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This paper introduces the Fej\'er-monotone hybrid steepest descent method(FM-HSDM), a new member to the HSDM family of algorithms, for solving affinelyconstrained minimization tasks in real Hilbert spaces, where convex smooth andnon-smooth losses compose the objective function. FM-HSDM offers sequences ofestimates which converge weakly and, under certain hypotheses, strongly tosolutions of the task at hand. Fixed-point theory, variational inequalities andaffine-nonexpansive mappings are utilized to devise a scheme that accommodatesaffine constraints in a more versatile way than state-of-the-art primal-dualtechniques and the alternating direction method of multipliers do. Recursionscan be tuned to score low computational footprints, well-suited for large-scaleoptimization tasks, without compromising convergence guarantees. In contrast toits HSDM's precursors, FM-HSDM enjoys Fej\'er monotonicity, the step-sizeparameter stays constant across iterations to promote convergence speed-ups ofthe sequence of estimates to a minimizer, while only Lipschitzian continuity,and not strong monotonicity, of the derivative of the smooth-loss function isneeded to ensure convergence. Results on the rate of convergence to an optimalpoint are also presented.

Fej\'er-monotone hybrid steepest descent method for affinely constrained and composite convex minimization tasks