This　paper　focuses　on　efficient　algorithms　for　finding　the　Dantzig　selector　which　was　first　proposed　by　Cande`\s　and　Tao　as　an　effective　variable　selection　technique　in　the　linear　regression.　This　paper　first　reformulates　the　Dantzig　selector　problem　as　an　equivalent　convex　composite　optimization　problem　and　proposes　a　semismooth　Newton　augmented　Lagrangian　(SSNAL)　algorithm　to　solve　the　equivalent　form.　This　paper　also　applies　a　proximal　point　dual　semismooth　Newton　(PPDSSN)　algorithm　to　solve　another　equivalent　form　of　the　Dantzig　selector　problem.　Comprehensive　results　on　the　global　convergence　and　local　asymptotic　superlinear　convergence　of　the　SSNAL　and　PPDSSN　algorithms　are　characterized　under　very　mild　conditions.　The　computational　costs　of　a　semismooth　Newton　algorithm　for　solving　the　subproblems　involved　in　the　SSNAL　and　PPDSSN　algorithms　can　be　cheap　by　fully　exploiting　the　second　order　sparsity　and　employing　efficient　techniques.　Numerical　experiments　on　the　Dantzig　selector　problem　with　synthetic　and　real　data　sets　demonstrate　that　the　SSNAL　and　PPDSSN　algorithms　substantially　outperform　the　state-of-the-art　first　order　algorithms　even　for　the　required　low　accuracy,　and　the　proposed　algorithms　are　able　to　solve　the　large-scale　problems　robustly　and　efficiently　to　a　relatively　high　accuracy.