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Towards the Solution of Mixed-Integer Nonlinear Optimization Problems using Simultaneous Convexification

  • 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 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 for the convex hull of constrained sets. It relies on determining the convex envelope of linear combinations of the constraints 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.

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Metadaten
Author:Frauke Liers, Alexander Martin, Maximilian Merkert, Nick Mertens, Dennis Michaels
Document Type:Preprint
Language:English
Date of Publication (online):2020/01/08
Date of first Publication:2020/01/08
Release Date:2020/01/08
Tag:Convex Envelope; Gas Network Optimization; Mixed-Integer Nonlinear Programming; Simultaneous Convexification
Pagenumber:36
Institutes:Friedrich-Alexander-Universität Erlangen-Nürnberg
Subprojects:A05
B06
B07
Z01
Licence (German):License LogoCreative Commons - CC BY-NC - Namensnennung - Nicht kommerziell 4.0 International