This very important paper is devoted to the description and application of a new iterative mesh optimization algorithm for the finite element solution of variational problem sets in finite-dimensional spaces. The optimality criterion is that the mesh should be such that the variational energy functional evaluated at the finite element approximation, be minimized. The chief merit of the procedure presented in this paper is that each node of the mesh, and the corresponding nodal value of the discrete approximation, are updated by solving sequentially local minimization problems with very few degrees of freedom: Main result: It is shown that this procedure reduces the energy functional monotonically, whithout the need to solve the global discrete problem at intermediate stages. The local nature of the algorithm makes it easy to devise safeguards against tangling without adversely affecting the overall performance. Applications to partial differential equations are considered. Finally, numerical results in two dimensions are obtained by incorporating the algorithm into \textit{R. E. Bank's} well-known PLTMG elliptic solver [PLTMG: a software package for solving elliptic partial differential equations. User's guide 7.0. (1994; Zbl 0860.65113)].