Solving mixed-integer nonlinear optimization problems using simultaneous convexification: a case study for gas networks

Frauke Liers (), Alexander Martin (), Maximilian Merkert (), Nick Mertens and Dennis Michaels
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Frauke Liers: Friedrich-Alexander University Erlangen-Nürnberg
Alexander Martin: Friedrich-Alexander University Erlangen-Nürnberg
Maximilian Merkert: Otto von Guericke University Magdeburg
Nick Mertens: Technical University of Dortmund
Dennis Michaels: Technical University of Dortmund

Journal of Global Optimization, 2021, vol. 80, issue 2, No 4, 307-340

Abstract: Abstract Solving mixed-integer nonlinear optimization problems (MINLPs) to global optimality is extremely challenging. An important step for enabling their solution consists in the design of convex relaxations of the feasible set. Known solution approaches based on spatial branch-and-bound become more effective the tighter the used relaxations are. Relaxations are commonly established by convex underestimators, where each constraint function is considered separately. Instead, a considerably tighter relaxation can be found via so-called simultaneous convexification, where convex underestimators are derived for more than one constraint function at a time. In this work, we present a global solution approach for solving mixed-integer nonlinear problems that uses simultaneous convexification. We introduce a separation method that relies on determining the convex envelope of linear combinations of the constraint functions and on solving a nonsmooth convex problem. In particular, we apply the method to quadratic absolute value functions and derive their convex envelopes. The practicality of the proposed solution approach is demonstrated on several test instances from gas network optimization, where the method outperforms standard approaches that use separate convex relaxations.

Keywords: Mixed-integer nonlinear programming; Simultaneous convexification; Convex envelope; Gas network optimization (search for similar items in EconPapers)
Date: 2021
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Citations: View citations in EconPapers (2)

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DOI: 10.1007/s10898-020-00974-0

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