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We consider a class of sparsity-inducing optimization problems whoseconstraint set is regularizer-compatible, in the sense that, the constraint setbecomes easy-to-project-onto after a coordinate transformation induced by thesparsity-inducing regularizer. Our model is general enough to cover, as specialcases, the ordered LASSO model and its variants with some commonly usednonconvex sparsity-inducing regularizers. The presence of both thesparsity-inducing regularizer and the constraint set poses challenges on thedesign of efficient algorithms. In this paper, by exploiting absolute-valuesymmetry and other properties in the sparsity-inducing regularizer, we proposea new algorithm, called the Doubly Majorized Algorithm (DMA), for this class ofproblems. The DMA makes use of projections onto the constraint set after thecoordinate transformation in each iteration, and hence can be performedefficiently. Without invoking any commonly used constraint qualificationconditions such as those based on horizon subdifferentials, we show that anyaccumulation point of the sequence generated by DMA is a so-called $\psi_{\rmopt}$-stationary point, a new notion of stationarity we define as inspired bythe notion of $L$-stationarity. We also show that any global minimizer of ourmodel has to be a $\psi_{\rm opt}$-stationary point, again without imposing anyconstraint qualification conditions. Finally, we illustrate numerically theperformance of DMA on solving variants of ordered LASSO with nonconvexregularizers.

Doubly majorized algorithm for sparsity-inducing optimization problems with regularizer-compatible constraints