# A Reduced-Space Algorithm for Minimizing $\ell_1hBcRegularized Convex Functions We present a new method for minimizing the sum of a differentiable convex function and an$\ell_1$-norm regularizer. The main features of the new method include:$(i)$an evolving set of indices corresponding to variables that are predicted to be nonzero at a solution (i.e., the support);$(ii)$a reduced-space subproblem defined in terms of the predicted support;$(iii)$conditions that determine how accurately each subproblem must be solved, which allow for Newton, Newton-CG, and coordinate-descent techniques to be employed;$(iv)$a computationally practical condition that determines when the predicted support should be updated; and$(v)\$ a reduced proximal gradient step that ensures sufficient decrease in the objective function when it is decided that variables should be added to the predicted support. We prove a convergence guarantee for our method and demonstrate its efficiency on a large set of model prediction problems.

## Citation

T. Chen, F. E. Curtis, and D. P. Robinson. A Reduced-Space Algorithm for Minimizing l1-Regularized Convex Functions. SIAM Journal on Optimization, 27(3):1583–1610, 2017.