Index selection with nonlinear profit function as a tool to achieve simultaneous genetic gain

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Abstract
Simultaneous improvement of several, and often negatively correlated, traits is frequently a desired objective in forest tree breeding. A profit function that includes a combination of both linear weights and weights for the cross-products of trait combinations facilitates the construction of a linear index, with an attractive response in all traits. A detailed algorithm for finding the index coefficients is provided, along with three examples of applications in tree breeding. The index is also a powerful tool in optimizing the selection for a ratio of two traits. It is argued that a more equal progress in several traits provides a safetey net when faced with economic uncertainties. The provided algorithm eliminates the need for direct search techniques. Existence of a dual set of linear weights means that the statistical properties of the index based on nonlinear profit functions are identical to those of the classical Smith-Hazel type of index.