Multi-objective optimization has become a very active area of research. It is an interdisciplinary field bringing together researchers from mathematics, computer science, engineering and economics. Multi-objective optimization problems arise frequently in applications when conflicting interests have to be considered. Multi-objective optimization problems are often solved according to the principle of efficiency or Pareto optimality: a solution is efficient if no other feasible solution exists that is better or equal in all objectives and strictly better in at least one objective. Each efficient solution corresponds to a possible compromise among the several objectives and is potentially relevant to a decision maker. Depending on the context, the goal is to compute either the set of all efficient solutions, its image in the objective space, or a representation of that image according to some measure of interest. The multidimensional nature of these problems raises relevant mathematical and algorithmic challenges.

The aim of this Special Issue is to collect the latest advances on exact and approximation methods with quality guarantees for multi-objective (mixed) integer optimization problems. The contributions cover both objective-space and decision-space approaches based on principles of branch-and-bound and column generation. New approximation results are also presented, as well as instance generators for benchmarking purposes.

Bazgan et al. (2023) investigate which types of partially exact approximation sets of polynomial cardinality are guaranteed to exist for general multi-objective problems. The authors also study minimum-cardinality partially exact approximation sets concerning (weak) efficiency of the contained solutions and relate their cardinalities to the minimum cardinality of an approximation set. Results concerning the polynomial-time computability of partially exact approximation sets are presented, showing that the (fully) polynomial-time solvability of the gap problem does not suffice for the computation of any type of partially exact approximation set in general.

Helfrich et al. (2023) unify existing positive results for minimization and negative results for maximization problems and show that, in principle, all multi-objective optimization problems can be approximated equally well by scalarizations using a transformation between objectives. Necessary and sufficient conditions for scalarizations that lead to a constant approximation quality are given, and, as a special case, norm-based scalarizations are investigated.

Bökler and Jasper (2024) study the complexity of the Multi-Objective Minimum Weight Minimum Stretch Spanner problem, which is a multi-objective generalization of the Minimum t-Spanner problem. The proposed multi-objective approach allows to find solutions that offer a viable compromise between cost and utility, which is a relevant feature, e.g., in the planning of transportation or communication networks. The authors show that this is intractable for degree-3 bounded outerplanar instances and prove that if \(P \ne NP\), the set of extreme points cannot be computed in output-polynomial time for instances with unit costs and arbitrary graphs.

Bauß and Stiglmayr (2024) present augmentations of bi-objective branch-and-bound algorithms. In particular, objective space information is used to improve the selection of the active node and adaptively chosen scalarizations are solved to integer optimality to improve upper and lower bound sets. The proposed improvements are evaluated on knapsack, assignment and facility location problems.

Das and Gzara (2024) propose flow-based and column-based models to solve the bi-objective gate assignment problem. The gate assignment problem compromises between passenger satisfaction and operational costs of airline and airport operators. Using a network representation the authors apply a column generation approach to approximate the Pareto front with a defined gap.

Teichert et al. (2024) investigate multi-objective optimization problems that are continuous in nature, but the outputs are mapped into discrete alternatives by means of a discrete utility function. The authors propose algorithms to solve those problems and discuss their application on a particular case study on radiotherapy planning.

Fallah et al. (2024) investigate the connection between the efficient frontier of a general multi-objective mixed-integer linear (MILP) optimization problem and the restricted value function (RVF) of a closely related single-objective MILP. The authors show that the efficient frontier of the multi-objective MILP is comprised of points on the boundary of the epigraph of the RVF and that any description of the efficient frontier suffices to describe the RVF and vice-versa. A generalized cutting-plane algorithm is proposed for constructing the efficient frontier of a multi-objective MILP that arises from an existing algorithm for constructing the classical MILP value function.

Pecin et al. (2024) present an algorithm to solve bi-objective mixed integer problems, extending the Boxed Line Method and the \(\epsilon \)-Tabu Method, and discuss its empirical performance on a wide range instances.

Bökler et al. (2024) present a outer approximation algorithm for multi-objective mixed-integer linear programming problems. The algorithm solves single-objective weighted-sum problems to determine the facets of convex hull in objective space. The authors show that the number weighted-sum computations is polynomially bounded w.r.t. the number of facets.

Eichfelder and Warnow (2023) consider multi-objective mixed-integer convex optimization problems. By decomposing the problem into multi-objective continuous convex subproblems, referred to as patches, the enclosure of the non-dominated set is computed. Moreover, the authors present a method to reduce the number of required patches to obtain a enclosure of predefined quality.

Multi-objective mixed-integer convex optimization problems are also considered by Lammel et al. (2024), who introduce an approximation algorithm that produces an inner and outer approximation of the nondominated front by combining the results of solving patch problems using simplicial sandwiching. A proof of convergence is also presented.

Dächert et al. (2024) propose an algorithm with an open-source implementation in C++ to solve to solve multi-objective integer programming problems. The objective space algorithm decomposes the overall problem into a series of scalar subproblems that can be solved with efficient single-objective IP solvers. Particular attention is paid to keeping the number of required subproblems small by avoiding redundancies. The algorithm can be combined with different scalarizations, also being able to scale up to higher dimensional problems.

Eichfelder et al. (2023) propose a new multi-objective mixed-integer test instances with a special structure that allows to construct instances with scalable numbers of variables, scalable numbers of objective functions, and a control over the resulting efficient and nondominated sets and over the number of efficient integer assignments.