Co-inertia analysis (CIA) is a multivariate statistical analysis method that can assess relationships and trends in two sets of data. Recently CIA has been used for an integrative analysis of multiple high-dimensional omics data. However, for classical CIA, all elements in the loading vectors are nonzero, presenting a challenge for the interpretation when analyzing omics data. For other multivariate statistical methods such as canonical correlation analysis (CCA), penalized least squares (PLS), various approaches have been proposed to produce sparse loading vectors via l1-penalization/constraint. We propose a novel CIA method that uses l1-penalization to induce sparsity in estimators of loading vectors. Our method simultaneously conducts model fitting and variable selection. Also, we propose another CIA method that incorporates structure/network information such as those from functional genomics, besides using sparsity penalty so that one can get biologically meaningful and interpretable results.