Sparse Optimization Theory and Methods

is a solution to the system (2.46). Therefore, it is the unique solution to this system since the coefficient matrix has full column rank. In summary, under the assumption of the theorem, the solution (v,t,α,β,r) of (2.11) is uniquely determined by (2.37), 2.38), 2.39), (2.40), (2.45) and (2.47). It is easy to see that this solution is exactly the one given by

$\left(v,t,\alpha ,\beta ,r\right)=\left(\rho e,\left|x\right|,\left|x\right|-x,\left|x\right|+x,0\right).$

Thus, the problem (2.11) has a unique solution. By Lemma 2.2.1, x must be the unique least ℓ₁-norm solution to the system Ax = b. □

Clearly, if (A_S+,A_S_) (equally, A_supp(x)) has full column rank, so is the matrix H defined in (2.29). Thus, an immediate consequence of Theorem 2.2.11 is the following result due to Fuchs [111].

Corollary 2.2.12 (Fuchs ...