A Verified Optimization Technique to Locate Chaotic Regions of Hénon Systems
Journal of Global Optimization
Journal of Computational and Applied Mathematics - Special issue: Scientific computing, computer arithmetic, and validated numerics (SCAN 2004)
Maximin Latin Hypercube Designs in Two Dimensions
Operations Research
INTLAB implementation of an interval global optimization algorithm
Optimization Methods & Software - GLOBAL OPTIMIZATION
Solving the problem of packing equal and unequal circles in a circular container
Journal of Global Optimization
A heuristic approach for packing identical rectangles in convex regions
Computers and Operations Research
Greedy vacancy search algorithm for packing equal circles in a square
Operations Research Letters
Efficiently packing unequal disks in a circle
Operations Research Letters
Patterns and pathways of packing circles into a square
International Journal of Computer Applications in Technology
Differential evolution methods based on local searches
Computers and Operations Research
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This paper presents a new verified optimization method for the problem of finding the densest packings of nonoverlapping equal circles in a square. In order to provide reliable numerical results, the developed algorithm is based on interval analysis. As one of the most efficient parts of the algorithm, an interval-based version of a previous elimination procedure is introduced. This method represents the remaining areas still of interest as polygons fully calculated in a reliable way. Currently the most promising strategy of finding optimal circle packing configurations is to partition the original problem into subproblems. Still, as a result of the highly increasing number of subproblems, earlier computer-aided methods were not able to solve problem instances where the number of circles was greater than 27. The present paper provides a carefully developed technique resolving this difficulty by eliminating large groups of subproblems together. As a demonstration of the capabilities of the new algorithm the problems of packing 28, 29, and 30 circles were solved within very tight tolerance values. Our verified procedure decreased the uncertainty in the location of the optimal packings by more than 700 orders of magnitude in all cases.