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Approximation of a Pareto Set Segment Using a Linear Model with Sharing Variables

Ping Guo, Qingfu Zhang, Xi Lin

TL;DR

The paper tackles approximating a local portion of the Pareto Set for continuous multiobjective optimization under a constraint of variable sharing. It introduces a two-part performance metric combining an expected Chebyshev-aggregation loss over a neighborhood of preferences and a variable-sharing penalty, then models the local PS with a sparse linear map $h_\theta(\lambda)=A(\lambda_{1:m-1}-\lambda_{1:m-1}^0)+b$ and enforces shared structure via the row-$2,1$ norm $\|A\|_{2,1}$. The optimization proceeds within a MOEA/D-LLA framework that alternates data-driven regression to learn $A,b$ from $(\lambda,x)$ pairs and population-based search, with solutions generated from both genetic operators and model-based sampling. Experiments on a none-shared MOZDT1 instance and standard ZDT/DTLZ benchmarks show the method can produce a local PS approximation with controllable variable sharing: increasing the weight $\gamma$ promotes sharing but may reduce optimality, while the linear-model predictions often outperform the baseline MOEA/D-DE in several settings. Overall, the work offers a practical path to reuse designs and reduce costs by preserving shared components while focusing search on a local Pareto-approximation, with future directions toward broader benchmarks and deeper learning-based strategies.

Abstract

In many real-world applications, the Pareto Set (PS) of a continuous multiobjective optimization problem can be a piecewise continuous manifold. A decision maker may want to find a solution set that approximates a small part of the PS and requires the solutions in this set share some similarities. This paper makes a first attempt to address this issue. We first develop a performance metric that considers both optimality and variable sharing. Then we design an algorithm for finding the model that minimizes the metric to meet the user's requirements. Experimental results illustrate that we can obtain a linear model that approximates the mapping from the preference vectors to solutions in a local area well.

Approximation of a Pareto Set Segment Using a Linear Model with Sharing Variables

TL;DR

The paper tackles approximating a local portion of the Pareto Set for continuous multiobjective optimization under a constraint of variable sharing. It introduces a two-part performance metric combining an expected Chebyshev-aggregation loss over a neighborhood of preferences and a variable-sharing penalty, then models the local PS with a sparse linear map and enforces shared structure via the row- norm . The optimization proceeds within a MOEA/D-LLA framework that alternates data-driven regression to learn from pairs and population-based search, with solutions generated from both genetic operators and model-based sampling. Experiments on a none-shared MOZDT1 instance and standard ZDT/DTLZ benchmarks show the method can produce a local PS approximation with controllable variable sharing: increasing the weight promotes sharing but may reduce optimality, while the linear-model predictions often outperform the baseline MOEA/D-DE in several settings. Overall, the work offers a practical path to reuse designs and reduce costs by preserving shared components while focusing search on a local Pareto-approximation, with future directions toward broader benchmarks and deeper learning-based strategies.

Abstract

In many real-world applications, the Pareto Set (PS) of a continuous multiobjective optimization problem can be a piecewise continuous manifold. A decision maker may want to find a solution set that approximates a small part of the PS and requires the solutions in this set share some similarities. This paper makes a first attempt to address this issue. We first develop a performance metric that considers both optimality and variable sharing. Then we design an algorithm for finding the model that minimizes the metric to meet the user's requirements. Experimental results illustrate that we can obtain a linear model that approximates the mapping from the preference vectors to solutions in a local area well.
Paper Structure (18 sections, 12 equations, 8 figures, 1 table, 2 algorithms)

This paper contains 18 sections, 12 equations, 8 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Performance Metric for Local Models
  3. Local Approximation Metric
  4. Shared Variable Metric
  5. Linear Sparse Representation of the Local PS
  6. Linear Model
  7. Variable Sharing and Sparsity of the model
  8. Method and Algorithm
  9. An Alternative Problem
  10. Algorithm Framework
  11. Initialization of the Preference Set
  12. Main Algorithm
  13. Sampling New Solutions
  14. Experimental Results
  15. Parameter Setting
  16. ...and 3 more sections

Figures (8)

  • Figure 1: Linear Model: Column vectors (e.g. $a_1, a_2$) in $A$ can be regarded as basis vectors that span the subspace of a hyperplane. This hyperplane can be used to approximate the local PS segment since we expect it to be linear under mild conditions.
  • Figure 2: Preference Vector Generation: This process is equivalent to sampling from $\mathcal{N}(\lambda^0, \sigma I)$ and projecting them on the simplex.
  • Figure 3: Linear Approximation for Local PS: An illustration of the population and the predictions of the model for a nonlinear Pareto set under different $\gamma$ in both decision space and objective space.
  • Figure 4: (a) The value of R-IGD calculated on the population from the last iteration and the predictions given by the model. The area within one standard deviation is shaded. (b) The variance of decision variables of the predictions. The model's predictions converge to a small area as $\gamma$ increases.
  • Figure 5: Mean squared errors between the true optimal solutions and the model's predictions for ZDT1 and DTLZ1. The results are the average value of 10 independent runs and are plotted on logarithmic axis.
  • ...and 3 more figures