Efficiently packing unequal disks in a circle

Placing non-overlapping circles in a smallest container is a hard task. In this paper we present our strategy for optimally placing circles in a smallest circle which led us to win an international competition by properly mixing local and global optimization strategies with random search and local moves.     

Global Optimization on Funneling Landscapes

Molecular conformation problems arising in computational chemistry require the global minimization of a non-convex potential energy function representing the interactions of, for example, the component atoms in a molecular system. Typically the number of local minima on the potential energy surface grows exponentially with system size, and often becomes enormous even for relatively modestly sized systems. Thus the simple multistart strategy of randomly sampling local minima becomes impractical. However, for many molecular conformation potential energy surfaces the local minima can be organized by a simple adjacency relation into a single or at most a small number of funnels. A distinguished local minimum lies at the bottom of each funnel and a monotonically descending sequence of adjacent local minima connects every local minimum in the funnel with the funnel bottom. Thus the global minimum can be found among the comparatively small number of funnel bottoms, and a multistart strategy based on sampling funnel bottoms becomes viable. In this paper we present such an algorithm of the basin-hopping type and apply it to the Lennard–Jones cluster problem, an intensely studied molecular conformation problem which has become a benchmark for global optimization algorithms. Results of numerical experiments are presented which confirm both the multifunneling character of the Lennard–Jones potential surface as well as the efficiency of the algorithm. The algorithm has found all of the current putative global minima in the literature up to 110 atoms, as well as discovered a new global minimum for the 98-atom cluster of a novel geometrical class.