We propose a new method for selecting a common subset of explanatory variables where the aim is to explain or predict several response variables. The basic idea is a natural extension of the LASSO technique proposed by Tibshirani (1996) based on minimising the (joint) residual sum of squares while constraining the parameter estimates to lie within a suitable polyhedral region. This leads to a convex programming problem for which we develop an e#cient interior point algorithm. The method is illustrated on a data set with infra-red spectrometry measurements on 14 qualitatively di#erent but correlated responses using 770 wavelengths. The aim is to select a subset of the wavelengths suitable to use as predictors for as many as possible of the responses. KEY WORDS AND PHRASES. Constrained least squares problem, constrained regression, convex programming, infra--red spectrometry, interior point algorithm, quadratic programming, subset selection, variable selection. # Department of Statistics...