  This paper introduces the Fej\'er-monotone hybrid steepest descent method
(FM-HSDM), a new member to the HSDM family of algorithms, for solving affinely
constrained minimization tasks in real Hilbert spaces, where convex smooth and
non-smooth losses compose the objective function. FM-HSDM offers sequences of
estimates which converge weakly and, under certain hypotheses, strongly to
solutions of the task at hand. Fixed-point theory, variational inequalities and
affine-nonexpansive mappings are utilized to devise a scheme that accommodates
affine constraints in a more versatile way than state-of-the-art primal-dual
techniques and the alternating direction method of multipliers do. Recursions
can be tuned to score low computational footprints, well-suited for large-scale
optimization tasks, without compromising convergence guarantees. In contrast to
its HSDM's precursors, FM-HSDM enjoys Fej\'er monotonicity, the step-size
parameter stays constant across iterations to promote convergence speed-ups of
the sequence of estimates to a minimizer, while only Lipschitzian continuity,
and not strong monotonicity, of the derivative of the smooth-loss function is
needed to ensure convergence. Results on the rate of convergence to an optimal
point are also presented.
