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The Cost of Not Knowing Enough: Mixed-Integer Optimization with Implicit Lipschitz Nonlinearities

Martin Schmidt(mar.schmidt***at***fau.de)
Mathias Sirvent(mathias.sirvent***at***fau.de)
Winnifried Wollner(wollner***at***mathematik.tu-darmstadt.de)

Abstract: It is folklore knowledge that nonconvex mixed-integer nonlinear optimization problems can be notoriously hard to solve in practice. In this paper we go one step further and drop analytical properties that are usually taken for granted in mixed-integer nonlinear optimization. First, we only assume Lipschitz continuity of the nonlinear functions and additionally consider multivariate implicit constraint functions that cannot be solved for any parameter analytically. For this class of mixed-integer problems we propose a novel algorithm based on an approximation of the feasible set in the domain of the nonlinear function - in contrast to an approximation of the graph of the function considered in prior work. This method is shown to compute global optimal solutions in finite time and we also provide a worst-case iteration bound. However, first numerical experiences reveal that a lot of work is still to be done for this highly challenging class of problems and we thus finally propose some possible directions of future research.

Keywords: Mixed-Integer Nonlinear Optimization, Global Optimization, Lipschitz Optimization, Gas Networks

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

Category 2: Global Optimization

Category 3: Applications -- Science and Engineering

Citation:

Download: [PDF]

Entry Submitted: 04/26/2018
Entry Accepted: 04/26/2018
Entry Last Modified: 04/26/2018

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