer nonlinear programming; Optimal value function; Optimization with differential equations; Parametric optimization
Page Number:16
Institutes:Friedrich-Alexander-Universität Erlangen-Nürnberg
Subprojects:A03
A05
B08
Licence (German):License LogoCreative Commons - CC BY - Namensnennung 4.0 International