Abstract : Most of the algorithms today available to compute the canonical decomposition of higher order arrays are either computationally very heavy, or are not guaranteed to converge to the global optimum. The solution we propose in order to keep the numerical complexity moderate is~i) to stop the latter algorithms once the solution belongs to the convergence region of the global optimum, and~ii) to refine the solution with a mere gradient descent algorithm. The case of fourth order hermitian positive semi-definite arrays with complex entries is considered. In fact, the hermitian symmetry constraint is taken into account by optimizing a higher order multivariate polynomial criterion. A compact matrix form of the gradient is then computed based on an appropriate framework allowing for derivation in C whereas the cost function is not complex analytic. This compact expression is perfectly suitable for matrix-based programming environments such as MATLAB where loops are to be avoided at all costs. Eventually, computer results show a good performance of the proposed approach.