A central goal for multi-objective optimization problems is to compute their nondominated sets. In most cases these sets consist of infinitely many points and it is not a practical approach to compute them exactly. One solution to overcome this problem is to compute an enclosure, a special kind of coverage, of the nondominated set. One … Read more
This paper proposes a general framework for solving multiobjective nonconvex optimization problems, i.e., optimization problems in which multiple objective functions have to be optimized simultaneously. Thereby, the nonconvexity might come from the objective or constraint functions, or from integrality conditions for some of the variables. In particular, multiobjective mixed-integer convex and nonconvex optimization problems are … Read more
We establish a broad methodological foundation for mixed-integer optimization with learned constraints. We propose an end-to-end pipeline for data-driven decision making in which constraints and objectives are directly learned from data using machine learning, and the trained models are embedded in an optimization formulation. We exploit the mixed-integer optimization-representability of many machine learning methods, including … Read more
We consider mixed-integer linear quantile minimization problems that yield large-scale problems that are very hard to solve for real-world instances. We motivate the study of this problem class by two important real-world problems: a maintenance planning problem for electricity networks and a quantile-based variant of the classic portfolio optimization problem. For these problems, we develop … Read more
In the first part of this paper, we present a unified framework for analyzing the algorithmic complexity of any optimization problem, whether it be continuous or discrete in nature. This helps to formalize notions like “input”, “size” and “complexity” in the context of general mathematical optimization, avoiding context dependent definitions which is one of the … Read more
This work addresses optimal control problems governed by a linear time-dependent partial differential equation (PDE) as well as integer constraints on the control. Moreover, partial observations are assumed in the objective function. The resulting problem poses several numerical challenges due to the mixture of combinatorial aspects, induced by integer variables, and large scale linear algebra … Read more
Inspired by the recently introduced branch-and-bound method for continuous multiobjective optimization problems from G. Eichfelder, P. Kirst, L. Meng, O. Stein, A general branch-and-bound framework for continuous global multiobjective optimization, Journal of Global Optimization, 80 (2021) 195-227, we study for a general class of branch-and-bound methods in which sense the generated terminal enclosure and the … Read more