This　paper　is　concerned　with　the　1,∞-norm　ball　constrained　multi-Task　learning　problem,　which　has　received　extensive　attention　in　many　research　areas　such　as　machine　learning,　cognitive　neuroscience,　and　signal　processing.　To　address　the　challenges　of　solving　large-scale　multi-Task　Lasso　problems,　this　paper　develops　an　inexact　semismooth　Newton-based　augmented　Lagrangian　(Ssnal)　algorithm.　When　solving　the　inner　problems　in　the　Ssnal　algorithm,　the　semismooth　Newton　(Ssn)　algorithm　with　superlinear　or　even　quadratic　convergence　is　applied.　Theoretically,　this　paper　presents　the　global　and　asymptotically　superlinear　local　convergence　of　the　Ssnal　algorithm　under　standard　conditions.　Computationally,　we　derive　an　efficient　procedure　to　construct　the　generalized　Jacobian　of　the　projector　onto　1　,∞-norm　ball,　which　is　an　important　component　of　the　Ssnal　algorithm,　making　the　computational　cost　in　the　Ssn　algorithm　very　cheap.　Comprehensive　numerical　experiments　on　the　multi-Task　Lasso　problems　demonstrate　that　the　Ssnal　algorithm　is　more　efficient　and　robust　than　several　existing　state-of-The-Art　first-order　algorithms.　©　2024　World　Scientific　Publishing　Co.