The author proposes a new algorithm of the ``projected gradient'' type to compute an \(\ell_{\infty}\)-solution of an overdetermined system of linear equations with linear constraints, i.e. to solve the problem: minimize \(F(x)=\max_{i\in B}| a^ T_ ix-b_ i|\) subject to \(a^ T_ ix=b_ i\) for \(i\in E\) and \(a^ T_ ix\geq b_ i\) for \(i\in C\), where B, E, C are finite index sets, \(x\in R^ n\) is the unknown vector, \(a_ i\) are given vectors in \(R^ n\), and \(b_ i\) are given real numbers. Instead of the usually applied penalty-function method the algorithm uses a ``traditional'' active set approach. To obtain the search (feasible-descent) direction, a linear least-squares subproblem is to be solved. The solution of the same subproblem subject to some simple constraints gives the steepest-descent direction for the source problem. This may be used to prevent cycling at degenerate dead points (if we meet anyone during the above calculation of the search direction). The results of numerical experiments for some test problems encourage to apply the presented approach. The author applied them also for the analogous \(\ell_ 1\) problem and linear programming [ibid. 6, 343-355 (1986; Zbl 0632.65074)].