algorithm approximating xargmin > (xwith logarithmic barrier inputan initial guess and stopping error outputthe approximate nonnegative minimizer xn of xb /sdx while kdxk and budget is not exhausted do compute the gradient and the hessian of at /ss /sw if ( diag( ) then /if cholesky successful compute the cholesky factor satisfying llh diag( dx - (computed by forward substitution dx -dx (computed by backward substitution else dx / /if cholesky failsdo steepest descent ds dx sdb dsa while min { ds do / /ensure nonnegative slack variables dxreturn xn dsb db further reading for an excellent introduction to convex optimization and lagrangian duality see [ classical text on optimization algorithms andin particularon quasi-newton methods is [ for more details on the alternating direction method of multipliers see [