The global optimization problem of a function satisfying the Lipschitz condition over a multidimensional hyperinterval with an unknown Lipschitz constant is considered. It is supposed that the function can be black-box, multiextremal, and hard to evaluate. Various approaches for solving this problem can be distinguished, e.g., from the following three viewpoints: (a) the way of obtaining the Lipschitz constant information; (b) the rule used to select subregions for partitioning; (c) the strategy for partitioning the chosen subregions. The main attention is dedicated to the last aspect (c). Particularly, the diagonal algorithms are examined, since they have a number of attractive theoretical properties and have proved to be efficient in solving applied problems. In these methods, the search hyperinterval is adaptively partitioned into smaller ones and the function is evaluated only at two vertices corresponding to the main diagonals of the generated hyperintervals. It is shown that traditional diagonal strategies are not so efficient due to many redundant function evaluations. A new adaptive diagonal partition strategy that allows one to avoid such computational redundancy is described. A number of multidimensional global optimization algorithms based on the new strategy are illustrated. They differ in both (a) the Lipschitz constant estimating and (b) the hyperintervals selection procedure. Results of their numerical verification on GKLS-generator test classes are discussed