An algorithm for complex linear approximation based on semi-infinite programming Complex linear approximation, even with Haar systems suffers from the lack of information about the characterising number of extreme points of the error function. In a first step, the authors re-write the complex approximation problem in terms of an optimization problem. This problem is linear, but not finite. Thus, a certain discretization is used which yields a finite problem. Eventually, by combining the primal and the dual optimization problem, the authors reduce the problem to a set of equations.par They describe how to find an initial guess and to improve that guess. The improvement is done by treating the dual problem (in the spirit of Remez) by an exchange of a single point which corresponds to a single step in the simplex algorithm which increases the lower bound for the maximal error. The authors present a convergence proof, thereby correcting a proof by {it P. T. P. Tang} [Ph. D. Thesis, Univ. of California at Berkeley (1987)].par Furthermore, the authors present two examples -- one is to find a Solotareff polynomial, the other consists of finding a certain approximation on an $L$-shaped region. The corresponding MATLAB code which was used is available. (Source: