Constructing new weighted ℓ₁-algorithms for the sparsest points of polyhedral sets

The ℓ₀-minimization problem that seeks the sparsest point of a polyhedral set is a longstanding challenging problem in the fields of signal and image processing, numerical linear algebra and mathematical optimization. The weighted ℓ₁-method is one of the most plausible methods for solving this problem. In this paper, we develop a new weighted ℓ₁-method through the strict complementarity theory of linear programs. More specifically, we show that locating the sparsest point of a polyhedral set can be achieved by seeking the densest possible slack variable of the dual problem of weighted ℓ₁ minimization. As a result, ℓ₀-minimization can be transformed, in theory, to ℓ₀-maximization in dual space through some weight. This theoretical result provides a basis and an incentive to develop a new weighted ℓ₁-algorithm, which is remarkably distinct from existing sparsity-seeking methods. The weight used in our algorithm is computed via a certain convex optimization instead of being determined locally at an iterate. The guaranteed performance of this algorithm is shown under some conditions, and the numerical performance of the algorithm has been demonstrated by empirical simulations.