*4.1. PolySCIP*

The strength of exact algorithms is the guarantee of reaching the global optimum, but the related computational cost can prevent their usage for large size NP-hard problems. Examples of classical exact algorithms are branch and bound, branch and cut, or A\*. There exist also commercial exact solvers such as IBM CPLEX [37] and AMPL [59] but to our knowledge, these algorithms and solvers are only able to solve single objective optimization problems. This limitation led to research for multi-objective exact solvers and in 2016, PolySCIP was proposed [58].

PolySCIP employs a "Lifted Weight Space Approach" [58]. This approach first optimizes the objectives lexicographically. The weighted (single objective) optimization problem from the first phase is optimized by using positive weight vectors. This guides the algorithm in exploring the Pareto front in the problem space. If the new non-dominated solution is found, the old solution (the one that has been dominated) is discarded and the process continues until all non-dominated solutions are found. As a result, the outcome is a Pareto front instead of just one solution. The method was proven mathematically in finding all global optima by the authors.
