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Standardization of Multi-Objective QUBOs

Loong Kuan Lee, Thore Thassilo Gerlach, Nico Piatkowski

TL;DR

The authors address the challenge of balancing multiple objectives in QUBO problems by proposing a variance-based standardization that scales each objective to unit variance, enabling fair equal-weight scalarization. They derive closed-form expressions for the mean and variance of objective values under a uniform binary distribution and present an $ obreakspace ext{O}(n^3)$ algorithm to compute the variance, enabling practical standardization. Empirical results across four QUBO problems show that standardization often improves the hypervolume of the Pareto set compared to unscaled or roof-dual-based normalization, highlighting its potential to yield more balanced trade-offs in no-preference settings. The approach offers a theoretically grounded, computationally efficient tool for multi-objective QUBO optimization with clear avenues for future work on constraints and alternative data distributions.

Abstract

Multi-objective optimization involving Quadratic Unconstrained Binary Optimization (QUBO) problems arises in various domains. A fundamental challenge in this context is the effective balancing of multiple objectives, each potentially operating on very different scales. This imbalance introduces complications such as the selection of appropriate weights when scalarizing multiple objectives into a single objective function. In this paper, we propose a novel technique for scaling QUBO objectives that uses an exact computation of the variance of each individual QUBO objective. By scaling each objective to have unit variance, we align all objectives onto a common scale, thereby allowing for more balanced solutions to be found when scalarizing the objectives with equal weights, as well as potentially assisting in the search or choice of weights during scalarization. Finally, we demonstrate its advantages through empirical evaluations on various multi-objective optimization problems. Our results are noteworthy since manually selecting scalarization weights is cumbersome, and reliable, efficient solutions are scarce.

Standardization of Multi-Objective QUBOs

TL;DR

The authors address the challenge of balancing multiple objectives in QUBO problems by proposing a variance-based standardization that scales each objective to unit variance, enabling fair equal-weight scalarization. They derive closed-form expressions for the mean and variance of objective values under a uniform binary distribution and present an algorithm to compute the variance, enabling practical standardization. Empirical results across four QUBO problems show that standardization often improves the hypervolume of the Pareto set compared to unscaled or roof-dual-based normalization, highlighting its potential to yield more balanced trade-offs in no-preference settings. The approach offers a theoretically grounded, computationally efficient tool for multi-objective QUBO optimization with clear avenues for future work on constraints and alternative data distributions.

Abstract

Multi-objective optimization involving Quadratic Unconstrained Binary Optimization (QUBO) problems arises in various domains. A fundamental challenge in this context is the effective balancing of multiple objectives, each potentially operating on very different scales. This imbalance introduces complications such as the selection of appropriate weights when scalarizing multiple objectives into a single objective function. In this paper, we propose a novel technique for scaling QUBO objectives that uses an exact computation of the variance of each individual QUBO objective. By scaling each objective to have unit variance, we align all objectives onto a common scale, thereby allowing for more balanced solutions to be found when scalarizing the objectives with equal weights, as well as potentially assisting in the search or choice of weights during scalarization. Finally, we demonstrate its advantages through empirical evaluations on various multi-objective optimization problems. Our results are noteworthy since manually selecting scalarization weights is cumbersome, and reliable, efficient solutions are scarce.

Paper Structure

This paper contains 11 sections, 2 theorems, 17 equations, 3 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Motivation
  3. Problem Setup
  4. Contribution
  5. Standardizing QUBO Objective Functions
  6. Experiment
  7. QUBO Problems
  8. Experimental setup
  9. Performance Measure: Hypervolume Indicator
  10. Results
  11. Conclusion

Key Result

Theorem 1

Assume we have the objective function $f(\boldsymbol{x}) = \boldsymbol{x}^{\top} \boldsymbol{Q} \boldsymbol{x}$ where $\boldsymbol{x}$ is a binary vector of length $n$ and $\boldsymbol{Q}$ is an $n\times n$ symmetric real-valued matrix. Furthermore, let $\mathbb{P}_{\boldsymbol{X}}$ be the uniform d and

Figures (3)

  • Figure 1: Plots over the distribution (b) and the objective space (a,c) of the objective functions $f(\boldsymbol{x})=\boldsymbol{x}^{\top}Q_{1}\boldsymbol{x}$ and $g(\boldsymbol{x})=\boldsymbol{x}^{\top}Q_{2}\boldsymbol{x}$. The $20\times20$ symmetric real-valued matrices $Q_{1}$ and $Q_{2}$ are randomly generated by the qubolite package mucke2025 with random_state seeds of $0$ and $1$ respectively. Additionally $Q_{2}$ is scaled up by a factor of $10$ as well.
  • Figure 2: Fast QUBO Variance in $\mathcal{O}(n^{3})$
  • Figure 3: Example of reference points (black) used for the Hypervolume calculations---that are later averaged over---in 2D objective space over problems MC$\{0,1\}$ and MC$[0,1]$.

Theorems & Definitions (2)

  • Theorem 1: Mean and variance of with uniform $\boldsymbol{X}$
  • Theorem 2: Variance of with independent Bernoulli distributions $\mathcal{B}(0.5)$