On Choosing Initial Values of Iteratively Reweighted ${\mathrm{\ell }}_1$ Algorithms for the Piece-wise Exponential Penalty

Computing the proximal operator of the sparsity-promoting piece-wise exponential (PiE) penalty $1-e^{-\mathrm{\mid }x\mathrm{\mid }\mathrm{/}\sigma }$ with a given shape parameter $\sigma >0$, which is treated as a popular nonconvex surrogate of ${\mathrm{\ell }}_0$-norm, is fundamental in feature selection via support vector machines, image reconstruction, zero-one programming problems, compressed sensing, etc. Due to the nonconvexity of PiE, for a long time, its proximal operator is frequently evaluated via an iteratively reweighted ${\mathrm{\ell }}_1$ algorithm, which substitutes PiE with its first-order approximation, however, the obtained solutions only are the critical point. Based on the exact characterization of the proximal operator of PiE, we explore how the iteratively reweighted ${\mathrm{\ell }}_1$ solution deviates from the true proximal operator in certain regions, which can be explicitly identified in terms of $\sigma$, the initial value and the regularization parameter in the definition of the proximal operator. Moreover, the initial value can be adaptively and simply chosen to ensure that the iteratively reweighted ${\mathrm{\ell }}_1$ solution belongs to the proximal operator of PiE.