Generator Sets for the Minkowski Sum Problem -- Theory and Insights

TL;DR

The paper analyzes the Minkowski Sum Problem (MSP) in multi-objective optimization, focusing on computing the global nondominated sum $\mathcal{Y}_{\textnormal{N}}$ as the ND sum of local sets. It introduces generator sets and minimum generator sets to identify a compact subset of local vectors that suffices to generate $\mathcal{Y}_{\textnormal{N}}$, and presents an IP-based method plus a bounding-set pruning approach to obtain such generators. Theoretical results show extreme supported local vectors are necessary components while non-extreme vectors can be redundant; computational experiments reveal exponential growth of $|\mathcal{Y}_{\text{N}}|$ with the number of objectives $p$ and substantial reductions in generator size achievable via bounding-set based redundancy detection, depending on local-set shapes. The work offers practical pathways to accelerate decomposable bi-objective optimization by pruning redundant subproblem vectors and integrating generator-set strategies into solution algorithms.

Abstract

This paper considers a class of multi-objective optimization problems known as Minkowski sum problems. Minkowski sum problems have a decomposable structure, where the global nondominated (Pareto) set corresponds to the Minkowski sum of several local nondominated sets. In some cases, the vectors of local sets does not contribute to the generation of the global nondominated set, and may therefore lead to wasted computational efforts. Therefore, we investigate theoretical properties of both necessary and redundant vectors, and propose an algorithm based on bounding sets for identifying unnecessary local vectors. We conduct extensive numerical experiments to test the the impact of varying characteristics of the instances on the resulting global nondominated set and the number of redundant vectors.

Figures (9)

• Figure 1: Classifications of two local sets and their MS. The number/index besides each local vector is used to identify which combinations are used to generate a VS, e.g. VS $(10, 6)$ is the sum of the vector with index 3 from $\mathcal{Y}^1$ and the vector with index 4 from $\mathcal{Y}^2$. Moreover, index $[1,6]$ gives the same VS. Vectors in grey are dominated.
• Figure 2: local sets generated using the three different methods l, m and u ($p=2$).
• Figure 3: Relative number of extreme vectors given number of objectives under different cardinalities of the local sets.
• Figure 4: The cardinality of $\mathcal{Y}_{\textnormal{N}}$ given local set cardinality and fixed configuration. Averages over the 5 instances for each configuration are illustrated as lines. Axis ranges vary for each subplot.
• Figure 5: Relative MGS cardinality given the relative number of unsupported ND local vectors. Trend lines for each combination of configuration and local set method are given. Moreover, the cardinality of the local sets are visualized using different point shapes.

Theorems & Definitions (22)

• Definition 2.1
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Corollary 3.1
• Proposition 3.3
• proof
• Definition 3.1
• Definition 3.2