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.Projekt DEAL 202
