This work investigates the topic of solving Bilinear Matrix Inequalities (BMIs) problems in the optimal control design field, using successive resolutions of properly defined Linear Matrix Inequalities (LMIs). This technique can be described as an 'LMI-based coordinate descent method'. Indeed the original (BMI) problem is solved independently for each coordinate at each step using a LMI optimization, while the other coordinate is fixed. No method based on this idea has been formally proved to converge to the global optimum of the BMI problem, or even a local optimum. This will be discussed using relevant results both from the mathematical programming and control design points of view. This discussion supports the algorithm proposed here which, thanks to a particular change of variables, leads to sequences of improving solutions. Also emphasized is a second improvement important to avoid in practice early convergence to suboptimal solutions instead of local optima. The control framework used is that of optimal output feedback design for linear time invariant (LTI) systems. An example using a random plant is drawn to illustrate the typical effectiveness of the algorithm.