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

Falk M. Hante (falk.hante***at***fau.de)
Martin Schmidt (martin.schmidt***at***uni-trier.de)

Abstract: 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.

Keywords: Optimization with differential equations, Optimal value function, Lipschitz continuity, Parametric optimization, Mixed-integer nonlinear programming

Category 1: Integer Programming ((Mixed) Integer Nonlinear Programming )

Category 2: Nonlinear Optimization (Systems governed by Differential Equations Optimization )

Category 3: Nonlinear Optimization

Citation:

Download: [PDF]

Entry Submitted: 12/05/2018
Entry Accepted: 12/05/2018
Entry Last Modified: 12/16/2019

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