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The design of sparse arrays able to radiate focused beam patterns satisfying a given upper-bound power mask with the minimum number of sources is a research area of increasing interest. The related synthesis problem can be formulated with proper constraints on the cardinality of the solution space, i.e., its l0-norm. Unfortunately, such a nonconvex constraint requires to solve an NP-hard problem. Interesting ideas to relax the above constraint in a convex way have been successfully proposed. A possible solution is based on the minimization of the l1-norm. This strategy is not always able to achieve a maximally sparse solution. In the following, an innovative synthesis scheme that optimizes both excitation weights and sensor positions of an array radiating pencil beam-patterns is discussed. The solution algorithm is based on sequential convex optimizations including a reweighted l1 -norm minimization. Numerical tests, referred to benchmark problems, show that the proposed synthesis method is able to achieve maximally sparse linear arrays, also compared to the best results reported in the literature, obtained by means of global optimization schemes.