# A selection index with minimal genetic relatedness for multi-trait data via binary quadratic programming

## TL;DR

This paper introduces a new method for plant breeding that selects superior individuals while maintaining genetic diversity by using a quadratic programming framework.

## Contribution

The novel contribution is a binary quadratic programming framework for multi-trait selection that optimizes genetic gain and minimizes genetic relatedness.

## Key findings

- The proposed QPMSI method outperformed LPMSI in balancing genetic gain and relatedness.
- QPMSI achieved at least a 53.8% improvement in the MV metric across five datasets.
- The framework is computationally efficient and suitable for sustainable breeding strategies.

## Abstract

Genomic selection (GS) in plant breeding aims to identify individuals with superior genetic merit while maintaining genetic diversity within populations. In plant breeding, considering multiple traits simultaneously makes optimizing selection complex, especially under genetic relatedness constraints. In this study, we propose a binary quadratic programming framework for constructing a multi-trait selection index that maximizes genetic gain while minimizing average pairwise relatedness appropriate for identifying superior candidates for advancement in the breeding pipeline. The approach combines estimated breeding values (EBVs) across multiple traits by applying trait-specific economic weights, while simultaneously accounting for coancestry through the genomic relationship matrix. By formulating the selection problem as a constrained Quadratic Programing Multi-trait Selection Index (QPMSI), our method enables the identification of a fixed number of candidate individuals that jointly optimize selection index values and control genetic relatedness. We evaluated the performance of the proposed method using five real genomic datasets and demonstrated that it provides a more effective balance between selection response and control of genetic relatedness than the Linear Programming Multi-trait Selection Index (LPMSI). In particular, the QPMSI consistently outperformed the LPMSI in terms of the MV metric (gain-to-degree of relatedness ratio), achieving improvements of at least 53.8%. This framework offers a practical and computationally efficient tool for sustainable breeding strategies in multi-trait selection contexts.