Realistic Constrained Multiobjective Optimization Benchmark Problems From Design

Abstract

Multi-objective optimization is increasingly used in engineering to design new systems and to identify design tradeoffs. Yet, design problems often have objective functions and constraints that are expensive and highly non-linear. Combinations of these features lead to poor convergence and diversity loss with common algorithms that have not been specifically designed for constrained optimization. Constrained benchmark problems exist, but they do not necessarily represent the challenges of engineering problems. In this paper, a framework to design electro-mechanical actuators, called MODAct, is presented and 20 constrained multi-objective optimization test problems are derived from the framework with a specific focus on constraints. The full source code is made available to ease its use. The effects of the constraints are analyzed through their impact on the Pareto front as well as on the convergence performance. A constraint landscape analysis approach is followed and extended with three new metrics to characterize the search and objective spaces. The features of MODAct are compared to existing test suites to highlight the differences. In addition, a convergence analysis using NSGA-II, NSGA-III and C-TAEA on MODAct and existing test suites suggests that the design problems are indeed difficult due to the constraints. In particular, the number of simultaneously violated constraints in newly generated solutions seems key in understanding the convergence challenges. Thus, MODAct offers an efficient framework to analyze and handle constraints in future optimization algorithm design.

A Survey on Evolutionary Constrained Multiobjective Optimization

“…test problems derived from actual problems [142], and equality constrained test sets (CZDT test set [55] and Eq-DTLZ test set [143]). They make substantial contributions to improving the performance of evolutionary algorithms.…”

“…Picard and Schiffmann [142] used the design of electroactuators to develop realistic benchmark CMOPs. Twenty problems are derived with four different constraint levels and up to five objectives.…”

Characterization of Constrained Continuous Multiobjective Optimization Problems: A Feature Space Perspective

“…Since the ratio of feasible boundary crossings does not consider objective values, it is a feature characterizing violation landscapes. Therefore, it can be used in multiobjective optimization without any modifications [3].…”

“…Currently, the benchmark suites are still unsatisfactory understood, and their characteristics remain unrevealed. In addition to characteristics describing the number of variables, objectives and constraints, type of objective and constraint functions, and geometric properties of Pareto front shapes, there are only few and limited techniques proposed to explore the search space of CMOPs, especially from the field of exploratory landscape analysis (ELA) [3]- [5].…”

“…There are only limited studies on ELA in the multiobjective combinatorial context [6]- [8], an initial study in continuous multiobjective optimization [9], and some first attempts to visualize bi-objective continuous problem landscapes with up to three variables [4], [10]- [12]. The situation is even worse in constrained optimization, with a single preliminary study on the characterization of constrained continuous singleobjective optimization problems [5] and an attempt to extend this work to multiobjective optimization [3]. The latter work studied five landscape features, including the feasibility ratio, disjointedness of feasible regions, and relationships between the objectives and constraints.…”