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Few for Many: Tchebycheff Set Scalarization for Many-Objective Optimization

Xi Lin, Yilu Liu, Xiaoyuan Zhang, Fei Liu, Zhenkun Wang, Qingfu Zhang

TL;DR

This work addresses the challenge of many-objective optimization where the Pareto set explodes combinatorially as the number of objectives grows. It introduces the Tchebycheff set (TCH-Set) scalarization and its smooth variant (STCH-Set) to find a small set of $K$ representative solutions that collaboratively cover $m$ objectives, with $K \ll m$. The authors provide theoretical guarantees linking the set-based scalarizations to Pareto optimality (STCH-Set offers stronger guarantees), and demonstrate via extensive experiments that STCH-Set achieves superior worst-case performance and competitive average performance on convex and noisy nonlinear tasks, while offering favorable runtime compared to MGDA-based or transport-based methods. The approach offers a practical pathway to handle many-objective problems in real-world settings by balancing solution economy with objective coverage, potentially enabling efficient model portfolios, design alternatives, and scalable decision-support in high-dimensional objective spaces.

Abstract

Multi-objective optimization can be found in many real-world applications where some conflicting objectives can not be optimized by a single solution. Existing optimization methods often focus on finding a set of Pareto solutions with different optimal trade-offs among the objectives. However, the required number of solutions to well approximate the whole Pareto optimal set could be exponentially large with respect to the number of objectives, which makes these methods unsuitable for handling many optimization objectives. In this work, instead of finding a dense set of Pareto solutions, we propose a novel Tchebycheff set scalarization method to find a few representative solutions (e.g., 5) to cover a large number of objectives (e.g., $>100$) in a collaborative and complementary manner. In this way, each objective can be well addressed by at least one solution in the small solution set. In addition, we further develop a smooth Tchebycheff set scalarization approach for efficient optimization with good theoretical guarantees. Experimental studies on different problems with many optimization objectives demonstrate the effectiveness of our proposed method.

Few for Many: Tchebycheff Set Scalarization for Many-Objective Optimization

TL;DR

This work addresses the challenge of many-objective optimization where the Pareto set explodes combinatorially as the number of objectives grows. It introduces the Tchebycheff set (TCH-Set) scalarization and its smooth variant (STCH-Set) to find a small set of representative solutions that collaboratively cover objectives, with . The authors provide theoretical guarantees linking the set-based scalarizations to Pareto optimality (STCH-Set offers stronger guarantees), and demonstrate via extensive experiments that STCH-Set achieves superior worst-case performance and competitive average performance on convex and noisy nonlinear tasks, while offering favorable runtime compared to MGDA-based or transport-based methods. The approach offers a practical pathway to handle many-objective problems in real-world settings by balancing solution economy with objective coverage, potentially enabling efficient model portfolios, design alternatives, and scalable decision-support in high-dimensional objective spaces.

Abstract

Multi-objective optimization can be found in many real-world applications where some conflicting objectives can not be optimized by a single solution. Existing optimization methods often focus on finding a set of Pareto solutions with different optimal trade-offs among the objectives. However, the required number of solutions to well approximate the whole Pareto optimal set could be exponentially large with respect to the number of objectives, which makes these methods unsuitable for handling many optimization objectives. In this work, instead of finding a dense set of Pareto solutions, we propose a novel Tchebycheff set scalarization method to find a few representative solutions (e.g., 5) to cover a large number of objectives (e.g., ) in a collaborative and complementary manner. In this way, each objective can be well addressed by at least one solution in the small solution set. In addition, we further develop a smooth Tchebycheff set scalarization approach for efficient optimization with good theoretical guarantees. Experimental studies on different problems with many optimization objectives demonstrate the effectiveness of our proposed method.
Paper Structure (59 sections, 4 theorems, 48 equations, 4 figures, 11 tables, 1 algorithm)

This paper contains 59 sections, 4 theorems, 48 equations, 4 figures, 11 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries and Related Work
  3. Multi-Objective Optimization
  4. Gradient-based Multi-Objective Optimization
  5. Multiple Gradient Descent Algorithm
  6. Scalarization Method
  7. Tchebycheff Set Scalarization for Many Objective Optimization
  8. Small Solution Set for Many-Objective Optimization
  9. Tchebycheff Set Scalarization
  10. Smooth Tchebycheff Set Scalarization
  11. Experiments
  12. Baseline Methods
  13. Experimental Setting
  14. Experimental Results and Analysis
  15. Convex Many-Objective Optimization
  16. ...and 44 more sections

Key Result

Theorem 1

There exists an optimal solution set $\bar{\boldsymbol{X}}^*_K$ for the Tchebycheff set scalarization optimization problem (eq_tch_set_scalarization) such that all solutions in $\bar{\boldsymbol{X}}^*_K$ are Pareto optimal of the original multi-objective optimization problem (eq_mop). In addition, i

Figures (4)

  • Figure 1: Large Set v.s. Small Set for Multi-Objective Optimization.(a)(b)(c) Large Set: Classic algorithms use $10$, $100$ and $1000$ solutions to approximate the whole Pareto front for $2$ and $3$-objective optimization problems. The required number of solutions for a good approximation could increase exponentially with the number of objectives. (d) Small Set: This work investigates how to efficiently find a few solutions (e.g., $5$) to collaboratively handle many optimization objectives (e.g., $100$).
  • Figure 2: Few Solutions to Address Many Optimization Objectives.(a)-(e):$5$ different solutions to tackle different optimization objectives in a complementary manner. (f): They together successfully handle all $100$ optimization objectives.
  • Figure 3: Different methods' performance for the same mixed nonlinear regression problem. We report the performance on the same set of $100$ randomly sampled objectives. STCH-Set can properly address all objectives and achieve the best overall performance. TCH-Set has a much better worst objective value than LS/TCH/STCH but is not reflected in this figure.
  • Figure 4: The effect of different smoothing parameters $\mu$ for the smooth minimization function:(a) The function $\mathtt{min} \{f_1(x), f_2(x), f_3(x) \}$ is highly non-convex even when all $\{f_i(x)\}_{i=1}^{3}$ are convex. (b) The smooth $\mathtt{smin}$ function has a better optimization landscape, especially with large $\mu$. When $\mu \rightarrow 0$, $\mathtt{smin}$ converges to $\mathtt{min}$.

Theorems & Definitions (12)

  • Definition 1: Dominance and Strict Dominance
  • Definition 2: (Weakly) Pareto Optimality
  • Definition 3: Pareto Set and Pareto Front
  • Definition 4: Pareto Stationary Solution
  • Theorem 1: Existence of Pareto Optimal Solution for Tchebycheff Set Scalarization
  • Theorem 2: Pareto Optimality for Smooth Tchebycheff Set Scalarization
  • Theorem 3: Uniform Smooth Approximation
  • Theorem 4: Convergence to Pareto Stationary Solution
  • proof
  • proof
  • ...and 2 more