Global Optimization in Geometry — Circle Packing into the Square

Global Optimization in Geometry | Circle Packing into the Squarep

Imbedded Circle Patterns with the Combinatorics of the Square Grid and Discrete Painlevé Equations

A discrete analogue of the holomorphic map z γ is studied. It is given by Schramm's circle pattern with the combinatorics of the square grid. It is shown that the corresponding circle patterns are imbedded and described by special separatrix solutions of discrete Painlevé equations. Global properties of these solutions, as well as of the discrete z γ , are established.

Packing circles in a square: a theoretical comparison of various convexification techniques

We consider the problem of packing congruent circles with the maximum radius in a unit square. As a mathematical program, this problem is a notoriously difficult nonconvex quadratically constrained optimization problem which possesses a large number of local optima. We study several convexification techniques for the circle packing problem, including polyhedral and semi-definite relaxations and...

Packing Unequal Circles into a Square Container by Partitioning Narrow Action Spaces and Circle Items

We address the NP-hard problem of finding a non-overlapping dense packing pattern for n Unequal Circle items in a two-dimensional Square Container (PUC-SC) such that the size of the container is minimized. Based on our previous work on Action Space-based Global Optimization (ASGO) that approximates each circle item as a square item to find large unoccupied spaces efficiently, we propose an opti...

Interval Methods for Verifying Structural Optimality of Circle Packing Configurations in the Unit Square

The paper is dealing with the problem of finding the densest packings of equal circles in the unit square. Recently, a global optimization method based exclusively on interval arithmetic calculations has been designed for this problem. With this method it became possible to solve the previously open problems of packing 28, 29, and 30 circles in the numerical sense: tight guaranteed enclosures w...