Monotonicity of the Sharpe ratio along the long-short LASSO path

Determine whether the Sharpe ratio must increase monotonically from corner to corner along the long-short LASSO path under a gross-exposure cap.

Background

The paper identifies the LASSO path with the efficient frontier for the mean–variance problem subject to a gross-exposure constraint. In the numerical test problem, the Sharpe ratio increases at every successive corner, from the first sparse portfolio to the unconstrained Kelly and maximum-Sharpe endpoint.

The authors explicitly state that they have not established whether this observed increase is necessary. Thus, the unresolved issue is whether monotonic Sharpe-ratio growth is a general property of the corner sequence generated by the long-short LASSO path, rather than an empirical feature of the particular test instance.

References

On the test problem the Sharpe ratio rises from corner to corner, from $0.539$ at the first to $\sqrt{\mu{-1}\mu}=2.120$ at the end; we have not shown that it must.

— The Efficient Frontier from a LASSO Solver  (2609.37108 - Schmelzer, 29 Sep 2026) in Section 4, “Long--short with a leverage cap” (Section \ref{sec:longshort})