We provide a comprehensive overview of the literature of algorithmic approaches for multiobjective mixed‐integer and integer linear optimization problems. More precisely, we categorize and display exact methods for multiobjective linear problems with integer variables for computing the entire set of nondominated images. Our review lists 108 articles and is intended to serve as a reference for all researchers who are familiar with basic concepts of multiobjective optimization and who have an interest in getting a thorough view on the state‐of‐the‐art in multiobjective mixed‐integer programming.

In the last decade, public and industrial research funding has moved quantum computing from the early promises of Shor’s algorithm through experiments to the era of noisy intermediate scale quantum devices (NISQ) for solving real-world problems. It is likely that quantum methods can efficiently solve certain (NP-) hard optimization problems where classical approaches fail. In our perspective, we examine the field of quantum optimization, that is, solving optimization problems using quantum computers. We provide an entry point to quantum optimization for researchers from each topic, optimization or quantum computing, by demonstrating advances and obstacles with a suitable use case. We give an overview on problem formulation, available algorithms, and benchmarking. Although we show a proof-of-concept rather than a full benchmark between classical and quantum methods, this gives an idea of the current quality and capabilities of quantum computers for optimization problems. All observations are incorporated in a discussion on some recent quantum optimization breakthroughs, current status, and future directions.

This article is dedicated to the weight set decomposition of a multiobjective (mixed-)integer linear problem with three objectives. We propose an algorithm that returns a decomposition of the parameter set of the weighted sum scalarization by solving biobjective subproblems via Dichotomic Search which corresponds to a line exploration in the weight set. Additionally, we present theoretical results regarding the boundary of the weight set components that direct the line exploration. The resulting algorithm runs in output polynomial time, i.e. its running time is polynomial in the encoding length of both the input and output. Also, the proposed approach can be used for each weight set component individually and is able to give intermediate results, which can be seen as an “approximation” of the weight set component. We compare the running time of our method with the one of an existing algorithm and conduct a computational study that shows the competitiveness of our algorithm. Further, we give a state-of-the-art survey of algorithms in the literature.

Scalarization is a common technique to transform a multiobjective optimization problem into a scalar-valued optimization problem. This article deals with the weighted Tchebycheff scalarization applied to multiobjective discrete optimization problems. This scalarization consists of minimizing the weighted maximum distance of the image of a feasible solution to some desirable reference point. By choosing a suitable weight, any Pareto optimal image can be obtained. In this article, we provide a comprehensive theory of this set of eligible weights. In particular, we analyze the polyhedral and combinatorial structure of the set of all weights yielding the same Pareto optimal solution as well as the decomposition of the weight set as a whole. The structural insights are linked to properties of the set of Pareto optimal solutions, thus providing a profound understanding of the weighted Tchebycheff scalarization method and, as a consequence, also of all methods for multiobjective optimization problems using this scalarization as a building block.

In this paper, we consider the inclusion of the solvency capital requirement (SCR) into portfolio optimization by the use of a quadratic proxy model. The Solvency II directive requires insurance companies to calculate their SCR based on the complete loss distribution for the upcoming year. Since this task is, in general, computationally challenging for insurance companies (and therefore, not taken into account during portfolio optimization), employing more feasible proxy models provides a potential solution to this computational difficulty. Here, we present an approach that is also suitable for future applications in quantum computing. We analyze the approximability of the solvency capital ratio in a quadratic form using machine learning techniques. This allows for an easier consideration of the SCR in the classical mean-variance analysis. In addition, it allows the problem to be formulated as a quadratic unconstrained binary optimization (QUBO), which benefits from the potential speedup of quantum computing. We provide a detailed description of our model and the translation into a QUBO. Furthermore, we investigate the performance of our approach through experimental studies.