Abstract. We propose a new class of primal-dual Fejér monotone algorithms for solving systems of composite monotone inclusions. Our construction is inspired by a framework used by Eckstein and Svaiter for the basic problem of finding a zero of the sum of two monotone operators. At each iteration, points in the graph of the monotone operators present in the model are used to construct a half-space containing the Kuhn-Tucker set associated with the system. The primal-dual update is then obtained via a relaxed projection of the current iterate onto this half-space. An important feature that distinguishes the resulting splitting algorithms from existing ones is that they do not require prior knowledge of bounds on the linear operators involved or the inversion of linear operators.Key words. duality, Fejér monotonicity, monotone inclusion, monotone operator, primal-dual algorithm, splitting algorithm AMS subject classifications. Primary 47H05; Secondary 65K05, 90C25, 94A081. Introduction. The first monotone operator splitting methods arose in the late 1970s and were motivated by applications in mechanics and partial differential equations [32,35,39]. In recent years, the field of monotone operator splitting algorithms has benefited from a new impetus, fueled by emerging application areas such as signal and image processing, statistics, optimal transport, machine learning, and domain decomposition methods [3,5,24,36,41,43,46]. Three main algorithms dominate the field explicitly or implicitly: the forward-backward method [38], the Douglas-Rachford method [37], and the forward-backward-forward method [47]. These methods were originally designed to solve inclusions of the type 0 ∈ Ax + Bx, where A and B are maximally monotone operators acting on a Hilbert space (via product space reformulations, they can also be extended to problems involving sums of more than 2 operators [9,45]). Until recently, a significant challenge in the field was to design splitting techniques for inclusions involving linearly composed operators, say