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.

Many real-life planning problems require making a priori decisions before all parameters of the problem have been revealed. An important special case of such problem arises in scheduling and transshipment problems, where a set of jobs needs to be assigned to the available set of machines or personnel (resources), in a way that all jobs have assigned resources, and no two jobs share the same resource. In its nominal form, the resulting computational problem becomes the assignment problem. This paper deals with the Robust Assignment Problem (RAP) which models situations in which certain assignments are vulnerable and may become unavailable after the solution has been chosen. The goal is to choose a minimum-cost collection of assignments (edges in the corresponding bipartite graph) so that if any vulnerable edge becomes unavailable, the remaining part of the solution contains an assignment of all jobs. We develop algorithms and hardness results for RAP and establish several connections to well-known concepts from matching theory, robust optimization, LP-based techniques and combinations thereof.