Summary: A descent algorithm of nonsmooth convex optimization is given, which is a modification and an extension of an algorithm introduced by M. Fukushima [Math. Program. 30, 163-175 (1984; Zbl 0545.90082)] and C. Lemarechal and R. Mifflin [ibid. 24, 241-256 (1982; Zbl 0505.90064)]. Global convergence for the algorithm is established under the assumption that the objective function is bounded from below. The authors also generalize the algorithm to constrained convex optimization and obtain convergence as for the unconstrained algorithm. If constraints are linear the algorithm is implementable.