A Polynomial-Time Inner Approximation Algorithm for Multi-Objective and Parametric Optimization

In multi-objective optimization, computing the entire non-dominated set (also known as the Pareto front or the Pareto frontier) is often intractable. However, for any multiplicative factor greater than one, an approximation set can be constructed in polynomial time for many problems. In this paper, we use the concept of convex approximation sets: Each point in the non-dominated set is approximated by a convex combination of images of solutions in such a set. Convex approximation sets can be used to efficiently approximate multi-objective optimization problems as well as parametric optimization problems. Recently, Helfrich et al. (2024) presented a convex approximation algorithm that works in an adaptive fashion and runs faster than all previously existing algorithms. We use a different approach for constructing an even more efficient adaptive algorithm for computing convex approximation sets of multi-objective mixed-integer linear programs. Our algorithm is based on the skeleton algorithm for polyhedral inner approximation by Csirmaz (2021). If the weighted sum scalarization can be solved exactly or approximately in polynomial time, our algorithm can find a convex approximation set for an approximation factor arbitrarily close to this solution quality. We demonstrate that our new algorithm runs faster than the current state-of-the-art algorithm from Helfrich et al. (2024) on instances of the multi-objective variants of the assignment problem, the knapsack problem, and the symmetric metric travelling salesman problem.