A　primary　challenge　in　fault　diagnosis　is　to　extract　multiple　components　entangled　within　a　noisy　observation.　Therefore,　this　paper　describes　and　analyzes　a　novel　framework,　based　on　convex　optimization,　for　simultaneously　identifying　multiple　features　from　superimposed　signals.　This　work　adequately　exploits　the　underlying　prior　information　that　multiple　faults　with　similar　frequency　spectrum　have　different　morphological　waveforms　that　can　be　sparsely　represented　over　the　union　of　redundant　dictionaries.　Within　this　framework,　prior　information　is　formulated　into　regularization　terms,　and　a　sparse　optimization　problem,　which　can　be　solved　through　the　alternating　direction　method　of　multipliers　(ADMM),　is　proposed.　Meanwhile,　the　convergence　and　computational　complexity　of　the　proposed　iterative　framework　are　profoundly　investigated.　Moreover,　sensitivity　analyses　and　adaptive　selection　rules　for　the　regularization　parameters　are　described　in　detail　through　a　set　of　comprehensive　numerical　studies.　The　proposed　framework　is　validated　through　performing　the　diagnosis　of　multiple　faults　for　gearbox　in　a　wind　farm.　The　comparison　with　respect　to　the　state　of　the　art　in　the　field　is　illustrated　in　detail,　which　highlights　the　superiority　of　the　proposed　framework.