7 research outputs found
PaMILO: A Solver for Multi-Objective Mixed Integer Linear Optimization and Beyond
In multi-objective optimization, several potentially conflicting objective
functions need to be optimized. Instead of one optimal solution, we look for
the set of so called non-dominated solutions. An important subset is the set of
non-dominated extreme points. Finding it is a computationally hard problem in
general. While solvers for similar problems exist, there are none known for
multi-objective mixed integer linear programs (MOMILPs) or multi-objective
mixed integer quadratically constrained quadratic programs (MOMIQCQPs). We
present PaMILO, the first solver for finding non-dominated extreme points of
MOMILPs and MOMIQCQPs. PaMILO provides an easy to use interface and is
implemented in C++17. It solves occurring subproblems employing either CPLEX or
Gurobi. PaMILO adapts the dual-benson algorithm for multi-objective linear
programming (MOLP). As it was previously only defined for MOLPs, we describe
how it can be adapted for MOMILPs, MOMIQCQPs and even more problem classes in
the future
An outer approximation algorithm for multi-objective mixed-integer linear and non-linear programming
In this paper, we present the first outer approximation algorithm for
multi-objective mixed-integer linear programming problems with any number of
objectives. The algorithm also works for certain classes of non-linear
programming problems. It produces the non-dominated extreme points as well as
the facets of the convex hull of these points. The algorithm relies on an
oracle which solves single-objective weighted-sum problems and we show that the
required number of oracle calls is polynomial in the number of facets of the
convex hull of the non-dominated extreme points in the case of multiobjective
mixed-integer programming (MOMILP). Thus, for MOMILP problems for which the
weighted-sum problem is solvable in polynomial time, the facets can be computed
with incremental-polynomial delay. From a practical perspective, the algorithm
starts from a valid lower bound set for the non-dominated extreme points and
iteratively improves it. Therefore it can be used in multi-objective
branch-and-bound algorithms and still provide a valid bound set at any stage,
even if interrupted before converging. Moreover, the oracle produces Pareto
optimal solutions, which makes the algorithm also attractive from the primal
side in a multi-objective branch-and-bound context. Finally, the oracle can
also be called with any relaxation of the primal problem, and the obtained
points and facets still provide a valid lower bound set. A computational study
on a set of benchmark instances from the literature and new non-linear
multi-objective instances is provided.Comment: 21 page
Output-sensitive complexity of multiobjective combinatorial optimization
We study output-sensitive algorithms and complexity for multiobjective combinatorial optimization problems. In this computational complexity framework, an algorithm for a general enumeration problem is regarded efficient if it is output-sensitive, i.e., its running time is bounded by a polynomial in the input and the output size. We provide both practical examples of MOCO problems for which such an efficient algorithm exists as well as problems for which no efficient algorithm exists under mild complexity theoretic assumptions
Multi-objective optimisation based planning of power-line grid expansions
German nuclear power phase out in 2022 leads to significant reconstruction of the energy transmission system. Thus, efficient identification of practical transmission routes with minimum impact on ecological and economical interests is of growing importance. Due to the sensitivity of Germany’s public to grid expansion (especially in case of overhead lines), the participation and planning process needs to provide a high degree of openness and accountability. Therefore, a new methodological approach for the computer-assisted finding of optimal power-line routes considering planning, ecological and economic decision criteria is presented. The approach is implemented in a tool-chain for the determination of transmission line routes (and sets of transmission line route alternatives) based on multi-criteria optimisation. Additionally, a decision support system, based on common Geographic Information Systems (GIS), consisting of interactive visualisation and exploration of the solution space is proposed
Multi-Objective Optimisation Based Planning of Power-Line Grid Expansions
German nuclear power phase out in 2022 leads to significant reconstruction of the energy transmission system. Thus, efficient identification of practical transmission routes with minimum impact on ecological and economical interests is of growing importance. Due to the sensitivity of Germany’s public to grid expansion (especially in case of overhead lines), the participation and planning process needs to provide a high degree of openness and accountability. Therefore, a new methodological approach for the computer-assisted finding of optimal power-line routes considering planning, ecological and economic decision criteria is presented. The approach is implemented in a tool-chain for the determination of transmission line routes (and sets of transmission line route alternatives) based on multi-criteria optimisation. Additionally, a decision support system, based on common Geographic Information Systems (GIS), consisting of interactive visualisation and exploration of the solution space is proposed