 L1 exact penalty

An L1 penalty can enforce a constraint exactly at a finite weight, rather than only in an infinite-weight limit. For a convex equality-constrained problem with an optimal multiplier $y_{\ast }$ and optimization Lagrangian $f-y^{T}c$, the lower bound $f(x)-y_{\ast }^{T}c(x)\ge f(x_{\ast })$ givesThus $\rho >\Vert y_{\ast }{\Vert }_{\infty }$ makes every penalty minimizer feasible and optimal. Local versions require appropriate regularity and second-order assumptions. The nonsquared norm is essential to this finite-threshold property; a quadratic penalty method usually approaches feasibility only as its weight diverges.