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The convex envelopes of the direct discrete measures, for the sparsity ofvectors or for the low-rankness of matrices, have been utilized extensively aspractical penalties in order to compute a globally optimal solution of thecorresponding regularized least-squares models. Motivated mainly by the ideasin [Zhang'10, Selesnick'17, Yin, Parekh, Selesnick'19] to exploit nonconvexpenalties in the regularized least-squares models without losing their overallconvexities, this paper presents the Linearly involved Generalized MoreauEnhanced (LiGME) model as a unified extension of such utilizations of nonconvexpenalties. The proposed model can admit multiple nonconvex penalties withoutlosing its overall convexity and thus is applicable to much broader scenariosin the sparsity-rank-aware signal processing. Under the generaloverall-convexity condition of the LiGME model, we also present a novelproximal splitting type algorithm of guaranteed convergence to a globallyoptimal solution. Numerical experiments in typical examples of thesparsity-rank-aware signal processing demonstrate the effectiveness of theLiGME models and the proposed proximal splitting algorithm.

Linearly Involved Generalized Moreau Enhanced Models and Their Proximal Splitting Algorithm under Overall Convexity Condition