Explicitly Multi-Modal Benchmarks for Multi-Objective Optimization

TL;DR

Explicitly Multi-Modal Benchmarks for Multi-Objective Optimization introduces 3BC, a benchmark framework that uses basin connectivity to specify the number, location, and connectivity of local Pareto fronts in a multimodal landscape. By encoding basins of attraction as a basin graph, 3BC can generate problem instances with known local and global optima and enable per-basin performance indicators such as basin-wise IGDX. The construction starts from a primitive uni-objective function f defined on a diamond-shaped domain and builds a rotated bi-objective objective, allowing explicit control of separability, Pareto front shape, and scaling to high dimensions. The paper also defines evaluation metrics, demonstrates with depth- and breadth-base basin graphs, and discusses how 3BC can benchmark diverse EMO solvers with explicit landscape specifications and non-separable, scalable test cases.

Abstract

In multi-objective optimization, designing good benchmark problems is an important issue for improving solvers. Controlling the global location of Pareto optima in existing benchmark problems has been problematic, and it is even more difficult when the design space is high-dimensional since visualization is extremely challenging. As a benchmarking with explicit local Pareto fronts, we introduce a benchmarking based on basin connectivity (3BC) by using basins of attraction. The 3BC allows for the specification of a multimodal landscape through a kind of topological analysis called the basin graph, effectively generating optimization problems from this graph. Various known indicators measure the performance of a solver in searching global Pareto optima, but using 3BC can make us localize them for each local Pareto front by restricting it to its basin. 3BC's mathematical formulation ensures the accurate representation of the specified optimization landscape, guaranteeing the existence of intended local and global Pareto optima.

Figures (14)

• Figure 1: A 2-objetive function which has local optima but can be solved by a greedy algorithm
• Figure 2: Graphs of $f_t$'s ($\dim X = 1$)
• Figure 3: Graph of $f$ ($\dim X=1$)
• Figure 4: Family $\{f_t\}_t$ as slices of $f$ ($\dim X = 1$)
• Figure 5: Basins of local minima