Enhanced Ideal Objective Vector Estimation for Evolutionary Multi-Objective Optimization

TL;DR

The paper addresses the challenge of estimating the ideal objective vector $\mathbf{z}^{ide}$ in multi-objective optimization under bias. It shows population-based estimates falter in distance-, position-, and mixed-biased spaces and introduces a biased test-problem generator to systematically evaluate this issue. The proposed Enhanced Ideal Objective Vector Estimation (EIE) framework uses $m$ extreme weighted-sum subproblems solved by PSA-CMA-ES in parallel to align the search with $\mathbf{z}^{ide}$, while dynamically allocating computational resources. Across 16 biased test instances and 55 bbob-biobj problems, EIE improves both the estimation accuracy and the MOEA’s approximation of the PF, validating its broad applicability and potential to improve MOEA performance in biased objective spaces.

Abstract

The ideal objective vector, which comprises the optimal values of the $m$ objective functions in an $m$-objective optimization problem, is an important concept in evolutionary multi-objective optimization. Accurate estimation of this vector has consistently been a crucial task, as it is frequently used to guide the search process and normalize the objective space. Prevailing estimation methods all involve utilizing the best value concerning each objective function achieved by the individuals in the current or accumulated population. However, this paper reveals that the population-based estimation method can only work on simple problems but falls short on problems with substantial bias. The biases in multi-objective optimization problems can be divided into three categories, and an analysis is performed to illustrate how each category hinders the estimation of the ideal objective vector. Subsequently, a set of test instances is proposed to quantitatively evaluate the impact of various biases on the ideal objective vector estimation method. Beyond that, a plug-and-play component called enhanced ideal objective vector estimation (EIE) is introduced for multi-objective evolutionary algorithms (MOEAs). EIE features adaptive and fine-grained searches over $m$ subproblems defined by the extreme weighted sum method. EIE finally outputs $m$ solutions that can well approximate the ideal objective vector. In the experiments, EIE is integrated into three representative MOEAs. To demonstrate the wide applicability of EIE, algorithms are tested not only on the newly proposed test instances but also on existing ones. The results consistently show that EIE improves the ideal objective vector estimation and enhances the MOEA's performance.

Figures (17)

• Figure 1: Plots of 10000 random solutions. The $PF$ of (b) is taken from tanabe2020easy. The $PF$ of (c) is obtained by the exact solver.
• Figure 2: Plots of random solutions on a two-objective problem with different biases. Letting $x_1,x_2\in[0,1]$, we have: (a) $f_1(x_1,x_2)=x_1+x_2^{0.3}$, $f_2(x_1,x_2)=(1-x_1)+x_2^{0.3}$; (b) if $x_1<0.5$, $f_1(x_1,x_2)=\frac{(2x)^{0.3}}{2}+x_2$ and $f_2(x_1,x_2)=1-\frac{(2x)^{0.3}}{2}+x_2$, or else $f_1(x_1,x_2)=1-\frac{(2-2x)^{0.3}}{2}+x_2$ and $f_2(x_1,x_2)=\frac{(2-2x)^{0.3}}{2}+x_2$; (c) if $x_1<0.5$, $f_1(x_1,x_2)=\frac{(2x)^{0.3}}{2}+x_2^{0.3}$ and $f_2(x_1,x_2)=1-\frac{(2x)^{0.3}}{2}+x_2^{0.3}$, or else $f_1(x_1,x_2)=1-\frac{(2-2x)^{0.3}}{2}+x_2^{0.3}$ and $f_2(x_1,x_2)=\frac{(2-2x)^{0.3}}{2}+x_2^{0.3}$.
• Figure 3: The comparison between the position-related bias and distance-related bias.
• Figure 4: Plots of the population-based estimated ideal objective vector on problems with distance-related bias.
• Figure 5: Plots of the population-based estimated ideal objective vector on problems with position-related bias.

Theorems & Definitions (2)

• Theorem 1
• proof