Dealing with Structure Constraints in Evolutionary Pareto Set Learning

TL;DR

This work presents Evolutionary Pareto Set Learning (EPSL), a model-based framework that learns a parametric Pareto set and can explicitly impose structure constraints on the entire solution set. By representing the Pareto set as $\boldsymbol{x}=h_{\boldsymbol{\theta}}(\boldsymbol{\lambda})$ and optimizing via evolutionary stochastic gradient methods under smooth Tchebycheff aggregation, EPSL can cover all preferences with a single model. The authors introduce several structure constraints—shared components, learnable variable relationships, and shape constraints (including polygonal chains)—and demonstrate that EPSL can closely approximate the full Pareto set while enabling labeled structure or simple, interpretable representations. Across 16 RE-engineering problems, EPSL achieves competitive hypervolume performance with MOEAs while offering direct sampling of a structured Pareto set, enabling more flexible and informative decision-making in practice.

Abstract

In the past few decades, many multiobjective evolutionary optimization algorithms (MOEAs) have been proposed to find a finite set of approximate Pareto solutions for a given problem in a single run, each with its own structure. However, in many real-world applications, it could be desirable to have structure constraints on the entire optimal solution set, which define the patterns shared among all solutions. The current population-based MOEAs cannot properly handle such requirements. In this work, we make the first attempt to incorporate the structure constraints into the whole solution set by a single Pareto set model, which can be efficiently learned by a simple evolutionary stochastic optimization method. With our proposed method, the decision-makers can flexibly trade off the Pareto optimality with preferred structures among all solutions, which is not supported by previous MOEAs. A set of experiments on benchmark test suites and real-world application problems fully demonstrates the efficiency of our proposed method.

Figures (14)

• Figure 1: Evolutionary Pareto Set Learning (EPSL):(a)The forward pass ($\color{forward} \rightarrow$) starts from the preference $\boldsymbol{\lambda} \in \mathbb{R}^m$ to the model $h_{\boldsymbol{\theta}}(\boldsymbol{\lambda})$ with parameter $\theta \in \mathbb{R}^d$ to the solution $\boldsymbol{x} \in \mathbb{R}^n$ to the objectives $\boldsymbol{F}(\boldsymbol{x}) \in \mathbb{R}^m$ and finally to the preference-conditioned subproblem scalar $g_{\text{stch}}(\boldsymbol{x}|\boldsymbol{\lambda}) \in \mathbb{R}^1$. The learnable model parameter$\theta$ is the only variable to optimize in this model. (b)The backward gradient $\nabla_{\boldsymbol{\theta}} g_{\text{stch}}$ ($\color{backward} \leftarrow$) from the subproblem scalar $g_{\text{stch}}(\boldsymbol{x}|\boldsymbol{\lambda}) \in \mathbb{R}^1$ with respect to the learnable model parameter $\boldsymbol{\theta} \in \mathbb{R}^d$ can be decomposed into a low-dimensional gradient $\nabla_{\boldsymbol{x}} g_{\text{stch}}$ and a high-dimensional Jacobian matrix $\frac{\partial h_{\boldsymbol{\theta}}(\boldsymbol{\lambda}) }{\partial \boldsymbol{\theta}}$, where the first term can be efficiently estimated by evolution strategies and the second term can be directly calculated by backpropagation.
• Figure 2: EPSL Model for solution set with shared components. The variables to optimize are the model parameters $\boldsymbol{\theta}$ and part of the decision variables $\boldsymbol{x}_{\boldsymbol{s}}=\boldsymbol{\beta}$ that shared by all trade-off solutions.
• Figure 3: EPSL Model for solution set with learnable variable relationship constraints. The variables to optimize are the model parameters $\boldsymbol{\theta}$ and the learnable relation expression $g_{\boldsymbol{\phi}}$ from some base decision variables $\boldsymbol{x}_{\boldsymbol{p}}$ to the dependent decision variables $\boldsymbol{x}_{\boldsymbol{s}}$.
• Figure 4: EPSL Model for solution set with shape structure constraint. The variables to optimize are the model parameters $\boldsymbol{\theta}$ and the parameters $\boldsymbol{\phi}$ for the shape function $g_{\boldsymbol{\phi}}(\boldsymbol{u})$.
• Figure 5: Convergence of our Proposed Evolutionary Pareto Set Learning (EPSL) Method: EPSL gradually pushes the set model to ground truth Pareto set via iteratively reducing the corresponding subproblems values of the sampled solutions. It does not require any Pareto-optimal solution just like the traditional MOEA counterparts which gradually push a set of solutions (e.g., population) to the ground truth Pareto set.

Theorems & Definitions (4)

• Definition 1: Pareto Dominance
• Definition 2: Strict Dominance
• Definition 3: Pareto Optimality
• Definition 4: Weakly Pareto Optimality