The Expanding Space Method in Sphere Packing Problem

A problem of packing unequal spheres, which radii are known, into containers of various shape are considered. An equivalent mathematical model to a standard sphere packing model is formed based on the assumption that the radii may be variable. In the new model formulated in a higher dimension space, a combinatorial structure is derived, and additional constraints are formed involving variable radii and providing its equivalence to the original packing problem. This approach of treating radii as variables allows improving local solutions of the original problem. Some practical tasks are indicated, which are reduced to packing unequal spheres.