Direct Search Methods for Nonlinear Optimization Problem used ART Theory

In this paper, the search is conducted along each of the coordinate directions for finding the minimum. If $e_i$ is the unit vector along the coordinate direction i, we determine the value a, minimizing f(a)= $f(x+ae_i)$, where a is a real number. A move is made to the new point $x+a_ie_i$ at the end of the search along the direction i. In an n dimensional problem, we define the search along all the directions as one stage. The function value at the end of the stage is compared to the value at the beginning of the stage in establishing the convergence. The gradient appears to be zero at point. We can safeguard this by introducing an acceleration step of one additional step along the pattern direction developed by moves along the coordinate directions.