The Application of ADMM Algorithm in Optimization Problem with Absolute Terms

Abstract

Optimizations for objective functions with absolute terms appear frequently in practical problems, like classical least square method with absolute penalty (lasso), least absolute deviation (LAD) regression and graphical lasso with absolute penalty all have absolute terms in their objective functions. Corresponding algorithms have been given when the problems were proposed, for example, least angle regression (LARS) and coordinate descent algorithms for lasso, linear programming for LAD and glasso for Gaussian graphical model Although they solve the problems correctly, they are not uniform and can be dramatically improved in efficiency. Using the alternating direction method of multipliers (ADMM) algorithms, we established a general framework to solve problems like these. And we have conducted simulation experiments under different parameter settings, and the simulation results showed that the efficiency of ADMM algorithm is higher than or comparable to, that of existing methods.