1. A method for L 1 constraint optimization under box constrains, comprising: providing a data processing device to implement the following steps: (I) accessing a vector v′ to be projected onto an L 1 -norm of value z, where v′ε ; (II) taking the absolute value of the elements of v′ to create a transformed vector v, where vε ; (III) accessing bound constraints [a,b], where [a] is a vector of lower bound constraints and [b] is a vector of upper bound constraints; (IV) transforming bound constraints [a,b] to [0,b] by identifying equivalent positive upper bounds; (V) identifying a projection vector solution, x, as the projection of vector v onto the L 1 -norm as limited by upper bounds b; (VI) defining a projection vector solution x′ by performing element-wise multiplication of x by the sign of v′ according to x′=x⊙sign (v′).