A test problem for visual investigation of high-dimensional multi-objective search

An inherent problem in multiobjective optimization is that the visual observation of solution vectors with four or more objectives is infeasible, which brings major difficulties for algorithmic design, examination, and development. This paper presents a test problem, called the Rectangle problem, to aid the visual investigation of high-dimensional multiobjective search. Key features of the Rectangle problem are that the Pareto optimal solutions 1) lie in a rectangle in the two-variable decision space and 2) are similar (in the sense of Euclidean geometry) to their images in the four-dimensional objective space. In this case, it is easy to examine the behavior of objective vectors in terms of both convergence and diversity, by observing their proximity to the optimal rectangle and their distribution in the rectangle, respectively, in the decision space. Fifteen algorithms are investigated. Underperformance of Pareto-based algorithms as well as most state-of-the-art many-objective algorithms indicates that the proposed problem not only is a good tool to help visually understand the behavior of multiobjective search in a high-dimensional objective space but also can be used as a challenging benchmark function to test algorithms' ability in balancing the convergence and diversity of solutions

A simplex-like search method for bi-objective optimization

We describe a new algorithm for bi-objective optimization, similar to the Nelder Mead simplex algorithm, widely used for single objective optimization. For diferentiable bi-objective functions on a continuous search space, internal Pareto optima occur where the two gradient vectors point in opposite directions. So such optima may be located by minimizing the cosine of the angle between these vectors. This requires a complex rather than a simplex, so we term the technique the \cosine seeking complex". An extra beneft of this approach is that a successful search identifes the direction of the effcient curve of Pareto points, expediting further searches. Results are presented for some standard test functions. The method presented is quite complicated and space considerations here preclude complete details. We hope to publish a fuller description in another place

Objective reduction in many-objective optimization: linear and nonlinear algorithms

The difficulties faced by existing Multi-objective Evolutionary Algorithms (MOEAs) in handling many-objective problems relate to the inefficiency of selection operators, high computational cost and difficulty in visualization of objective space. While many approaches aim to counter these difficulties by increasing the fidelity of the standard selection operators, the objective reduction approach attempts to eliminate objectives that are not essential to describe the Pareto-optimal Front (POF). If the number of essential objectives are found to be two or three, the problem could be solved by the existing MOEAs. It implies that objective reduction could make an otherwise unsolvable (many-objective) problem solvable. Even when the essential objectives are four or more, the reduced representation of the problem will have favorable impact on the search efficiency, computational cost and decision-making. Hence, development of generic and robust objective reduction approaches becomes important. This paper presents a Principal Component Analysis and Maximum Variance Unfolding based framework for linear and nonlinear objective reduction algorithms, respectively. The major contribution of this paper includes: (a) the enhancements in the core components of the framework for higher robustness in terms of-applicability to a range of problems with disparate degree of redundancy; mechanisms to handle an input data that is mis-representative of the true POF; and dependence on fewer parameters to minimize the variability in performance, (b) proposition of an error measure to assess the quality of results, (c) sensitivity analysis of the proposed algorithms for the parameters involved and on the quality of the input data, and (d) study of the performance of the proposed algorithms vis-à-vis dominance relation preservation based algorithms, on a wide range of test problems (scaled up to 50 objectives) and two real-world problems